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Complex Systems - Jean-Philippe Bouchaud -Lecture 9: Choice Theory, Ising paradigm & Schelling model — Transcript

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0:00this conference will now be recorded
0:08okay so it's uh unfortunately on our
0:11last lecture went very quickly this year
0:14I thought
0:15and it's a little sad not to see you I
0:17have four people in the room though
0:19which is a change
0:21um so
0:24today that there's one lecture and then
0:27the pity as usual
0:30and then next week I guess that uh you
0:33don't have anything so you're supposed
0:34to work on your
0:37on your revisions but the the problem is
0:41that we don't know yet exactly the
0:43circumstances in which the exam will
0:45take place
0:46we hope to know that soon
0:49in any case what I usually do in the in
0:51this last week is I propose a kind of
0:53Open Session
0:55um so if you want I'll send the link
0:57I'll see that with Valentina I'll send
1:00the link and for next Wednesday uh nine
1:03o'clock the usual time
1:05and then you can connect and we can chat
1:07about anything really if you have
1:09questions
1:10about the lectures or whatever it helps
1:15you I mean there's no obligation but I
1:17usually do that in
1:19in real so we can do that also
1:23through go to meeting or Zoom or
1:25whatever
1:27okay so
1:29um last time it was always already two
1:30weeks ago sorry for uh
1:33the slides changed last week true
1:38um so I'm in the last chapter chapter
1:40four interactions and Collective texts
1:42of course I had talked about Collective
1:44effects in the random field icing model
1:46example but here I want to I wanted to
1:50dwell more on this so I told you about
1:53Choice Theory and detailed balance I'm
1:55going to recall a little bit uh what I
1:57said because it's very important for
1:59today
2:00then I
2:02told you about the the icing Paradigm so
2:05the fact that
2:07you can have spontaneous appearance of a
2:10collective choice if uh interaction is
2:13strong enough or if
2:15temperature or irrationality is low
2:19enough
2:21and today I want to generalize The
2:25Simple Choice theory that I gave you
2:28last time to the case of multi-agents
2:32interacting multi-agents
2:34and this will allow me to speak about a
2:37very well-known model in economics or
2:41sociology I don't know how you want to
2:43call it but anyway uh shelling how much
2:46selling goods and over five in economics
2:49so I guess uh must be thought of as
2:51economics propose the model to
2:54understand
2:55segregation effects in cities
2:58and you'll see that this is amenable to
3:00kind of statistical mechanics type of
3:03treatment and it shows a very
3:06counter-intuitive uh effect
3:08and then you know I of course this is
3:11going to be very quick uh this session
3:14is only an hour and a half a little more
3:17so I won't have time to speak a lot
3:19about spin glasses but I want to tell
3:21you uh very few things about the general
3:24phenomenology of what's called in
3:26physics spin glasses which has many uh
3:29incarnations also in economics and
3:32social sciences and to show you that
3:35some of the things that I told you about
3:37optimization I maybe you remember this
3:40idea of fragile optimization as soon as
3:42you change a little bit some of the
3:44parameters you can completely change the
3:46solution that this actually occurs in
3:49skin glasses in a very paradigmatic
3:52manner
3:54okay so that's the outline for today
3:57so let me recall what I said last time I
3:59I was considering a single agent and
4:03this single agent had a certain number
4:05of possible choices Alpha Gamma and so
4:08on
4:09to which is associated the certain
4:11utility function U of alpha
4:15and then the rule of the game in
4:19decision theory is that agents can
4:21revise their decision and change their
4:23mind
4:24and go from one choice to another
4:27and this is with a sudden rate w
4:31Alpha to gamma
4:33which is
4:35the probability to change your mind from
4:38alpha to gamma between t and t plus DT
4:41and this is equal to a certain
4:44base rate gamma
4:46Capital gamma divided by one plus
4:50exponential of beta and there's always a
4:55sign here to get right U Alpha minus U
4:58gamma
4:59foreign
5:00[Music]
5:04it's in physics it's the inverse
5:06temperature in economics it's a measure
5:10of irrationality or sometimes it's it's
5:13thought of as a measure of the
5:16uncertainty you have on your own utility
5:19you don't exactly know what you want
5:20this is well known in life
5:23and uh and the sign so let's run through
5:26it again so if you gamma is larger than
5:29U Alpha so you're more happy with Choice
5:31uh gamma then this is negative
5:35and exponential of beta times the
5:37negative number uh when beta is large is
5:41very small and so you you actually
5:44change
5:46nearly deterministically once you've
5:48decided to change your mind with with
5:50raid gamma then you do go for the better
5:54alternative
5:56so that explains the sign here okay and
5:59uh what we've seen is that this
6:02particular choice which again is a
6:05choice that can be justified from first
6:07principles in physics
6:09where U is the analog of the energy
6:13in social sciences it's it's really more
6:16uh a convenient Choice it's something
6:19that you know goes in the right
6:21direction of course you make choices
6:23that are on average favorable
6:25but the detailed shape of this hopping
6:29rate of the rate of change is very
6:32arbitrary and is only motivated by
6:35mathematical convenience so that that's
6:38that's really a problem in a sense
6:39because one doesn't know whether the
6:42results that one gets from this
6:44particular choice uh are generic or not
6:48and I'll give you a little bit uh more
6:51discussion on that later on but in any
6:54case what makes everything
6:56work is that these hopping rates or
7:00these
7:01transfer rate obey
7:05what's called again in physics detail
7:07balance
7:09which is that there exists a sudden
7:11function
7:13h of alpha
7:20such that
7:23the ratio of the rate to go from alpha
7:26to gamma over the inverse rate is given
7:30by such a
7:33a form for any Alpha and gamma
7:38and in this particular case it's very uh
7:42it's very easy to show that this is the
7:45case when u h is is minus U but we'll
7:48see that this is not necessarily the
7:50case and it's going to be one of the
7:53main points of today but then whenever
7:57this is true when I have a detailed
7:59balance Falls then we know what's going
8:02to happen at long times at long times
8:04this x this random exploration of choice
8:07of the possible choices is going to lead
8:10to an invariant probability measure an
8:13invariant distribution which is that
8:15this the probability to find agent
8:19um in Choice Alpha probability to find
8:22that an agent has made Choice Alpha in
8:24equilibrium or stationary state is uh
8:28one over some normalization exponential
8:31of minus
8:34beta h of alpha okay
8:36and this is of course the traditional
8:38boltzmann Gibbs weight
8:42in physics
8:44so again in physics we have a pretty
8:46pretty detailed understanding of why
8:48detailed balance holds and of course
8:51detail balance is is a way to recover
8:54uh the Bolson Lake
8:56okay now I want to
8:59put this in a slightly more General
9:01context
9:03which is a
9:05a context where there's not only one
9:07agent making choices but many agents
9:10making choices
9:12and the utility function of each agent
9:14depends on the choice of others
9:16possibly
9:18so there are many agents many choices
9:20and
9:22um
9:25and they are possibly interacting or not
9:27we'll see so I'm going to now
9:32right away use my camera
9:35don't forget okay
9:52okay so now I'm considering
9:55as I said many ages I equal one to n say
10:01and each of these agents has a certain
10:03set of choices
10:05and the configuration of all these
10:07people together
10:09is described by the set of all the
10:12choices they've made so she currently C
10:15is going to be a configuration a
10:18configuration of choice so
10:21Alpha One is the choice made by agent
10:24one which can be anything in his set of
10:28choices or her set of choices
10:30Alpha 2
10:32Alpha I
10:34alpha n okay
10:37so it's a pretty big thing
10:39uh
10:40in the simplest case where which I
10:43talked about last time and also within
10:46the random field icing model
10:48each agent has the binary choice
10:51and and so this is a space of Dimension
10:53I mean there are two to the end
10:55configuration in that case but if Alpha
10:59is something different and we'll see an
11:01example in the sharing model then the
11:03space can be even larger Alpha can even
11:06be continuous variable whatever
11:10so this is one configuration and
11:14we'll assume that at each time Step
11:18One agent changes uh his or her decision
11:23uh to something else
11:26and this is not this is done one at a
11:28time so uh in the in the analog of of
11:33this rate here we're only going to
11:35consider cases where only one agent
11:38changes between t and t plus DT and so
11:42I'm going to call this agent I
11:46foreign
11:50the target configuration
11:53is going to be C Prime which is the same
11:56as C except that one of the agent has
11:59changed to from alpha to gamma okay
12:04of course this is again this is a
12:05notation that is not necessarily very
12:08explicit Alpha One doesn't Alpha One
12:10Alpha 2 doesn't mean that all agents
12:12take the same decision alpha alpha one
12:15is the label of the of the decision made
12:18by one but it's not necessarily the same
12:20as the decision made by two in in most
12:22General generality these Alphas don't
12:25necessarily need to live in the same
12:27space anyway they can describe
12:29completely different things anyway so
12:32it's a maybe a slightly confusing
12:35notation
12:36so what I'm going to assume is that each
12:39agent does it does this change of uh
12:42decision based on his or her own utility
12:46function only so I'm going to assume
12:49that the probability that the system
12:51goes from C to C Prime
12:54is given by something very similar to
12:56what I wrote here which is one plus one
13:01over I mean gamma gamma
13:03over one plus exponential beta and here
13:08I'm only taking into account the change
13:10of utility of agent I so UI of
13:16uh well
13:18uh so UI of e Prime
13:26yes UI same notation is here uift minus
13:31Qi of C Prime
13:36okay so this looks very similar to that
13:39okay
13:42so each agent looks at at its own or her
13:47own I mean I don't know what to use his
13:50or it's
13:52um so agent I look at its current uh
13:56satisfaction uifc looks at the
13:59satisfaction you would have in the next
14:01configuration where he changes from
14:04alpha to gamma and then this depending
14:06on this difference he decides or he
14:09decides to do it or not okay
14:12so now what's maybe unexpected is that I
14:17cannot assume right away that these WCS
14:22T to C Prime although they're written
14:24exactly the same way as here
14:27I cannot assume right away that detailed
14:30balance will hold
14:33it will hold in a trivial manner if
14:37the choices the utility functions are
14:40independent that is if the choice of
14:43agent I doesn't affect at all the choice
14:45of other agents but in in the case where
14:49agents interact it's not at all obvious
14:52that in general these this Choice allows
14:58detailed balance to hold and I'm going
14:59to show this explicitly on an example
15:03yes so there's a question
15:11yeah every time you took an agent
15:13randomly but you you just take one at a
15:16time
15:17so here it's I but yeah I haven't
15:19specified that you right but it can be
15:22any agent can change his mind also her
15:24mind but at each time step it's taken
15:26random B and then you you compute this
15:28to know whether you're going to go to
15:30the next consideration
15:33okay so the question I'm asking now is
15:37can I find a sudden function H now of of
15:41the whole configuration
15:47stats the WC to C Prime obey detailed
15:51balance which is that WC to C Prime will
15:54follow something analog to this except
15:57that instead of having a single choice
15:59of a single agent I have here a function
16:02of the whole configuration H okay and so
16:06what I'm saying is that in some cases
16:09so in Easy cases
16:12it will be when h of C
16:15is simply so there's a spine difference
16:18uh
16:19just to keep the fact that in physics
16:22we're used to boltzmann gives uh having
16:24an exponential of minus beta H and think
16:27of H as an energy whereas in economics
16:30it's more a utility so people tend to
16:32maximize their utility whereas physical
16:34systems tend to minimize their energy
16:36but that's uh that's that's just a
16:39detail so in the easy case we would have
16:43that h of
16:44T is that the sum over I of u i of
16:49of of of Alpha I
16:53okay so in this case where the use only
16:57depends on uh your own choice
17:01then uh you have this easy rule that
17:04detailed balance is obeyed just by
17:07summing individual utility functions and
17:10then people evolve independently from
17:12one another and so it's not surprising
17:14that the whole system the whole problem
17:16goes back to the uh single agent case
17:21and you see that if H is the sum of UI
17:24then the probability the stationary
17:28probability is the product of
17:30exponential of UI that is it's a product
17:32of independent uh individual probability
17:35so this is this is the easy case
17:41but let me give you an example which
17:43already is not that trivial which is
17:46again uh the case where there are two
17:52possible decisions for each agent
17:56so
17:59back to the I think model back to the
18:02binary Choice decision imagine that
18:04Alpha I is s i equal plus or minus one
18:10and that UI
18:14of the whole configuration is
18:19h
18:21Plus
18:22h i
18:25s i
18:27and here I'm using exactly the same
18:29notation as in the random field icing
18:31model Plus
18:33some
18:37over J different from i j i j s j
18:41times s i
18:44okay
18:46and so if you want to maximize your
18:50utility then it means that s i your
18:54choice must be in the direction of
18:58the sum of the external field which we
19:01we call the the common news the
19:04endosyncratic yield and the influence of
19:06others okay
19:10so
19:11um
19:12so this is the individual utility
19:14function which we are going to use
19:17in this uh hopping rate in in this
19:21transfer rate from T to C Prime but what
19:24you can show is that
19:26and we'll we're going to show it is that
19:29h
19:30is function of the configuration that
19:33we're looking for
19:35is equal actually to minus
19:39sum over I of H
19:42as h i s i
19:45so this system which is of course the
19:48independent contribution the what you
19:51see independence of what others do this
19:54you just sum over I as as usual but then
19:57the next term is not the sum over I of
20:01this one it's one half of that
20:03so plus one half
20:06of sum over I and J of s i g i g
20:12s j
20:14okay and so in general this is not equal
20:19to minus the sum
20:22over I of u i
20:25of of S5
20:30there's a one-half here
20:33so where does it come from well we just
20:35have to check what's going on
20:39uh we're going to check that
20:41um
20:42h of C Prime minus h of C
20:45as the correct form in order to ensure
20:48that this obeys detail balance so let me
20:53do it uh quietly so first of all let's
20:56notice that in this notation here
21:00it means that each jij appears only once
21:05okay each
21:07appears
21:11only one
21:15because of course you know for example
21:18one say J12 I can be equal to one and J
21:22equal to two but
21:25um but it can be the other way around so
21:27they are actually in this sum there's uh
21:30there's twice
21:31the contribution of gij but because of
21:33the one half here
21:35it's it's uh it should actually say
21:37appear I should just contribute
21:41foreign
21:48so let me compute h of C Prime
21:52minus h of t
21:58so h of T Prime minus h of C so what is
22:02e Prime and what is C here C Prime
22:06is the same as D so it's
22:09S1
22:11S2
22:13minus f i
22:16s n
22:19okay so this is what I mean
22:21in this General uh
22:26formalism here when Alpha can only take
22:30two two values then changing your
22:34decision is going from s i to minus s i
22:38Okay so
22:40you see that
22:43in this sum here the only term that will
22:46contribute is
22:49okay so maybe I should
22:53change my notation
22:56not to have either everywhere
23:00I'm changing the index of the sum to K
23:06and then I'm picking an I here so among
23:10all these K there's one I that
23:12corresponds there's one k that
23:14corresponds to I and for this one I need
23:17to change SSI into minus this I whereas
23:21all the other ones will be the same so
23:23when I take the difference here all the
23:26terms that are not equal to I in this
23:28term don't change and they cancel out
23:32and what I get in the end is twice
23:37h plus h i
23:40times s i
23:44okay
23:45twice because
23:47I went from s i to minus s i so the
23:50difference is 2si
23:52or minus 2si but there's a minus sign
23:55in front of that okay so the all the
23:58signs and the these details uh you
24:01should you know
24:02sit down and think about them but I hope
24:05I haven't made a sign mistake here and
24:07then in this sum here I should look at
24:11all the terms where one s i appears okay
24:15so uh s i k can be equal to I or J can
24:20be equal to 2i
24:22and but in any case as I said each
24:26link here appears only once and so
24:29because of that what you find is that
24:32the change of this interaction term when
24:35I change SI to minus SI is also twice
24:41um the sum over J
24:44of 3 i j s j s i
24:52okay each gij occurs wants so there's I
24:56mean again contributes once but because
24:59I change f i to minus s i the country
25:02the associated contribution changes by a
25:05factor 2 times s i
25:08so that's what we get
25:14and what I'm claiming is that and this
25:17is really obvious that this is also
25:19equal
25:20to minus
25:24UI of C Prime
25:27minus q i of C
25:31and this is Trivial because if you just
25:34look at at this term
25:36and you change S5 to minus s i you
25:38immediately find find this okay so what
25:42I'm saying is that if I choose H to be
25:45this function of spins with again a
25:48factor of one-half which is not the sum
25:50over I of UI then I'm make sure that if
25:55I take the ratio
25:58so
26:01I make sure because I can identify
26:04this UI C minus UIC Prime that this WC
26:08to C Prime is also equal to x amount 1
26:13over 1 or gamma sorry again one plus
26:16exponential of beta
26:19h of C
26:22Prime
26:24minus h
26:30okay and this is true whatever I I take
26:35so I have a function that is such that I
26:39obey Global balance at the level of the
26:41whole configuration space the
26:44configuration space being
26:46the set of all choices
26:50Okay so
26:51this example is very easy in a sense
26:54because I can explicitly construct
26:57the HSC that makes the choice of a scale
27:01balance but what I insist on is that you
27:04see that the interaction term
27:06makes the problem non-trivial makes the
27:09problem normative and of course this is
27:12expected because we know that in the
27:13presence of interaction the stationary
27:16State cannot be a product of individual
27:19choices because otherwise it would mean
27:22that people are not interacting
27:24so there's an interaction term and this
27:26interaction term can completely change
27:29uh the outcome
27:31and that's what I want to show now
27:35so this is
27:36this formalism that I just told you is a
27:39way to generalize the random field icing
27:42model uh to uh to non-zero temperature
27:46or to uh
27:48non-infinite values of beta you we we
27:52know how we would make the whole thing
27:54work because of course it's very similar
27:56to spins in physics
27:59but now I want to change a little bit
28:04the framework and tell you about
28:07the sharing model
28:10and read the setting model within this
28:12General formalism
28:21and so the the aim will be start to
28:24define the model and second will be to
28:29find this famous function HST that
28:33allows
28:34the tools of statistical mechanics to
28:37describe the equilibrium state of the
28:39system
28:44okay so as I said the shelling model
28:48was invented uh by Thomas shelling in
28:52the 70s
28:59and if you want to read more there's a
29:03literature that has developed in the
29:05physics
29:07Community recently but Thomas Jenning
29:10himself has a very nice book called uh
29:12from micromotives to macro Behavior
29:16where he's interested exactly in this
29:19transition from single agents to uh
29:23Collective effects and so what he what
29:26he was trying to understand was
29:28the fact that in U.S cities in
29:32particular there's a very strong racial
29:34segregation
29:36and and what he was puzzled by is that
29:39if you make surveys at the level of
29:41individual people
29:44um
29:44you know of course you know not all
29:47states are equivalent but in many states
29:50where there's a strong segregation
29:51people individually don't feel
29:54intolerant they don't feel that they
29:57want to live in their own Community uh
30:00their own uh in a community made by the
30:04same race
30:06so there's a kind of contradiction
30:08between individual preferences where
30:11people are not necessarily opposed to
30:13living in a mixed neighborhood and the
30:16actual observation that that he is tend
30:18to be extremely segregated
30:22and so he came up with a a very simple
30:24model and what I'm going to tell you
30:27about is the model that's inspired by
30:30the initial model is not exactly the
30:32same but it has the same uh flavor
30:36so what I'm going to imagine
30:39is a city made of
30:42neighborhoods
30:45so each Square here
30:48is going to be a neighborhood and the
30:51neighborhood is going to be
30:53labeled by the position of the center of
30:56the neighborhood for example
30:58and each neighborhood is going to be
31:01occupied by a certain number of people
31:05and there will be a number of vacancies
31:08or empty spaces or unconstructed sites
31:11the way you want to think of it and so
31:14I'm going to describe each neighborhood
31:16by the density of people in that
31:18neighborhood
31:20so row of R is equal to so if I if I
31:25want to be really
31:26physical
31:28each neighborhood is going to be used to
31:30assume to be assumed to have a square
31:34shape with a linear size L and so row of
31:37R is the number of people living in
31:40neighborhood r divided by L squared okay
31:47and what I'm saying is that uh here
31:50there's no question of race there's just
31:53a question of occupancy of density of
31:57population
31:58and what I'm going to assume is that
32:01people don't like to live in crowded
32:04neighborhoods because everything I mean
32:07you know I can tell you the story but
32:08you can imagine that you should live in
32:10an overcrowded neighborhood everything
32:11is difficult parking your car noise blah
32:15blah but people don't want to live
32:17either in uh discounted neighborhoods if
32:21you live in a neighborhood where there's
32:24no nobody and then there's no coffee
32:27houses there's no Cinemas blah blah so
32:30you don't want that either so we're
32:32going to assume that people have a
32:35preference
32:36for living in a kind of half filled the
32:41happy middle of half filled
32:43neighborhoods so I'm going to assume
32:45that row of r that you know this is
32:48normalized in such a way that this is
32:51something that belongs to zero one zero
32:53means there's nobody of course and one
32:55means that it's fully occupied
33:00so
33:02um what is the utility of agent I
33:05living in neighborhood r
33:09so here you see the choice of Agents
33:12will be the neighborhood so it's not a
33:15binary Choice it's a it's a choice that
33:18you know can have the infinite values if
33:21the city expands forever uh but it's uh
33:25it's a location here so this is the
33:27equivalent of the choices that I talked
33:30about before
33:32uh U of r
33:35is going to be equal to
33:38uh
33:39to you star
33:42times row of r
33:45if
33:47row is less or equal to a half
33:51and a two U star
33:54one minus row of r
33:59is uh row is placed or equal to
34:05so what what does it mean it means that
34:07if I plot
34:09U as a function of rho
34:13I have a tent shape
34:15function
34:18that Peaks
34:19around one half
34:21one half
34:24n is maximum is U star
34:29okay so that's that's what this function
34:31encodes and the choice here of a tent
34:34shape is just to make the calculation uh
34:37easier but what I'm going to tell you
34:39about does not depend on this particular
34:42choice I mean you can choose other
34:44functions so for example you might not
34:46like this
34:48discussed at one half so I could have
34:51chosen a function that does like this
34:57it doesn't really matter I mean the this
34:59function tent shape function is just
35:02there to illustrate the point is not a
35:04major uh aspect of the modeling
35:07Choice following procedure
35:11but so what I'm heading at here is that
35:15people in sheddings uh
35:19analog are you know they don't want to
35:22live in other in either extreme so it's
35:25like in the racial problem of shedding
35:27people don't want
35:29actually people are tolerant they they
35:31they actually prefer living in half uh
35:34filled neighborhoods so they're they're
35:36happy to be uh in in a mixed
35:39neighborhood in that sense
35:46so that's the setup of the model and
35:48what I'm going to look at is a case
35:51where the average
35:54density which I'm going to call rho Bar
35:57which is the total number of people
36:00divided by the total area of the city if
36:03you want the average density is exactly
36:06equal to one half
36:08so that's the choice that I'm going to
36:09make in order to amplify
36:13uh the point that's going to get out of
36:15the calculation is that on average
36:19there's enough people and even enough
36:22spaces such that all neighbors all
36:26neighborhoods could be filled at exactly
36:28one half okay so in in principle
36:32everything everybody can be happy in
36:34this model because
36:36there's exactly the right size of the
36:39city if you want to accommodate
36:40everybody in half space a half-filled
36:43neighborhood okay
36:45and in principle because everybody wants
36:47that this naturally should be what you
36:50find right
36:52so let's see what happens in the model
36:55and in order to understand what happens
36:57I need to give a rule for
37:00uh what people do
37:03and so I'm going to choose the choice
37:06theory that I've exposed earlier
37:09so what I'm going to assume is that the
37:12probability for an agent to go from r to
37:16R Prime
37:18okay
37:19is
37:21gamma divided one by one plus
37:24exponential of beta
37:28you
37:31I of
37:33uh rho of r
37:36minus u i of rho
37:40of our Prime
37:46Okay so
37:48if you want
37:49this is
37:51index I means that it's agent I who is
37:54moving is moving from R to R Prime and
37:57what he wants to see is whether the
37:59neighborhood at which he decides I mean
38:02that he you know visits to know whether
38:05he's going to move
38:08is the better suited than the current
38:11neighborhood
38:13so he looks at this function here and if
38:16he if he see the neighborhood where U is
38:18the higher he goes there
38:21with higher probability if if not he
38:24still may go there but with lower
38:25probability
38:27so if beta goes to Infinity he really
38:30systematically goes to neighborhoods
38:32that are better from his point of view
38:37but you see that again his decision or
38:40her decision is made based on only what
38:44happens to his or her utility function
38:49the choice doesn't take into account the
38:52fact that when you leave a neighborhood
38:55you're going to lower the popular the
38:57density obviously and therefore you're
39:00going to change the utility function of
39:02others because suddenly people will find
39:04themselves in a less populated
39:07environment but that you don't care you
39:10do it whatever
39:11and so people here uh you know take
39:15selfish decisions in that sense is that
39:17they only care about their own utility
39:19but they don't care about what they
39:21leave behind
39:22and we'll see how we could take that
39:25into account to uh obtain a better
39:28social outcome
39:29so clearly you know what I'm going to
39:33show you you've guessed because this is
39:35the whole uh Knack of the shedding model
39:38is the fact that although we've put
39:40everything in the model apparently to
39:43get the right social outcome so the
39:46right social outcome would be that
39:47everybody lives in half-based
39:49neighborhood then the dynamic this
39:52stochastic Dynamic here is going to lead
39:55to completely different outcome and it's
39:57going to lead to neighborhoods that are
39:59completely empty and neighborhoods that
40:02are more crowded than one half which is
40:06very surprising
40:08but in order to show this we have now
40:12the tools that we need which is quite
40:14remarkable because we're going to be
40:15able to map the problem in a sense to a
40:19problem of the liquid gas transition
40:24okay
40:27so
40:29now let's look
40:32at how the general framework that I've
40:35tried to tell you about is construction
40:37of the H function how does this work
40:41in the in the current situation
40:56I think that one microphone is on
41:00I think yes thank you
41:08okay so let me again try to understand
41:12what's going on so if I have One agent
41:18moving
41:21from
41:23R to R Prime
41:26how does it change the densities
41:29well clearly
41:31C Prime C is is
41:35the set of all the row of
41:39of
41:41Texas
41:43okay
41:45this is the what
41:48uh
41:49describes a configuration it's the it's
41:52the that's what I'm going to choose to
41:54describe your configuration I'm not I
41:56could choose to describe the
41:57configuration by the position of all the
41:59agents but actually the only thing that
42:01is going to matter is the density okay
42:05so if one agent goes from R to R Prime
42:08then uh C Prime
42:12will be the same density
42:15rho of x
42:18when X is different
42:20from R and R Prime
42:24okay and then
42:27row of r
42:30will have decreased a little bit by one
42:34unit so by 1 over L squared
42:39and R Prime on the contrary will have
42:42increased by a little bit
42:44which is 1
42:46over alphabet
42:49okay
42:52so that's my new configuration
42:55and what I want what I need to construct
42:57is
42:59what is the what is it that I want
43:02exactly as in the ising case I want to
43:04construct h of t
43:08such that
43:12HSC Prime
43:15minus h of B
43:19is equal
43:21to minus UI of uh
43:25R minus UI
43:29of R Prime of rho of r
43:35minus UI of rho of R Prime
43:39okay
43:41if I manage to construct such a function
43:44which in a sense now I think the kind of
43:48functional of all these densities
43:50then as I've shown in the icing case I
43:54ensure that the choice at the individual
43:57level corresponds to detailed balance at
44:00the global level okay
44:03so I'm going to assume that there exists
44:06such a function so I'm going to assume
44:09that h of C is a sudden function
44:13of all the rows
44:22and I need to understand how does this
44:25function evolve when I go from C to C
44:27Prime so
44:29[Music]
44:31if I assume that
44:33L is sufficiently large so that one over
44:36L squared
44:37is very small compared to one
44:41I can make a kind of failure expansion
44:44if you want
44:45of how the change of density on R and on
44:49R Prime is going to change the function
44:52H that depends on all the density
44:55and so what you find is that
44:57h of e Prime
44:59minus h of C
45:03and all that the rows that have not
45:06changed don't contribute at all
45:09and only the change that I've uh
45:12imposed to the the density at R and that
45:15R Prime will contribute to this change
45:17so what I get is
45:20uh
45:25derivative of H with respect to rho of r
45:32times uh
45:34-1
45:38or let me set it that way dh0 prime time
45:431 over L squared
45:45minus
45:47DH
45:50e row of r
45:54times one over Rel Square
45:58I'm assuming that H is a function of all
46:00the densities is the density is changed
46:03by a little bit then I can tailor expand
46:06the variation so I have the derivative
46:09of each but with respect to rho of R
46:11Prime Times the change one over L
46:13squared plus the derivative of the H
46:16with respect to rho times the change
46:18minus one over real squared okay
46:22and as I said this must be equal to what
46:24it must be equal to minus
46:30U of rho of r
46:33Prime
46:35minus
46:37U of Rover
46:51okay
46:53so
46:54big thanks to this equation here I can
46:58you know
46:59guess what H should be
47:02if I take H to be
47:05uh the anti-derivative of U with respect
47:09to rho it's going to work so let's let
47:13me write it down and then we'll check
47:16so this is the function this is the
47:18equation
47:20which must be true for all row of R rho
47:22of R Prime
47:24that ensures that if I can find a
47:26function H that obeys this equation this
47:29differential equation in row if you want
47:32then I'm sure that I will have detailed
47:35balance okay I will have a detailed
47:38balance although
47:39people act individually and only follow
47:42their use
47:44uh the function H will ensure that
47:46globally the system away details balance
47:48even taking into account the interaction
47:52this is exactly the same story as for
47:54the icing model where you can find an
47:58age
47:58which is not the sum of the use but
48:02which ensures that skill balance is a
48:05base
48:05okay so let me give you the solution and
48:09then we'll see by eye that it works
48:12foreign
48:26[Music]
48:28so what I'm claiming is that if I choose
48:31h
48:32of all the rows
48:34of x
48:40equal
48:42L squared
48:45thumb over all r's
48:50or X's
48:53of
48:55the integral from 0 to rho of x
49:01is a minus sign that I shouldn't forget
49:04U
49:08of
49:09um row Prime
49:11hero Prime
49:15and then plus an arbitrary constant a
49:18times
49:19wrote
49:21okay
49:33so this is the general solution of the
49:36detailed balance uh Criterion because
49:39you see that if I take the derivative of
49:42this function of all rows with respect
49:45to a certain row of r
49:47I'm going to pick in this sum
49:50r equal to X and I will take the
49:53derivative with respect to rho which is
49:55easy because it's the
49:57anti-derivative so the derivative
49:59respect row of R will give me U of uh
50:03row of R which is exactly what I need
50:06okay
50:08and this will give a constant and the
50:10constant will disappear because if I'm
50:13subtracting uh the two derivatives
50:16computed at different points a just
50:19cancels out okay
50:21so this is the solution I'm looking for
50:25and you see right away that what is
50:28interesting is that this is not equal
50:32to uh
50:35integral
50:37sum of Rex
50:40of rho of x
50:44U of of row of x
50:50that should be maybe an elsewhere
50:52normalizing but anyway it's it's a
50:55different
50:56uh functional of row this would be the
51:00naive thing you know if each agent has a
51:03utility u of rho then the number of
51:06Agents having that utility is number of
51:09people in neighborhood X rho of x times
51:12U so this is the this would be the
51:15independent
51:19assumption
51:22but you see clearly that what I have
51:24here is not row U of rho is the the
51:27anti-derivative
51:29of you
51:32okay
51:35great so that's the solution I was
51:38looking for
51:40and now what can I say about the
51:43long-time evolution of the system
51:45so let me recall what uh we know we know
51:49that if
51:50we call
51:53if
51:54w c to C Prime
51:58over WC Prime to C
52:01obey detail balance
52:07then at long time
52:10I know that the equilibrium distribution
52:13the probability to find the sun
52:15configuration is one over Z exponential
52:19of minus beta HSE
52:24so now I'm going to use this result to
52:27try to understand what this result tells
52:31me in terms of the densities I'm going
52:33to try to interpret what it means
52:35but uh that's the whole power of having
52:40a model that obeys detailed balance is
52:42the fact that you know right away what
52:44the stationary state is going to be
52:47but at this stage I should put a big
52:51warning sign
52:55which is the possibility of slow dynamic
53:01okay so it's always the case that you
53:04know when you read papers on
53:07uh
53:09Monte Carlo Dynamics or Master equations
53:12or Markov chain Evolutions then it's
53:15great because you can get a stationary
53:18State that's explicit which would not be
53:21the case
53:22for arbitrary choices of the W's but on
53:26the other hand you know you don't know
53:28at all
53:29if you're going to reach the stage 3
53:31State quickly or slowly
53:34and this is the problem that sometimes
53:37is not present that is on reasonable
53:41time scales you reach the equilibrium
53:43but in some cases
53:46and in particular in this model when you
53:48try to make the numerical simulation you
53:50realize that although the stationary
53:52state is uh that one then you can have a
53:57very slow Dynamic setting in and uh and
54:00so you know there's a whole discussion
54:01that should be made and this often
54:03eluded in uh
54:06in in the literature in particular in
54:09economics literature is how fast do you
54:12reach Equity room and in this question
54:14of the speed at which you reach
54:16equilibrium is of course absolutely
54:19crucial to know whether what you're
54:21talking about is Meaningful or
54:23meaningless and there's a lot of uh
54:25models in economics where people assume
54:27equilibrium without
54:29really trying to answer the question of
54:31is it reasonable to think that the
54:33Dynamics is going to lead me to
54:34equilibrium quickly or slowly and in
54:37particular well in in the physics the
54:40problem there's a lot of examples where
54:43uh the system has a boltzmann
54:46distribution in equilibrium but never
54:50reaches it like glasses glassy systems
54:53or the spin blasters that I'm going to
54:55talk about there are special systems in
54:58the sense that although you know their
55:00equilibrium state in reality If You
55:03observe the system even over you know
55:07if
55:09astronomical time or geological times
55:11they are actually out of equilibrium so
55:14you know beware of these General
55:17statements that look great but actually
55:20sweep a lot of issues under the rug
55:25okay so having said that
55:27let me apply
55:32this General
55:34result to the problem at hand
56:01so what I want to know is what is the
56:04stationary
56:05distribution of the equilibrium
56:07distribution
56:09of some
56:13set of row of x okay so again a
56:17configuration is
56:21described by
56:24a set of densities you should know all
56:26the densities in all the neighborhoods
56:28row of row one row two Row three and so
56:32on
56:33and this gives me uh
56:37a description of the coarse grain
56:39description of the equilibrium because
56:41at this stage I've lost
56:43the information of who lives who
56:46I just have a an information about the
56:50densities in each neighborhood
56:52and the probability to observe a certain
56:55configuration Row one road to row three
56:57well I know it is
57:01going to be given by
57:04the Wilson Way so there's a
57:05normalization
57:08there's a boltzmann weight
57:10that I've
57:12written here so
57:15it's going to be given by
57:18exponential of minus beta
57:21L squared
57:24sum over X
57:28of V
57:30of row of x
57:35where I've introduced the notation
57:40here I'm going to call the
57:42antiderivative of U
57:45V so well
57:50let's write it here
57:52so V is such that D Prime of rho is
57:56equal to U
58:04and importantly because there's a minus
58:07sign in H
58:09and the minus in front of the
58:13Aviation the Bossman way there's a plus
58:15in the end that comes here
58:19um the sum of rects of rho of X this is
58:22the the total number of people in the
58:25city so this doesn't change so a is uh
58:28is arbitrary constant and it actually
58:31you can reabsorb it in the normalization
58:34it doesn't play any role I'm dropping it
58:38but there's something that you should uh
58:42be familiar with which is the fact that
58:46this is
58:48the weight that comes from
58:50uh the people the choice of the people
58:53but there's on top of that in the
58:56property distribution of all the row of
58:57X the entropy term which comes from the
59:00fact that as I've said I'm describing
59:02the configuration in terms of densities
59:05and knots in terms of who lives where
59:07and therefore for a given choice of rho
59:11there's a there's a community
59:13combinatorial problem which is which
59:16leads to an entropy term on top of that
59:18which is given by I'm going to write it
59:21exponential of s of rho
59:25which is the number of possible choices
59:28to put my all my individuals in the city
59:32with the correct
59:34uh choice of densities and S of row is
59:38given by the familiar
59:41entropy term which is the
59:44the ideal gas entropy if you want so
59:47it's sum of Rex
59:49of row of x
59:53log
59:56of 4 of x
59:58Plus
1:00:001 minus rho of x
1:00:03log
1:00:04of 1 minus 4.
1:00:08and there's an overall minus sign
1:00:11so if you want to be you know very
1:00:14precise about
1:00:16going from individuals to densities you
1:00:19should take into account this entropy
1:00:22which you can see it is a kind of
1:00:24Jacobian in the transformation
1:00:26from people's uh
1:00:29physician to the dentist eco
1:00:32but the entropy is going to actually as
1:00:36usual at low temperature the entropy is
1:00:38not going to matter so I'm going to
1:00:40focus
1:00:43on
1:00:45beta large
1:00:49such that
1:00:51s is negligible
1:00:57one can actually do the theory
1:00:58completely also with the entropy term
1:01:01and I'll give you the results at the end
1:01:03but what I'm going to focus on is just
1:01:06this term here
1:01:08uh and I'm going to disregard the
1:01:11entropy contribution
1:01:14so okay so now I have my the probability
1:01:17of a given set of densities
1:01:21which is given by the exponential of
1:01:23beta the sum of Rex of some function V
1:01:25of the density
1:01:28right okay so what what should I do I
1:01:32mean this is uh this is what the
1:01:34probability of observing a certain set
1:01:36of density is but if I want to make this
1:01:39more concrete more visible I should you
1:01:43know try to speak about the most
1:01:45probable uh configuration
1:01:48and in particular when beta goes to
1:01:50Infinity
1:01:51this is everything that's going to
1:01:53matter is what are the most probable
1:02:00the most probable
1:02:02configuration
1:02:04configuration
1:02:20foreign
1:02:32X and try to find the configuration of
1:02:36row that maximize such an object with
1:02:40the definition that D Prime is equal to
1:02:41U and so if I want to give you the
1:02:45explicit
1:02:47function D of rho is equal to
1:02:50you star rho squared
1:02:54when rho is less than one half
1:02:57and U star
1:03:00two row minus row squared minus one half
1:03:05when rho is greater or equal to the half
1:03:10and what I need to do is
1:03:12um
1:03:13to try to find the configurations of the
1:03:15rows that maximize
1:03:18this V of row
1:03:20with the constraint
1:03:22that the average value of rho is equal
1:03:24to one half
1:03:30okay this is the case that I've chosen
1:03:32because it's the most uh uh
1:03:36spectacular case if you want
1:03:39so V of rho has a certain shape here
1:03:41that I could draw but it doesn't really
1:03:43matter uh this is the explicit form and
1:03:47here I'm left with a problem that we
1:03:51used to in uh the statistical mechanics
1:03:56which is you know in a sense the problem
1:03:59of a liquid gas transition where
1:04:03the there's an entropy contribution that
1:04:05I'm discarding anyway and then there's
1:04:07an energy contribution telling you uh
1:04:10you know what is the typical energy of a
1:04:14gas without with a sudden density
1:04:19and so by analogy
1:04:23and one can show that it indeed the case
1:04:26but by analogy one can look for a
1:04:29solution where
1:04:31rho takes two values
1:04:42so
1:04:45look for a solution
1:04:49such that
1:04:52rho is equal to
1:04:54row plus
1:04:55greater than one half
1:04:58with probability
1:05:02p
1:05:04and row minus less than one half
1:05:07with probability
1:05:09one minus t
1:05:11okay
1:05:13so
1:05:15if I do this I see that the sum over X
1:05:19of V of rho of x
1:05:25is equal to the number of neighborhoods
1:05:37times
1:05:39p V of row Plus
1:05:421 minus t
1:05:46of rho minus
1:05:49and I should optimize over
1:05:52p rho plus and row minus with uh the
1:05:57constraint
1:05:59so
1:06:01I'm looking for the maximum of this
1:06:06with a constraint that P row plus plus
1:06:09one might one minus P row minus must be
1:06:13equal to one half
1:06:15or more generally to robot
1:06:20okay
1:06:21so you know if I do this well maybe I I
1:06:25will find that in the end row plus is
1:06:27equal to row minus or that P is equal to
1:06:29one so that would be a uniform solution
1:06:33but I can also find solution where row
1:06:36plus and row minus are different and P
1:06:38is neither equal to uh zero node to a
1:06:42half
1:06:43and
1:06:44again in the limits where beta goes to
1:06:47Infinity in the low temperature limits
1:06:49and for this particular choice of V of
1:06:52rho I'm
1:06:55I'm not going to you know give you the
1:06:57calculation which is not very difficult
1:06:59but a little heavy
1:07:01um one has to you know distinguish
1:07:03different cases but in the end what one
1:07:06finds is that the configuration that
1:07:08maximizes
1:07:10so the optimal configuration
1:07:13or the most probable configuration
1:07:15because you see that if I maximize this
1:07:18object
1:07:21again I'm insisting on the fact that
1:07:23most probable configurations this amount
1:07:26to maximizing this object so if I
1:07:29maximize the object I'm optimizing this
1:07:31object I'm also maximizing the
1:07:34probability of observing such a
1:07:36configuration and the optimal Choice the
1:07:38optimal uh uh configuration
1:07:45are such that
1:07:48rho minus equals zero
1:07:51rho plus equals
1:07:53uh
1:07:54square root of two over two
1:07:58and this is also equal to p
1:08:03it turns out that you know don't give
1:08:05too much meaning to this but it turns
1:08:08out that row plus is square root of two
1:08:09over two which is 0.7
1:08:12and uh rho minus the zero so you see by
1:08:16you know inspection that P times row
1:08:18plus is one half
1:08:21and because row minus is zero this
1:08:23doesn't contribute so I'm satisfying the
1:08:25constraint but you also see that the
1:08:28optimal configurations the
1:08:29configurations that you'll see most
1:08:31often are such are the following
1:08:34so if I draw my little city again with
1:08:37neighborhoods
1:08:39I will have with probability
1:08:410.3 completely empty spaces
1:08:45and with probability 0.7
1:08:50neighborhoods that are over crowded with
1:08:54a density row plus which is
1:08:56square root of two over two
1:08:59and this is the most likely
1:09:01configuration
1:09:03and so this is the you know very
1:09:05surprising result that
1:09:08um
1:09:09shelling
1:09:11was able to
1:09:13show experimentally I mean it's quite
1:09:16remarkable the way shedding did shedding
1:09:18simulated uh this his model with coin
1:09:23and he was making you know the the the
1:09:25changes without any computer at the time
1:09:29he was making these changes by hand and
1:09:32he was seeing that systematically by
1:09:34following rules similar to this he was
1:09:37you know uh led to uh segregation of
1:09:40these coins coins of different colors
1:09:42and so it was a very also visual uh way
1:09:46to to see it
1:09:48but in any case uh the the transposition
1:09:52of the shedding model to the physics
1:09:54language is due to a paper by clover
1:10:02Berta
1:10:06at all
1:10:08that you can find easily on on the net
1:10:10that I can also give you I can give you
1:10:13the reference but it's it's in tnaf
1:10:15proceedings of National Academy of
1:10:18Sciences and uh and so they they
1:10:21essentially do what I've uh told you
1:10:24today
1:10:25so what is nice is that we have this
1:10:28completely paradoxical resolve
1:10:31if you want you know the story of uh
1:10:33Adam Smith Adam Smith
1:10:36thinks that if people act act selfishly
1:10:39uh and optimize their own welfare then
1:10:43the society as a whole is going to
1:10:45benefit and here in this case you see an
1:10:48example of the exact opposite that
1:10:50people follow what they want to do and
1:10:53everybody you know is doing something
1:10:55that he or she feels is good for him or
1:10:59her but in the end Collective leads a
1:11:02disaster
1:11:03and it's a disaster because there's this
1:11:08the the fact that the rule of the game
1:11:09doesn't take into account
1:11:11uh the pain you leave behind Okay the
1:11:14fact that when you change your your
1:11:16position the people living in your ex
1:11:19neighborhood are worse off and so they
1:11:22won't they will try to leave too and so
1:11:25on and this is the mechanism by which uh
1:11:28the configuration where everybody lives
1:11:30in the same neighborhood is actually
1:11:31unstable Dynamics if you start by a
1:11:34configuration where all the
1:11:36neighborhoods are filled with the
1:11:37density one half you can show that you
1:11:40know accidentally someone who is going
1:11:41to leave and then the whole situation
1:11:44will evolve how this situation although
1:11:47maybe on long time scale
1:11:49so it's a very beautiful model I think
1:11:51in that sense
1:11:53and let me give you a few more
1:11:57information about the model
1:12:00uh one is what happens if
1:12:04beta is non-infinite if beta is
1:12:09is if we're not in the low temperature
1:12:12limits well in the Lo if you increase
1:12:16temperature
1:12:19you decrease beta
1:12:23and you see that in the limit where beta
1:12:26goes to zero but a very high temperature
1:12:30only the entropy matters
1:12:32and clearly as you all know the entropy
1:12:36is maximized for row equal one-half
1:12:39and so obviously if people take choices
1:12:42completely at random
1:12:44then the density is going to be uniform
1:12:47right because if you completely choose
1:12:49randomly then all neighborhoods are
1:12:51completely equivalent and in the end
1:12:53you'll end up with a uniform
1:12:55distribution and this is driven by
1:12:58entropy exactly as in the liquid gas
1:13:00transition as you increase temperature
1:13:02at one point energy won't matter anymore
1:13:05and you'll recover
1:13:07a gas phase which is the uniform in
1:13:11space but if you lower temperature you
1:13:13know that in the liquid gas transition
1:13:15there's a space stress separation the
1:13:16liquid goes on one side and leaves the
1:13:19void behind and so in a sense it's the
1:13:22same thing that we see here
1:13:25um and so you can show that in this
1:13:27model that there exists a critical
1:13:30temperature basic C
1:13:32such that at higher temperature so for
1:13:35data greater than beta C there's
1:13:37segregation
1:13:42and for beta less than beta C there's
1:13:45the the solution is uniform
1:13:51so that's an Insight that we know very
1:13:54well from physics but uh which is maybe
1:13:57more surprising if you don't know the
1:13:59nominology of this transition and so on
1:14:02so
1:14:04we're in in known territories here we
1:14:08know that uh Collective effects
1:14:10can completely change what you uh within
1:14:15what your intuition would tell you from
1:14:18microscopic elements here we see that
1:14:21microscopically people want to live in
1:14:23half neighborhood a half cell
1:14:25neighborhood but collectively because of
1:14:28this systematic application systematic
1:14:31iteration of the dynamical rule that is
1:14:33the one that I've given you you end up
1:14:35in uh
1:14:37in Dire Straits
1:14:40okay so is there a way to improve
1:14:58the situation so
1:15:00as the question also that is addressed
1:15:03in the paper that I've
1:15:06that I've referred to
1:15:08is there
1:15:10a solution
1:15:15to this conundrum
1:15:26so is there a way that maybe the state
1:15:29or maybe
1:15:32social
1:15:34solution can help the system coordinate
1:15:37and find a more acceptable uh maximum
1:15:43configuration
1:15:45optimal configuration
1:15:47so the idea that um
1:15:50they propose in the paper that the
1:15:52office that I've uh mentioned proposed
1:15:55in the paper is to say well let's
1:15:59help people in doing the right thing by
1:16:03imposing a kind of tax
1:16:05which is that
1:16:09um
1:16:10as I've shown you there's h
1:16:14but there's also
1:16:16the total U
1:16:19which is uh
1:16:21the sum over X of rho of x
1:16:26U of rho of x
1:16:31so this is something that people don't
1:16:34really know about individually the total
1:16:37satisfaction of people but
1:16:40a superstructure like the state could
1:16:43measure This Global utility and
1:16:47as people make their choice they should
1:16:50be aware of the change in total utility
1:16:53that they their choice
1:16:55uh
1:17:01induces so instead of only looking at
1:17:04the change of individual utility you can
1:17:07think that maybe if I in the in the
1:17:11decision rule if I change Delta U
1:17:14of rho R minus U of rho of R Prime into
1:17:18Delta U plus Theta
1:17:22times
1:17:24Delta capital u minus Delta U
1:17:32so sorry maybe my capital u is not
1:17:36different enough from small you so what
1:17:39I'm saying here is that there's a
1:17:40parameter data which on top of the
1:17:43change of your own utility
1:17:45adds some contributions from the change
1:17:48of the global utility that you'll move
1:17:50uh provokes okay
1:17:53so if data is equal to zero you recover
1:17:56the previous model
1:17:58if Theta is equal to one you don't take
1:18:01into account your own utility you only
1:18:03take into account the global utility
1:18:07so
1:18:08you can think of that as a kind of tax
1:18:11right if you change it you think you
1:18:13have others you must pay something and
1:18:15therefore it's going to reduce the
1:18:17probability of making your your move
1:18:21so Theta here is a parameter that you
1:18:23can interpret as a tax
1:18:27and you can redo the whole calculation
1:18:29that I've done
1:18:31uh it's actually quite easy because in
1:18:34the end what you find if you redo the
1:18:37whole step is you find that V of row the
1:18:40empty derivative of U that I found
1:18:43before this one is just transformed into
1:18:46one minus data V of rho
1:18:51plus Theta
1:18:53row U of row
1:18:57and this is not surprising it's just uh
1:19:00you know looking at what I've uh defined
1:19:04here I see that there's the part that
1:19:06comes from the V of row that I had
1:19:10before but a part that also comes from
1:19:13the total utility row UFO
1:19:16so you turn the crank you do the same
1:19:19calculation as the one I've sketched for
1:19:22you you look for the maximum uh probably
1:19:25the configurations with maximum
1:19:27probability and what you find is that
1:19:31as a function of
1:19:33data
1:19:37the total utility per agent
1:19:41U divided by n
1:19:45is going to have this the following
1:19:47shape so this is one here
1:19:56red you don't like so
1:20:00choose blue for example
1:20:02so what happens is that for data equals
1:20:050
1:20:07um with the values of row plus equals
1:20:09square root of 2 over 2 that I've given
1:20:11you you can compute
1:20:13um
1:20:14the maximum the the utility
1:20:17per agent is given by
1:20:21I don't know blue you shouldn't either
1:20:25maybe yellow
1:20:28or green
1:20:32so you start from a value that's around
1:20:350.3 U star
1:20:37foreign
1:20:41by the way the the completely uniform
1:20:44solution
1:20:53you can also compute
1:20:55its utility it
1:20:58it's U star over four
1:21:04so Point 25 you star if you if you're
1:21:07all mixed
1:21:09but the segregated solution has a better
1:21:11you start per person which is 0.3
1:21:14instead of 0.25 so it's just re-saying
1:21:17what I was saying that the system will
1:21:20segregate but what's really interesting
1:21:23is that
1:21:26as you increase data
1:21:31there's a value of data which is one
1:21:33third where you reach
1:21:37uh
1:21:39U star over 2.
1:21:45sorry U star
1:21:52so
1:21:54um what you see is that by introducing
1:21:57this extra tags at first you improve the
1:21:59situation but it still remains
1:22:01sub-optimal
1:22:02and then at one point you reach
1:22:06um
1:22:08uh you reach the optimal state
1:22:11of everybody segregates everybody living
1:22:14in in the same neighborhood
1:22:22so that's an example where the the
1:22:25um Adam Gibbs Adam Smith
1:22:29invisible hand fails but you can help it
1:22:32by having
1:22:34um
1:22:35these taxes that I talked about
1:22:38okay
1:22:41so
1:22:43that was the the main message what I
1:22:45wanted to finally end on this the
1:22:48problem is that I've assumed uh in the
1:22:52choice Theory I've assumed a very
1:22:54particular shape of hopping rate
1:22:57you remember that I've insisted on that
1:23:00from the beginning
1:23:02um I've assumed that the W's
1:23:05are given by this the so-called logistic
1:23:11rule
1:23:23w c to C Prime
1:23:27is given by gamma over one plus
1:23:30exponential beta Delta U
1:23:34so this has allowed me to find an H
1:23:37function that
1:23:39is such that the whole system of a
1:23:42detailed balance and thanks to detail
1:23:44balance I can find the stationary
1:23:46distribution and show that the system is
1:23:49going to segregate
1:23:50but what happens if I take different
1:23:54choices for this transition rate
1:23:58as I've said in physics it's something
1:24:01that is Justified from first principle
1:24:04but
1:24:05in social sciences you know why on Earth
1:24:08would people follow exactly that rule
1:24:11uh and here is a little bit of an open
1:24:13question is whether
1:24:15the results that I've shown you today
1:24:17are uh robust against different choices
1:24:20of this individual uh hopping rate or if
1:24:26this whole phenomenology is going to
1:24:29completely disappear if I change uh
1:24:32sufficiently this rule so for example I
1:24:35could choose I don't know if I if I go
1:24:38from this particular shape to the sun
1:24:41function of Delta U
1:24:45which is General
1:24:49will this change completely the the
1:24:51final results or will the final results
1:24:54be you know qualitatively similar
1:24:59are the results
1:25:05robust
1:25:10and in a sense this is a very general
1:25:12question because the
1:25:14if you lose detailed balance
1:25:17then we're in the realm of uh
1:25:19non-equilibrium statistical physics in a
1:25:21sense where many dynamical systems don't
1:25:24obey detailed balance
1:25:26and there's no general theory for that
1:25:28so uh phase transitions in some cases in
1:25:33some cases they resist the existence of
1:25:36uh of non-detailed balance effects in
1:25:40other cases the the phenomenology is
1:25:43completely different as soon as you
1:25:45introduce a little bit of
1:25:48the violation of detail balance so
1:25:51because of that intuition it I I draw
1:25:54your attention to the fact that in
1:25:56social sciences
1:25:57there's a you know there's the this
1:26:00question of knowing how robust are the
1:26:03results is very uh rarely posed because
1:26:07it's difficult because we lack tools and
1:26:10I think it's a it's a very interesting
1:26:12question in general to know whether
1:26:15these segregation effects for example
1:26:17all these phase transitions they exist
1:26:21beyond the realm of detailed balance and
1:26:23Bolston Gibbs or if they're very fragile
1:26:27to any change of the retail balance
1:26:30condition
1:26:31okay so that's uh what I wanted to say
1:26:35on the shedding model
1:26:44and what I like to finish on is the
1:26:48oh my
1:26:51you don't see any more the outline of
1:26:53the lecture but it doesn't matter
1:26:56so I want to finish on a few words on
1:26:59spin glasses
1:27:06so the spin glass problem is uh
1:27:10is the problem of spins
1:27:13which is given by an energy which I call
1:27:17H
1:27:19of the configurations s i
1:27:23which is simply the term
1:27:28of interaction between bins
1:27:31and for the for now I'm completely
1:27:33neglecting all the uh individual terms
1:27:37I'm putting to zero h i n capital h
1:27:45so there's only interaction and the jijs
1:27:49are random
1:27:52for example gaussian
1:27:57variables
1:28:01of zero mean
1:28:06okay
1:28:08so what does it mean to have jij's
1:28:11random of Euro mean it means that for
1:28:14some pairs of spins
1:28:17the J is positive and therefore spins
1:28:21want to be aligned either up
1:28:24or or down
1:28:29but if jij is negative then spins want
1:28:33to be anti-aligned
1:28:38okay
1:28:39and so there's a mixture of that
1:28:42some spins want to be aligned with one
1:28:44another some spins want to be
1:28:46anti-aligned to one another
1:28:48and this causes a headache in a sense so
1:28:51it's called frustration
1:28:53and so the typical little graph that you
1:28:56can draw is a triangle where you have
1:28:58for example J negative here J negative
1:29:01here uh and
1:29:04J positive here and J positive here
1:29:07so if this spin is up
1:29:10this pin wants to be up but this one
1:29:13doesn't know what to do okay
1:29:18so you have a problem with many
1:29:20constraints and these constraints are
1:29:23contradictory to one another so this is
1:29:25the archetype example of an optimization
1:29:28problem with constraints that are not
1:29:31compatible with one another okay
1:29:34and it leads to a very very interesting
1:29:37set of uh a phenomenon that I'm going to
1:29:40summarize in a few moments but uh you
1:29:44know you you can think of this problem
1:29:46in a in a social context
1:29:49in the following way I mean it's of
1:29:51course a toy example but I think it's a
1:29:54very visual one
1:29:55so imagine that you have a a class of 50
1:29:59students and you want to organize a boat
1:30:02trip
1:30:03but you have only boats with 25 capacity
1:30:06so you you want to split your
1:30:09your your class in two groups
1:30:15so groups that are going to a group that
1:30:17is going to go to
1:30:18both a and a group that's going to go to
1:30:21both d
1:30:23and you make a little survey beforehand
1:30:26and you try to know who likes whom in
1:30:28your class
1:30:29okay and so you have 50 individuals and
1:30:33you're going to map out all the 50 times
1:30:3749 divided by two if you assume that
1:30:41liking is a symmetric
1:30:44uh condition which is unfortunately not
1:30:46always the case but anyway so if uh if
1:30:50gij is are symmetric then um
1:30:53you're looking for 50 times 49 over 2 uh
1:30:58information about your class
1:31:00pieces of information and if jij is
1:31:04positive it means that
1:31:06the two people in question like each
1:31:08other and would prefer being on the same
1:31:10boat and if jij is negative
1:31:15then they would like to be on separate
1:31:17boats okay
1:31:19uh well the question of
1:31:23optimizing this uh function over the SI
1:31:28so you you need to find which sis are
1:31:32plus and which S5 are minus which will
1:31:34correspond to which people you put on
1:31:36the a boat and which people you put on
1:31:38the E-boat in such a way that the global
1:31:43satisfaction is maximized
1:31:45so that what it would correspond in
1:31:48terms of uh of this city example and the
1:31:52problem is that that although the the
1:31:55problem although the equation seems
1:31:58incredibly simple
1:32:00finding the ground state finding the
1:32:02optimal configuration for a given choice
1:32:05of gij is extremely difficult
1:32:07algorithmically
1:32:09the the algorithms allowing you to find
1:32:14the true ground say the true optimal
1:32:16configuration
1:32:17uh are exponentially long in N so as you
1:32:22grow the system size uh the time you
1:32:25need to spend to find the real ground
1:32:27state is going to grow exponentially
1:32:29with the size of the system we don't
1:32:30know
1:32:31yet maybe
1:32:33one day we will
1:32:36show that there's very little hope of
1:32:38that that this problem is polynomial in
1:32:42the number of bins but everybody
1:32:44believes that it's actually not and that
1:32:47even the best algorithms won't be able
1:32:49to
1:32:50find a solution in a time that
1:32:53polynomial in the in the system side so
1:32:56it's an incredibly complicated problem
1:32:58to solve
1:32:59but at the same time there are many
1:33:01configurations that are quasi-optimal so
1:33:04if you're really insisting on finding
1:33:06the optimal configuration is very
1:33:08difficult but on the other hand there's
1:33:11a lot of configurations that are locally
1:33:13optimal so what does it mean that it's
1:33:16locally optimal it means that
1:33:20locally optimal
1:33:24configuration
1:33:27or one spin flip stable it means that
1:33:31each spin s i
1:33:33is in the direction of the local field
1:33:37so you remember
1:33:41the random clizing model where there was
1:33:44on top of that
1:33:45an idiosyncratic scale and external
1:33:47field but
1:33:49here I'm I'm having a simplified view on
1:33:52that where there's only the interaction
1:33:54term and if
1:33:56everybody every spin is in such a
1:34:00configuration it means that you know I
1:34:02cannot improve that if I flip SI it's
1:34:06going to be worth so nobody wants to
1:34:08change individually of course there
1:34:10might be the collective move improving
1:34:14uh uh the energy or improving the
1:34:17utility of everybody but individually
1:34:20people cannot do that people are locally
1:34:22happy if you want
1:34:25so you can look at the number of
1:34:27solutions of this problem
1:34:29and what you find is that the number of
1:34:31solutions the number of configurations
1:34:33touched that this is true
1:34:35is
1:34:36is exponentially large in n
1:34:39so the number of configurations in a
1:34:41spin problem is of course 2 to the N
1:34:44okay
1:34:46and the number of configurations that
1:34:48satisfy such a constraint when J's are
1:34:52random gaussian variables of zero mean
1:34:55is
1:34:56exponential of 0.2 times n
1:35:01so of course 2 to the N is much greater
1:35:04than exponential of 0.2 times n
1:35:08because log 2 is 0.7 and not uh 0.2 but
1:35:14um but you see that you still have an
1:35:16enormous amount of possible choices
1:35:19such that this is uh satisfied so
1:35:24there's a lot of metastable states in
1:35:26the system a lot of configurations that
1:35:28are locally stable and a lot means
1:35:30really a lot because you have a
1:35:33combinatorial type of explosion of the
1:35:36number of solutions
1:35:39so it's a it's an interesting problem
1:35:43where
1:35:45although the the true solution is is
1:35:48hard to find there's a lot of
1:35:51secondary Minima if you want or
1:35:53secondary Maxima if you want to think of
1:35:56it in terms of utility
1:35:58secondary Minima in terms of energy
1:36:01technology
1:36:02Maxima in terms of utility
1:36:07and so let's look a little bit at
1:36:10graphically at what happens
1:36:15so it's traditional to draw the graph
1:36:18that I'm going to draw although it
1:36:21doesn't make sense really because
1:36:24the configurations P space that spins is
1:36:28uh is not a one-dimensional space at all
1:36:30you see that you know for example if I
1:36:34want to look at the configuration of
1:36:36n equal to spins it's a
1:36:39it's a square
1:36:41up up
1:36:43down
1:36:45and down down
1:36:47up
1:36:49so this is the configuration space
1:36:53for n equal to
1:36:55and for n equals three it's the it's a
1:36:58cube and if you go to higher Dimension
1:37:00it's the hypercube of Dimension n
1:37:02uh so I'm going to you know plot
1:37:06this
1:37:08space which is you know logically
1:37:11completely different from a line I'm
1:37:13still going to draw it as a line because
1:37:15we can't do anything
1:37:17else on the board but you know be
1:37:19careful that this is very misleading by
1:37:21many accounts anyway so it's a
1:37:23traditional way to make the point so if
1:37:25I plot H as a function of configuration
1:37:27what you find for the spin glass problem
1:37:30is an extremely rough and complex
1:37:32landscape
1:37:35let me try to do it right
1:37:39so there's a lot of it's very rough so
1:37:41as soon as you change the configuration
1:37:42a little bit you change the energy in a
1:37:46kind of random way
1:37:47but also if you look at so here I'm
1:37:50speaking in terms of physics where I'm
1:37:53looking at the low energy State and you
1:37:55should flip this up down if you want to
1:37:58think of it in terms of utility but you
1:38:01see that on my graph I guess that this
1:38:04is the absolute minimum
1:38:07okay
1:38:08so this is the ground state I should
1:38:10choose if I'm a rational person
1:38:14but
1:38:15or if you want at zero temperature this
1:38:17is where the system should be because
1:38:19that zero temperature it should be in
1:38:21the absolute ground state but what you
1:38:22see is that very close to the ground
1:38:24state but very far very close in energy
1:38:28but very far in configuration space
1:38:30there are other
1:38:32uh candidate
1:38:34that are nearly as good
1:38:36but they are very different in terms of
1:38:38the configuration of spin
1:38:41and so this is the kind of landscape
1:38:44that you need to think about to
1:38:46understand in this in the context of
1:38:48physics what's going to happen
1:38:50dynamically what's going to happen
1:38:52dynamically is that the system is never
1:38:54going to reach equilibrium is going to
1:38:56get stuck in in valleys that are not the
1:39:00optimal values but that are you know
1:39:03good enough and the time it needs to
1:39:06reach the true Crown state is going to
1:39:08grow uh like the exponential of the size
1:39:11of the system for such a problem
1:39:14so in physics it's called you know aging
1:39:18and out of the equilibrium Dynamics but
1:39:20in terms of uh what I warned you before
1:39:23in terms of optimization in terms of
1:39:25finding the optimal solution uh using
1:39:28Monte Carlo for example it means that
1:39:31the algorithm is going to never contact
1:39:34uh actually
1:39:36but on the other hand it might find
1:39:38solutions that are good enough
1:39:40okay it might find solutions that are
1:39:43not the true best one but are close in
1:39:47terms of energy or utility to the to the
1:39:50best one so they are good enough
1:39:52Solutions
1:39:53uh satisfying Solutions
1:40:01and the problem is that you don't really
1:40:03know which one you're going to find
1:40:04because depending on your algorithms
1:40:07maybe you'll end up here maybe you'll
1:40:08never there maybe you'll end up there
1:40:10and so what is interesting from a
1:40:13philosophical point of view uh you know
1:40:16thinking that agents are rational and
1:40:17try to optimize their utility function
1:40:19is that if you're confronted with such a
1:40:22complicated problem to solve then you
1:40:25don't know what others are going to do
1:40:28because some of them are going to choose
1:40:30one solution which is good enough and
1:40:33some others are going to choose another
1:40:34solution so you cannot use rationality
1:40:36as a way to infer what other people are
1:40:39doing so I think that in a philosophical
1:40:42sense it's a very important uh model
1:40:45that shows that the reality can be
1:40:48extremely compact something else related
1:40:51to this
1:40:52is that this is uh the the landscape for
1:40:57a given set of gij
1:41:00but maybe you know maybe I don't measure
1:41:04the jic correctly maybe there's a little
1:41:06bit of noise in the gijs or maybe gijs
1:41:08are evolving with time
1:41:10so maybe today jij is equal to this and
1:41:15tomorrow a gij will be changed by some
1:41:18small
1:41:21uh
1:41:22perturbation Delta jij
1:41:25or maybe you know I've not measured the
1:41:28jic correctly so there's a little bit of
1:41:30error and what happens is that because
1:41:33of this very small perturbation you can
1:41:37have some of the
1:41:38secondary Minima that become
1:41:43the new Minima
1:41:46okay so when you change Delta i j a
1:41:48little bit you can have
1:41:50a change of what you call the ground
1:41:53state in a chaotic manner so
1:41:57when the system size goes to Infinity
1:41:59the Delta gij you need to change the
1:42:03order of the different configuration is
1:42:06going to zero so this this is called
1:42:08chaos and uh and I've alluded to that
1:42:11earlier in my lecture so it's a kind of
1:42:14again of a fragile optimization
1:42:21in the sense that
1:42:24if you change a little bit the value of
1:42:26the parameters
1:42:27you can completely change the structure
1:42:30of the solution so again this is a very
1:42:32I think this is a very important uh
1:42:34Paradigm to understand that optimization
1:42:37alone often is misleading because it's
1:42:40the it's it's it's useless in a way to
1:42:44understand what people are going to do
1:42:45because depending on what they actually
1:42:48assume about the gij they might end up
1:42:51even if they are able to solve the
1:42:53problem in the the they might end up in
1:42:56very different considerations
1:42:58and finally not only you can have these
1:43:01small changes but because of small
1:43:04perturbations in the gigs some of the
1:43:06Minima that existed before May
1:43:09completely disappear so for example
1:43:11because of a Delta IJ that's very small
1:43:14this minimum here can suddenly
1:43:18you know be bypassed completely and
1:43:21disappear
1:43:22and so you know you might be in a
1:43:24situation which is locally stable
1:43:26for some gigs and because you've changed
1:43:29the Delta gig a little bit you're not
1:43:31locally stable anymore
1:43:33so uh this is again often called chaos
1:43:40in a sense a little different from chaos
1:43:43in dynamical systems it's chaos with
1:43:45respect to the uh to the data you give
1:43:48to the optimization problem
1:43:56right so that's
1:43:59of course the I mean spin glass
1:44:01literature has exploded in the last 50
1:44:04years in the in physics and there's
1:44:07there's a lot of ramifications of this
1:44:10the I think very beautiful example that
1:44:13I've only touched upon very quickly in
1:44:16the last 15 minutes but I I think it's
1:44:19an important uh item to add in these
1:44:23lectures for you to understand the well
1:44:26where I think we're going in terms of uh
1:44:30transposing problems from physics into
1:44:33uh
1:44:35economics and sociology
1:44:37and so on that note I'm ending the set
1:44:40of lectures of course there's a lot I
1:44:42would have liked to talk about and uh
1:44:44for some reason I haven't had time so I
1:44:46don't know if it's because uh you're not
1:44:49in the room so I'm I was a little slower
1:44:52than usual I don't know but anyway
1:44:55um this is what I roughly what I want to
1:44:57speak about to you this year
1:45:00uh I hope that the exam will be
1:45:03interesting and that we can hold it in
1:45:07normal conditions and I'm and I'm
1:45:10reiterating my proposal to speak with
1:45:13any of you next week Wednesday I'll send
1:45:16a link I won't be in the room but I'll
1:45:19send the link and if you have any
1:45:21anything you want to chat about then
1:45:24feel free to connect
1:45:26that's it folks
1:45:28any question now I don't know
1:45:31is the
1:45:35hey Valentina is here
1:45:38I mean Valentina is on the
1:45:41not physically in the room but she's
1:45:43present
1:45:46so in 15 minutes you'll have Valentina
1:45:48but until then any comment question
1:45:52so she's here
1:45:59no
1:46:07in the room questions
1:46:13I'm surprised I mean that's one of the
1:46:15big changes is that they are really
1:46:17really very few questions I don't know
1:46:20what you're
1:46:21taking out anyway
1:46:27okay
1:46:32well I'll end the recording session and
1:46:35wish you good luck for the exam
1:46:38this conference will now be recorded
1:46:42Okay so
1:46:45welcome back to everybody to the last
1:46:47day
1:46:48So the plan for today is to discuss a
1:46:52little bit uh the model which is given
1:46:55in the paper that I I think was in the
1:46:58uh yes folder I hope or in any case I
1:47:01just sent it to you via email
1:47:05so this is a paper of uh 2015 and it is
1:47:10a little bit an excuse for us to discuss
1:47:12about uh
1:47:14say checking the stability of solutions
1:47:18of focus blank equations
1:47:20so let me first tell you what is the
1:47:22idea of the paper that included first
1:47:28and then I will tell you what we are
1:47:31actually going to discuss
1:47:32so the idea is to write down a simple
1:47:36model of interactive firms so we will
1:47:38just think at the model in terms of
1:47:41firms indeed but if you look at the
1:47:43paper towards the end and there is a
1:47:45discussion on how to interpret the model
1:47:47uh for example then epidemiological
1:47:51models or or as a model of interacting
1:47:54banks in a financial market and so on so
1:47:58there are plenty of possible
1:47:59interpretation
1:48:00and the interesting thing is that there
1:48:03are regimes in which this the behavior
1:48:07of the system shows persistent
1:48:09oscillations which are called We Will
1:48:12Call some synchronization phenomena and
1:48:16now this is interesting because uh
1:48:18somehow it's telling you that you have a
1:48:20model where you go from having a moment
1:48:23or period of prosperity if you think of
1:48:26this in terms of of an economy where
1:48:29everything seems to to go well and then
1:48:32suddenly you have
1:48:34the negative oscillation you have abrupt
1:48:37crisis in your economy and this goes on
1:48:40in a cyclical way and this is
1:48:43interesting because this crisis again
1:48:45happen without any external shocks which
1:48:48perturbs your economy but are really
1:48:51built in in the choice of parameters of
1:48:53your model
1:48:55so what we're going to do today is not
1:48:57really uh go into detail of this phase
1:49:00uh where we where we have oscillations
1:49:02but rather we are going to
1:49:04estimate and compute what is the
1:49:07boundary of stability of the phase
1:49:09without uh oscillation so what we want
1:49:11to do uh today so these are
1:49:16introductory comments if you want
1:49:19and this starts so if you have the paper
1:49:23with you if you look at people two
1:49:26just so I have an idea
1:49:27what the paper is about so figure two
1:49:30gives you the behavior in time of a
1:49:33quantity that I will introduce in a
1:49:35minute that is essentially
1:49:37the fraction of firms in your model that
1:49:41are uh
1:49:43that go bankrupt so which have financial
1:49:46problems if you want and you see from
1:49:49that figure that there are two different
1:49:50regimes of parameters so in one regime
1:49:52of parameter you find that the behavior
1:49:54of this dynamical quantity uh looks like
1:49:57this so you have some initial
1:50:00oscillation a transient but then you
1:50:01converge to some positional value for
1:50:04this fraction and this happens for the
1:50:07parameter five that I will introduce in
1:50:08a minute which is all
1:50:11efficiently small whereas you have
1:50:14another phase if you want where this
1:50:16type of oscillations appears so if you
1:50:18look at this plot at that plot so I
1:50:20don't change colors because I know that
1:50:23is very hard to see them on the screen
1:50:26but somehow in the second phase you see
1:50:29that you have this
1:50:30Collective Global synchronized
1:50:33oscillations in the number of firms that
1:50:37are bankrupt and these items
1:50:42is large
1:50:44and so the idea is that we are not going
1:50:46as I said to compute explicitly the
1:50:49solution so the time behavior of this
1:50:51fieste but we want uh somehow to
1:50:54estimate
1:50:55right
1:50:59estimate
1:51:01when
1:51:04the
1:51:05express stationary solution
1:51:09which is not postulating
1:51:13becomes unstable
1:51:23and by when I mean uh of course for
1:51:25which values of parameter uh this first
1:51:28regime becomes dynamically unstable
1:51:32so what does it mean technically well
1:51:34technical it means that we are going to
1:51:36try to solve for the stationary state of
1:51:39a focal blank equation which describes
1:51:42our model
1:51:43and then so this would come into one
1:51:47and then we are going to look at the
1:51:50stability of this focal Planck equation
1:51:53so let me add a comment on this
1:51:58so what does it mean to look at the
1:52:00stability well let's
1:52:03um think about a very simple example
1:52:06that we discussed also last time so
1:52:08let's think about
1:52:09the case in which for instance we are
1:52:11looking at some Optima of some function
1:52:16like the one that I'm drawing here
1:52:19and you can have different points which
1:52:22are stationary points one which is for
1:52:23instance a local minimum and one which
1:52:25is a local maximum and they are
1:52:27distinguished by their properties of
1:52:29stability so of course the local minimum
1:52:32is stable and the local maximum is not
1:52:33stable
1:52:35and one way to check this that we are
1:52:37going to generalize the function today
1:52:39is just to uh to do the following so you
1:52:43sit into your point which is a solution
1:52:45of your equation in this case the
1:52:47equation is the potential the equation
1:52:50will just be
1:52:51that we are asking the derivative of the
1:52:55potential uh is equal to zero so you
1:52:58have different solutions and to check
1:52:59the stability what you can do is to do a
1:53:02little bit
1:53:03of distribution around your uh your
1:53:05stable point and to check whether doing
1:53:08the preservation and then letting the
1:53:10system evolve the system goes back to
1:53:12your original solution or it goes
1:53:15somewhere else of course in this case
1:53:18with with gradient descent like dynamics
1:53:21that we discussed last time if we go
1:53:23back so you preserve a little bit under
1:53:25the system relaxes back to the local
1:53:28minimum so this is
1:53:30table
1:53:32whereas if you see them here as you
1:53:35easily realize as soon as you do
1:53:36preservation your system flows somewhere
1:53:39else and therefore this point will be
1:53:44so the idea is to use the same type of
1:53:47reasoning today but for full functions
1:53:50which are the solutions to uh
1:53:54to our Focus line equation as I said
1:53:57and now perhaps let's make a third
1:54:00comment and then we
1:54:02start
1:54:03and the third comment is about
1:54:06[Music]
1:54:08current and stationary States so
1:54:16and and this introduces a little bit uh
1:54:19the problem of today so the difference
1:54:21with respect to the soccer Planet
1:54:23equation that we are discussing today
1:54:24and the one that we discussed in the
1:54:27last day is that today we are going to
1:54:30have a system with uh
1:54:32what people call some sources or some
1:54:36things
1:54:38they
1:54:42we are having a soccer blank equation
1:54:49we've
1:54:51accuracies
1:54:53and things
1:54:55because this will become clear and once
1:54:58we write it down
1:54:59but I just wanted to anticipate it
1:55:01because I want to comment on a
1:55:04difference with respect to what we have
1:55:06seen last time
1:55:07so last time
1:55:11and in the previous study we were always
1:55:14looking at soccer blank equations which
1:55:16we could write in the in the form of a
1:55:18continuity equation so we had expression
1:55:21of the following form so let's think
1:55:24about one dimension
1:55:25we have the derivative of our
1:55:27probability was
1:55:29minus the derivative with respect to
1:55:33the position let's say of a current
1:55:39and the current was depending itself on
1:55:42the probability and on its derivative
1:55:43over time
1:55:45so this is this is what is usually
1:55:47called the continuity
1:55:51equation
1:55:55and then when we looked at the
1:55:57stationary points uh the session and
1:56:00state sorry what we were doing is well
1:56:02if the stationary state
1:56:04stationary it means that the derivative
1:56:07over time has to be equal to zero and
1:56:10the derivative over time being equal to
1:56:12zero because of this relation was the
1:56:16same as asking the current
1:56:19is equal to constant
1:56:24and we always assume that we could
1:56:27choose this constant to be
1:56:30to be equal to zero
1:56:32and there are arguments so in some cases
1:56:35there are arguments to to make this
1:56:37choice for instance if you have a focal
1:56:40plant equation which is defined uh in
1:56:43the space which goes from minus infinity
1:56:45to infinity and you have currents which
1:56:48are usually functions of the probability
1:56:51it says
1:56:53and of the derivative
1:56:55then you can say well in order for the
1:56:57probability to be well normalized
1:57:00if the probability itself and the
1:57:02derivatives have to go to zero to
1:57:04Infinity but if they are constant and
1:57:06they are 0 to Infinity then they have to
1:57:07be equal to 0 everywhere
1:57:10which is true if you have an unbounded
1:57:12interval if you have a boundary interval
1:57:15like the one that we discussed in the
1:57:18exercise about the model or the
1:57:22Kirman and more than last time you can
1:57:25still argue in a similar way so you just
1:57:28say okay I expect that there is no
1:57:30current in my stationary solution and
1:57:33therefore I set this to zero and setting
1:57:36these two zero gives me then an equation
1:57:37that I can solve very easily with with
1:57:41the separation of variables that we saw
1:57:44now today this is going to be a little
1:57:46bit different because precisely because
1:57:48we will have this sources and change
1:57:51so this means that our equation will be
1:57:54defined in some intervals of X and we
1:57:57will have special points
1:57:59such that whenever
1:58:02our variable reaches this point it
1:58:05either dies in some way so it disappears
1:58:09from the model and this will be the sink
1:58:11or it is injected back at that
1:58:14particular point in our model and this
1:58:17will be assert so it is like having an
1:58:20open system in which you have some
1:58:22special points where you start
1:58:24inserting a new probability for your
1:58:27variable and points where you
1:58:30say eject the probability from your
1:58:33model and therefore you can easily
1:58:35understand that if I start increasing
1:58:38probability here and taking it back in
1:58:41here there might be even in the
1:58:43stationary State some constant parents
1:58:45in my system which goes current of
1:58:48probability which goes in this direction
1:58:51and therefore we should not put this J
1:58:54equal to zero but we will
1:58:56expect it to be gone
1:58:58and this is what is going to happen in
1:59:00this in the example of today
1:59:04okay so this was a little bit verbal but
1:59:06maybe it becomes more clear
1:59:09when doing the uh the exercise
1:59:12so let me see
1:59:17the direction
1:59:25okay
1:59:26it just it is
1:59:36let's introduce the model and the
1:59:39equation and then I think
1:59:42all of this talking will be a little bit
1:59:46more transparent
1:59:58so the model looks like this you can
2:00:01look at the paper if you want
2:00:05a more detailed description but the idea
2:00:07is as follows so you have n firms
2:00:15it was um
2:00:18from one to one
2:00:21and you have a variable which describes
2:00:23each of these firms which is the
2:00:25so-called fragility
2:00:32which you can think of as a measure of
2:00:35how bad this sperm is doing so this is
2:00:39the ratio between the dabs
2:00:42that the sperm uh Escape
2:00:49the bank or
2:00:51whoever
2:00:53gives blown to the firm I divided by the
2:00:56total asset so this is the amount of
2:00:59money if you want
2:01:01of the Easter so of course the larger is
2:01:05the dead so let's say if uh if you have
2:01:08a lot of that then this variable
2:01:10X will be negative and the more negative
2:01:14is this variable the worse is
2:01:17is the state of your firm
2:01:21now this firm uh have a Dynamics so the
2:01:24idea of the model is to assume that the
2:01:26Dynamics is like around the wall so you
2:01:28have
2:01:29diffusion
2:01:33this agility space of the firms with
2:01:37some diffusion content B and then you
2:01:40have also some brief terms the constant
2:01:43velocity
2:01:46which is given by B which are parameters
2:01:51and then you have uh two special points
2:01:55that based on what I said before can be
2:01:58interpreted as a surf and as a sink so
2:02:03the surf the sink will be
2:02:08at
2:02:09a value of x which we call let's say
2:02:13minus Theta
2:02:15and it is as follows so you have your X
2:02:18variable
2:02:20yes zero somewhere and then you put a
2:02:23special at some particular point of your
2:02:26choice is minus Theta
2:02:29and you have all of these points which
2:02:32are performing some diffusion in this
2:02:34our x-axis and we say that whenever
2:02:37the point
2:02:39has adapt which becomes as large as
2:02:43a negatives
2:02:46these are now negatively smaller than uh
2:02:49than my Theta so if you if you have a
2:02:52firm which crosses this particular
2:02:55threshold in here
2:02:56then you say that the depth is too large
2:02:59and therefore the firm goes bankrupt
2:03:03threshold
2:03:06for
2:03:10bankrupt
2:03:17so what this means is that healthy firms
2:03:19we live in this part of your
2:03:22configuration space
2:03:25these are the firms which are active
2:03:29and then as soon as you cross the
2:03:32special your firm becomes uh say
2:03:35inactive
2:03:38so they hire for bankruptcy and
2:03:41therefore they are in some sense no
2:03:42longer uh they freeze they are no longer
2:03:45into your model
2:03:47so in terms of the soccer plan equation
2:03:49this will be translated into the
2:03:51presence of an absorbing boundary so
2:03:54whenever you reach that point then the
2:03:57corresponding probability has to go to
2:03:58here we will see this so we have this
2:04:01special value which is uh if you want
2:04:03the sink
2:04:04of our model and then we also have a
2:04:07surf and the source
2:04:11is
2:04:13at x equal to zero
2:04:16so what this means is that you have some
2:04:18firms which to die or become inactive
2:04:22but then you also decide that with a
2:04:24given rate you can take one of these
2:04:26terms and give some loan or give some
2:04:29money to it and revive it so put it back
2:04:32into the model and you do this by
2:04:35putting it back at this particular value
2:04:39of agility which is which is you know
2:04:43so this means that with some frequencies
2:04:45I can take one of the firms which are
2:04:47that and I can arrange like them into my
2:04:50uh
2:04:51say Axis or regime of active firms
2:04:54precisely F0
2:04:57and so this will be the source
2:04:59where some firms will appear with a
2:05:02certain frequency
2:05:04and then you have a next ingredient the
2:05:07final one which is interactions
2:05:14so these firms perform their run work
2:05:17with a given list and with the diffusion
2:05:20sometimes they die sometimes they are
2:05:21they injected but they also have some
2:05:24sort of interaction that takes the form
2:05:26of feedback
2:05:30which happens whenever one firm dies or
2:05:34becomes inactive so the idea is that
2:05:36when the firm becomes so bad that it
2:05:40reaches the values minus Theta then what
2:05:43happens is that the fact that this sperm
2:05:45is failing influences negatively all of
2:05:48the other sperms because you have a
2:05:51depth that this firm had to pay and that
2:05:53now it's unable to pay anymore because
2:05:55it is sales that gets redistributed to
2:05:58all of the other firms which are alive
2:06:01so what this means
2:06:03uh
2:06:05maybe I should write it in words and
2:06:07then we look at the formulas
2:06:09so the for
2:06:12uh
2:06:15yeah
2:06:18terms
2:06:21the tail
2:06:27that that's
2:06:29which is of the order of theta is
2:06:31redistributed
2:06:40to
2:06:42all of the others
2:06:46uh
2:06:48Optics first
2:06:53and the way we model this is by
2:06:55modifying the drift of those firms which
2:06:58are active
2:07:00towards let's say the negative axis
2:07:02whenever one of those reaches this
2:07:06actual value might be
2:07:08so let's look at this with formulas
2:07:10which I hope
2:07:14it's perhaps the best thing to do
2:07:17so all of these ingredients
2:07:19that I mentioned enter into some
2:07:22chocolate blank equation
2:07:24that is written in the paper for this
2:07:26model
2:07:371.1
2:07:41which is to justify the focal blank
2:07:43equation that is given in the paper so
2:07:45the focal blank equation looks like this
2:07:46so now this is of course the probability
2:07:49who have a firm which has a fragility X
2:07:52at a given time t
2:07:55which of course depends on time
2:07:58so you have a drift term
2:08:01that
2:08:03depends on time itself that I call BLT
2:08:10multiplied by the space derivative so if
2:08:12you remember
2:08:14uh well okay let me comment later
2:08:18then you have the diffusion curve
2:08:21recognized
2:08:25and then you have to we have to model
2:08:27this sort so the fact that we are
2:08:30sometimes pre-injecting ferns at zero
2:08:34and we model it in the following way so
2:08:37first I forgot to tell you something so
2:08:39from now on let me see
2:08:43the x axis and then we find it
2:08:492x minus beta so I shift it in such a
2:08:52way
2:08:54I shift everything forward in such a way
2:08:57that now the threshold for bankruptcy
2:08:59becomes zero and I have active firm in
2:09:02the positive synapses
2:09:05and so the injection which before was
2:09:07Zero now that I shifted happens at the
2:09:09point x which is equal to Beta
2:09:13so curious if you are
2:09:17at that particular point
2:09:19of the Delta function s like
2:09:22the poinsettia you reject your firms
2:09:26with a given rate that is this parameter
2:09:28five
2:09:30and the rate is multiplied by
2:09:33the fraction of firms
2:09:36which are inactive
2:09:39so which are in the negatives can I ask
2:09:41this every given time T which I call 1
2:09:43minus Phi of t
2:09:45so what is Phi of t
2:09:47I of T is the fraction of a live firm so
2:09:51this would be the integral
2:09:53in the position my Axis once I shifted
2:09:55everything off
2:09:58the elections
2:10:01so this is a fractional
2:10:06off
2:10:09ER
2:10:13which of course is not constant in the
2:10:16model so they put a number of firms is
2:10:18constant so if you if you want if you
2:10:20expand this integration to the fully
2:10:22interval and take into account also
2:10:24those which are inactive
2:10:26your probability is normalized one but
2:10:28if you focus only on the sub interval
2:10:31which corresponds
2:10:33firms which are really participating to
2:10:36the economy then you have a quantity
2:10:37which increases
2:10:41so this explains the
2:10:44here which is another source
2:10:50and then where is uh the interaction uh
2:10:53hidden the interaction that I mentioned
2:10:55well this is hidden in this drift
2:10:58coefficient e of e which has a
2:11:02particular form foreign
2:11:07is equal to some constant drift that is
2:11:10the B that I introduced before
2:11:14and then you have an extra term which
2:11:16accounts for this redistribution of the
2:11:18debt uh whenever somebody dies
2:11:22will become
2:11:24bankrupt
2:11:26which is of the following form so beta
2:11:28here is uh you want another
2:11:30phenomenological concept of the model it
2:11:33is the strengths
2:11:34of the interaction if you want
2:11:37or of the feedback
2:11:40of one firm
2:11:44on all the others
2:11:49which is telling you that of course in
2:11:51an economy whenever one firm uh fails
2:11:54this is not good for the others as well
2:11:56maybe because they were depending on the
2:11:58product of that firm and which they will
2:12:01have available anymore or because they
2:12:04they were exchanging with it and so on
2:12:06and so forth so there is a negative
2:12:07feedback between the different firms
2:12:10with the strengths encoded in this
2:12:12system
2:12:14then you have Theta so this accounts for
2:12:17how much depth is ready to be has to be
2:12:20redistributed and between all of the
2:12:22other firms
2:12:23and then in here to uh to control this
2:12:27feedback you have P times the derivative
2:12:33space
2:12:34your probability distribution
2:12:36distribution evaluated
2:12:38h0
2:12:40so at the point where where the firms
2:12:44are disappearing from from the economy
2:12:48and why does this term has this form so
2:12:52first of all what is this so the way you
2:12:55can interpret this term in here is as a
2:12:58flux so this is the flux
2:13:02of
2:13:05ability
2:13:11or the flags of firms
2:13:14okay
2:13:18the speech
2:13:26so uh what do I mean by your snaps while
2:13:29a flax is telling you what is the
2:13:31probability for unit time and for uh
2:13:35unit of surface in this case uh if you
2:13:39focus on a small interval
2:13:41uh around
2:13:43minus beta that now became zero because
2:13:46we have
2:13:47is with everything so this flux controls
2:13:51uh if you want what is the current of
2:13:54probability that crosses a little
2:13:57interval of
2:13:59size DX around the special Point here so
2:14:02it is measuring
2:14:04if you want how much firms are are
2:14:07failing in a given unit of time
2:14:11and well maybe if you
2:14:15okay let me add the first comments
2:14:19and then we explain this term a little
2:14:21bit better
2:14:22so for us the idea is I hope it's clear
2:14:25so what happens is that if you have a
2:14:28lot of firms which are uh which are
2:14:32failing then you start getting a
2:14:35velocity that is more and more negative
2:14:38so all of the other firms start all
2:14:41together to breathe themselves towards
2:14:44the absorbing point because of this
2:14:46feedback term
2:14:48and what happens at the observing point
2:14:50so this is the last thing that we need
2:14:53for defining the model
2:14:56well the observing Point as I just
2:14:58mentioned is an absorbing point so which
2:15:00means that the probability
2:15:03any time at the point zero has to be set
2:15:08to zero
2:15:11and with all of these terms we have
2:15:14somehow defined
2:15:16models that we are going to okay
2:15:20so this is often called an absorbing
2:15:22boundary
2:15:24in the focus which
2:15:27so let me perhaps
2:15:29make a little bit of a comment uh about
2:15:33this flux
2:15:34right
2:15:45backwards
2:16:01foreign
2:16:07why can we interpret this as a slacks or
2:16:10what is the idea
2:16:12but the idea is a little bit as follows
2:16:16so you have again your x-axis you have
2:16:19this point
2:16:20zero and here you have some
2:16:24absorbing point so whatever goes beyond
2:16:27this point uh
2:16:30diminutive so it's no longer in your
2:16:32border
2:16:33you have this special Point uh Theta
2:16:36where you start injecting firms so you
2:16:38somehow expect that you will have an
2:16:40access
2:16:41of probability around this point because
2:16:45that is where you put
2:16:47your firms with a given rate
2:16:51Phi so I'm now drawing what one can
2:16:53expect for the shape of this probability
2:16:56and then you will have something that
2:16:58you impose
2:16:59has to go to zero in here and then it
2:17:03has to remain exactly equal to zero
2:17:07for whatever value of x which is
2:17:10negative because these are no longer in
2:17:12your mother so you don't track them with
2:17:15your Dynamics equation and then at
2:17:17Infinity you should go down in such a
2:17:19way that they are normalizable
2:17:22but the way you go to zero here is with
2:17:24the derivative which is uh which is
2:17:26non-finite and and somehow this gives
2:17:29you what is the probability flux to go
2:17:32to zero so you will have a current of
2:17:34probability that flows uh towards this
2:17:37point
2:17:38and the way you can make sense uh of of
2:17:42this expression here I think it follows
2:17:45so you look at the focal flank equation
2:17:47you should describe your model
2:17:49and then you try to integrate this in a
2:17:52small interval uh DX which is around
2:17:55your special point
2:17:57let me take the left hand side
2:18:01and then we integrate
2:18:03this derivative in here with respect to
2:18:06X which goes from
2:18:08Epsilon
2:18:11so you don't have to write this noun
2:18:13it's just to
2:18:14try to motivate
2:18:16this term here
2:18:19so if I do this I take the derivative
2:18:20over time outside and what do I get well
2:18:23I get the derivative
2:18:26over time of the probability
2:18:29uh
2:18:32well
2:18:33if I approximate this
2:18:36as if I assume that P is somehow
2:18:39constant in this middle interval or let
2:18:41me let me write it like this
2:18:44so this already tells you what I want to
2:18:46say so this is how much it changes the
2:18:49amount of probability that you have in a
2:18:51small interval around zero
2:18:53and using the right hand side you do the
2:18:55same thing so you integrate
2:18:57now this expression over X in this case
2:19:00you don't have to worry about this Delta
2:19:02function because this Delta function is
2:19:04far away at this value of theta which is
2:19:06not around
2:19:07an Infinity
2:19:10close to zero so I can forget about the
2:19:12source term
2:19:14and I can integrate whatever remains
2:19:16which has the form of a total derivative
2:19:19and if I integrate what do I get I get a
2:19:22contribution which is
2:19:24P of t
2:19:26of sine over P minus P of minus
2:19:32and then I get the contribution from the
2:19:34diffusion which is Plus
2:19:37these
2:19:38the derivative
2:19:40in x
2:19:42Cylon
2:19:43minus the derivative of minus 5.
2:19:49I hope you see this okay
2:19:53and then I take Epsilon to zero and if I
2:19:56take Epsilon to zero well actually if
2:19:59you are with the last of these intervals
2:20:02as minus the challenge we are assuming
2:20:04that everything is equal to zero so this
2:20:05will be zero but
2:20:07oh
2:20:08and the derivative will also be zero
2:20:12and if I take
2:20:14flash Epsilon
2:20:17which goes towards zero as well what I
2:20:20realized is that because of the
2:20:21absorbing boundary this is also a
2:20:24converging to zero so in the limit of
2:20:26Epsilon small which is what we're
2:20:28interested in because
2:20:30flux then this term will disappear and
2:20:33what you are left with is precisely
2:20:36this derivative of the probability with
2:20:39respect to X which is what we are
2:20:41putting it here
2:20:44so this was just to motivate why the
2:20:46term which appears
2:20:48in the driest has has this particular
2:20:52form
2:20:54okay so so for the model uh we are more
2:20:58or less there so you see the source is
2:21:01here
2:21:02the sink if you want is encoded in this
2:21:05absorbing boundary and the interaction
2:21:06is encoded in this time dependence drift
2:21:11for our shorts
2:21:13and now what we have to do is to try to
2:21:15solve for the stationary state of this
2:21:18program and then look
2:21:20at the stability
2:21:24which are points uh so this was point
2:21:26one motivate the model and now let's do
2:21:29point
2:21:30uh two and three and I just realized
2:21:33that
2:21:35backwards
2:21:43back here
2:21:45okay
2:21:49so before we start for the stationary
2:21:51state
2:21:52let's try to organize a little bit all
2:21:55of the parameters that we have
2:21:57which are many
2:22:15foreign
2:22:17ters
2:22:21in the model when we have essentially
2:22:23five parameters so we have the
2:22:26drift B and diffusion
2:22:28as in a user wrap work then we have the
2:22:32strength of the interactions or the
2:22:33feedback which will be stopping beta
2:22:36then we have the threshold for failure
2:22:40which is
2:22:41parameters
2:22:44and then what did I forget then we have
2:22:47this High which is the rate at which you
2:22:49range at the firms into the model
2:22:54so these are many but what you can show
2:22:58by solving the model is that
2:23:01essentially everything will depend on a
2:23:04combination of this
2:23:05of this parameter so you can reduce
2:23:08everything to the behavior of three
2:23:10parameters
2:23:14one is beta which remains is
2:23:18then you can introduce a ratio which is
2:23:21called the clear number that is
2:23:23something which appears when you study
2:23:26transport of his shoes
2:23:30that will not appear in what we are
2:23:33going to discuss but let me introduce
2:23:35it anyway
2:23:37the papers so this is the ratio between
2:23:40beta theta over the diffusion constant
2:23:44and it is if you want a measure of
2:23:47the drift versus the diffusion
2:23:50of your work where the interaction does
2:23:53not enter
2:23:56and then we have another parameter which
2:23:57is instead important for us that I call
2:24:00is that
2:24:01here
2:24:03which is a ratio of time
2:24:06and in particular it is the ratio of the
2:24:10injection time the rate at which one
2:24:13over the rate at which you
2:24:15research firms in your model so this is
2:24:17one over five
2:24:20divided by uh the time which is the time
2:24:24space at which typically your firms are
2:24:28suppressed from the model so this is
2:24:30your firm dies and and this is computed
2:24:34in the limits
2:24:35when beta is particularly small
2:24:39so when we can neglect this feedback
2:24:43then what is the typical science case at
2:24:46which a firm dies well this is basically
2:24:49controlled uh just by the drift constant
2:24:52B so I
2:24:53at a given time P0 I range Act a firm at
2:24:57C time then I ask what is the typical
2:25:00time which is required for each for each
2:25:03to reach zero and therefore disappears
2:25:05from the model and I can estimate this
2:25:08if I can neglect beta if I can neglect
2:25:10the diffusion essentially as
2:25:13uh as B divided by so I want these what
2:25:19B is a velocity
2:25:21will be the in order to go from here to
2:25:23here I have to cover a distance that is
2:25:26equal to Theta with a velocity B so what
2:25:29is the time that I need to do this well
2:25:32it is related to B by by this
2:25:35relationship
2:25:36but now that I'm interested in is just
2:25:42okay and this is what I put in here
2:25:45but
2:25:46one over five use a time and then I
2:25:48divide by another time which is
2:25:51Theta over B
2:25:53which is my parameter be time this is
2:25:55that sorry and that is what will appear
2:25:58in our calculation in a minute
2:26:02so it could be introduce it
2:26:04and it's good to take or to have in mind
2:26:08a limit
2:26:09that will be useful later on
2:26:12which is the limit
2:26:15when
2:26:17that is going to zero and again
2:26:20God is having mine for instance that
2:26:22data is very very small
2:26:24so when that is going to uh
2:26:28zero
2:26:31then what we are saying so so this is
2:26:33the time at which we reinject
2:26:36versus uh the time uh for for the first
2:26:41guy and so the idea is that whenever we
2:26:45take this limit we should
2:26:47uh
2:26:48we have essentially that every time
2:26:51somebody dies it gets immediately
2:26:53reinject it so you don't have to wait a
2:26:55lot in order to see it rejected so in
2:26:59this limit
2:27:00any
2:27:04Burns
2:27:06but
2:27:08nice I say nice but of course and it
2:27:11means it goes
2:27:13bankrupt
2:27:17is immediately
2:27:21Ranger
2:27:27and we will use this fact in a minute to
2:27:31fix some boundary condition
2:27:33or to select some solutions
2:27:36of office consistent equation so just
2:27:39this is just a common keep in mind this
2:27:41limit because it will be useful in the
2:27:44following
2:27:47okay so now that we have all of this we
2:27:51can now try to solve
2:27:53for the stationary state of this
2:27:56particular equation
2:27:59and I don't know what the best space
2:28:03uh Solutions so
2:28:10I will keep the equation to this point
2:28:13three
2:28:15and these I will erase
2:28:18but we will use it all the time
2:28:22keep in mind all of this definition
2:28:31absolutely
2:28:35okay
2:28:38so now let's look for this stationary
2:28:41solution
2:28:44so what does stationary means
2:28:46well
2:28:48in general it means that you won't
2:28:53seems not to depend
2:28:55sometimes but to be constant
2:28:57time
2:28:59so the first thing that you can ask is
2:29:01that your fraction
2:29:03of
2:29:05active terms which we Define that
2:29:08Phi of t
2:29:09in the stationary state is time
2:29:12Independence and it is just equal to a
2:29:15constant that I call five zero
2:29:21and with a very similar reasoning
2:29:26you can also ask that all of the other
2:29:30time dependent quantities that you have
2:29:33in your model reach a stationary state
2:29:35where as a human depend on time so for
2:29:38instance it will ask that b of t
2:29:41will
2:29:44reach a constant value which is d0
2:29:49and what is b0 well
2:29:51if you remember what was this fraction
2:29:53for B of t
2:29:55will be zero was B plus
2:30:02and then you add B times the derivative
2:30:06which following the notation of the
2:30:08paper I will call so the flux at 0 and
2:30:12we call it J of T so this is d times the
2:30:15derivative of e
2:30:19evaluated at x equal to zero
2:30:25so this is a quantity which in general
2:30:26depends on time which enters in my
2:30:28definition of d of T but of course if I
2:30:30ask that b does not depend on time
2:30:32anymore then J should have submerged
2:30:35or be equal to a constant we check for
2:30:37j0
2:30:40and of course
2:30:48and of course
2:30:50you have another quantity in here which
2:30:52depends on time which is the itself so
2:30:54the fourth thing that you have to ask is
2:30:58that your field
2:31:00X and E
2:31:02is something which does not depend on
2:31:04time which I will call be stationary
2:31:06effects
2:31:11okay now how can this be true that you
2:31:14reach a Time independent value for this
2:31:17fraction of a live terms
2:31:20whenever you have certain I think
2:31:24so whenever your system is open
2:31:26at this particular point of zero and
2:31:30Theta
2:31:31well in order for the fraction of firms
2:31:33not to change what you have to ask
2:31:36is that uh somehow the the Flux Of firms
2:31:41that you inject
2:31:42at the point x equal to Theta so the
2:31:45incoming flux of probability has to be
2:31:48equal to the flux which goes out from
2:31:51your system at the point here
2:31:53which was exactly given by this jfp
2:31:57so with this stationarity uh implies or
2:32:00what we have to impose in addition to
2:32:03this is that the fluxes
2:32:12are equally
2:32:14so the rate at which firms 9 is the same
2:32:17rate which firms are introduced into our
2:32:20model in such a way that the total
2:32:21number of firms for the fraction remains
2:32:25so what this means is that
2:32:29J of T which is the slot of
2:32:32outgoing firms
2:32:35which we assume be equal to a constant
2:32:38in the stationary state
2:32:40has to be equal to the Flux Of incoming
2:32:42terms which is given by this Search
2:32:46terms here
2:32:47so it has to be equal
2:32:49y
2:32:511 minus
2:32:535 0.
2:32:56so in principle you have Phi of T but in
2:32:58the stationary State we assume that all
2:33:00of them
2:33:02so we have to impose this uh particular
2:33:05relation in order for this
2:33:08assumption here to make sense
2:33:13okay
2:33:14so now given this let's try to plug this
2:33:19into the equation so let's try to
2:33:23impose this and solve the four fourth DP
2:33:28of x t already t
2:33:30equal to zero as we did also last time
2:33:35with this assumption
2:33:41Here and Now
2:33:42as I commented at the beginning
2:33:46so DP DT equal to zero means that the
2:33:49right hand side of our equation is zero
2:33:51and before we could
2:33:53translate this
2:33:56or in simpler models we could translate
2:33:59it or if you want in simpler model we
2:34:01could write
2:34:02the right hand side in the form of a
2:34:05continuity equation so as the total
2:34:07derivative something and then we add a
2:34:09current and then we could set the
2:34:10current to zero but now we have that
2:34:12Delta function which
2:34:18and how embed this term
2:34:21between some continuity equation
2:34:25so we have to do something else
2:34:28let's see looks a little bit different
2:34:29but actually you will see it more or
2:34:32less the same thing
2:34:34for
2:34:36a specialized state
2:34:41so instead of putting the current to
2:34:43zero what we do is to integrate
2:34:47our soccer plan equation with respect to
2:34:49X
2:34:51it is essentially the same way in which
2:34:53you get a current so if you remember
2:34:56in the usual continuity equation you
2:34:58have something like this DP over DC
2:35:00equal to something
2:35:02you set it to zero and now getting the
2:35:04current means that you're integrating
2:35:06the right side with respect to X and if
2:35:09you do this this is the anti-derivative
2:35:11so you just get J which then you said
2:35:14you know so here we are not setting it
2:35:16to zero but
2:35:17uh we are not setting J
2:35:20directly to zero but we are exciting the
2:35:24integral of the focused blank equation
2:35:27so these were many words
2:35:30pretty easy
2:35:31actually so this means
2:35:35again dbvt of XP
2:35:38equals zero
2:35:40and now what I do
2:35:42this is also equal to the
2:35:44right side and then I integrate both
2:35:48sides
2:35:49with respect to X
2:35:55and because I have a data function I
2:35:57have to split two different cases so if
2:36:00I integrate
2:36:02in an interval which does not contain
2:36:05Theta where the function will not
2:36:07contribute or as if I integrate in an
2:36:09interval which contains Theta it will
2:36:12actually continue let's split the two
2:36:15cases so let me fix the value of x
2:36:18which is smaller than Theta
2:36:20and then let me integrate the right hand
2:36:23side from x0 to this particular value of
2:36:26x
2:36:28and we will choose x0
2:36:32so if I do it the left hand side is just
2:36:35the integral of 0 which is 0 and then
2:36:37what do I have I have
2:36:39P of T times V integral of the
2:36:43derivative I hope you can see
2:36:46of course not
2:36:48uh
2:36:51the focal plant equation
2:37:00sorry what is x0
2:37:03yes x0 is the is the horizontal is
2:37:06arbitrary but
2:37:07I will later on
2:37:09so you can well
2:37:12we can choose it directly now let me
2:37:14leave it we will choose it to be zero so
2:37:16for the moment I just take my focus
2:37:18equation and I integrate over an
2:37:20interval which goes from some point x 0
2:37:23largely zero to some arbitrary point x
2:37:27smaller than Theta
2:37:28okay
2:37:31I can choose a 0 and it will be
2:37:33convenient to choose it equal to zero
2:37:35because there and we know that P has to
2:37:37be equal to zero so this is what I'm
2:37:39doing in a minute
2:37:41so sorry I don't think you you see the
2:37:43equation but I hope you have it in the
2:37:45notes I'm just integrating the right
2:37:47hand side and I get the following
2:37:50so this term was multiplied by a space
2:37:52derivative so if I integrate I just have
2:37:54P of x p minus P of x 0 t
2:37:59and then I have the diffusion term
2:38:04which
2:38:05I have two derivatives I integrate one
2:38:08and I get
2:38:10the other one
2:38:12so so far this looks like the normal
2:38:13copper plank equation because
2:38:16X is smaller than T times so the Delta
2:38:18is not
2:38:19giving any
2:38:23and then I choose as you pointed out x0
2:38:26equal to zero so if I choose
2:38:29x videos arbitrary so if I choose x 0
2:38:31equals to zero
2:38:33this term here is Vanishing because I
2:38:35have my absorbing boundary condition f x
2:38:380.
2:38:40and what is this term in here well I
2:38:44have remember that this is equal
2:38:46to this uh quantity J
2:38:50uh
2:38:53that I defined before so J of P was
2:38:57e times the derivative of P
2:39:01of x e the X evaluated Steel
2:39:06which is precisely disturbing here
2:39:12and now of course I'm using uh the wrong
2:39:15notation because if I put a zero on the
2:39:18left hand side it means that I'm already
2:39:19considering the stationary state so
2:39:23this is actually be stationary
2:39:26wax
2:39:28there should be no time dependent
2:39:30this is the stationary attack zero this
2:39:33is the derivative
2:39:34with respect to work stationary
2:39:38variable Vision this was the derivative
2:39:40of the stationary
2:39:46compute and therefore this product in
2:39:48here will just give me under my
2:39:51stationary assumption the constant Json
2:39:57okay so all together and this is under
2:40:00my stationary assumption between
2:40:04so the equation reads the zero the
2:40:07stationary events
2:40:10Mr banishing
2:40:13plus b
2:40:15derivative of the stationary of x
2:40:20minus j0
2:40:22equal to zero
2:40:27and now I can use the usual trick
2:40:32that I use whenever I want to find a
2:40:34Stationary State and indeed
2:40:36as long as Theta is X is smaller than
2:40:39Theta because it's just the user for the
2:40:40Planck equation so what I can do to
2:40:42solve this equation is
2:40:44the usual separation of variables that
2:40:47we have discussed last time
2:40:52so let me write it
2:40:55in a faster way
2:41:00actually
2:41:02write it as
2:41:08so this is just a little bit of algebra
2:41:10that's going to change Zero mine
2:41:14zero
2:41:16stationary events
2:41:21divided by D
2:41:26okay
2:41:31and then
2:41:34do the separation of variables I have
2:41:37remember I have bring everything which
2:41:39depends on p on one side and everything
2:41:41which depends on X on the other side so
2:41:44this in differential form
2:41:46can be read recent as
2:41:49deep stationary divided by
2:41:520 over D minus d0 over d
2:41:57ictionary
2:41:59equal to X
2:42:03okay
2:42:07then I integrate
2:42:11with it last time again from some
2:42:13arbitrary
2:42:150 to X more than beta
2:42:200.
2:42:26okay
2:42:29and as usual what you get on the left
2:42:32hand side is a logarithm and what you
2:42:34get on the right hand side is just
2:42:37X
2:42:49now we have to track the minus sign
2:42:52foreign
2:42:56side you would have
2:42:58minus the logarithm
2:43:01if I do the integrand
2:43:03I do therefore
2:43:06minus the logarithm of
2:43:10j0 over D minus
2:43:17the other side
2:43:23so
2:43:25this I will have to evaluate
2:43:27from X to Zero and then I have another
2:43:30constant which is
2:43:34front which is minus B over B right
2:43:40so if I take the so this is the
2:43:42anti-derivative now if I take the
2:43:43derivative of the log I get 1 over this
2:43:46and then I have
2:43:48a constant which is minus P0 over D
2:43:50which I have to count to this Factor
2:43:53and I have to evaluate it from X
2:43:56zero or if you want P of x
2:44:00V of zero
2:44:02this should be CLI
2:44:08speed of 0
2:44:14so if I do this I will just get the log
2:44:17of this evaluated at X the minus the log
2:44:20of this evaluated that's zero which I
2:44:22can write as the log of a ratio
2:44:26and the P stationary S 0 we know that is
2:44:30equal to because I think
2:44:32once you get something like this for the
2:44:35left side
2:44:36if we check and the right hand side is
2:44:38easy it is just a factor of x
2:44:43okay
2:44:47so now let me bring this constant on the
2:44:50other side so that I have minus is 0
2:44:53over d
2:44:55x negative is
2:44:58and now I can exponentiate so if I take
2:45:01the exponential of this expression what
2:45:03do I get
2:45:05yeah
2:45:060 over d
2:45:09nine would be 0 over d p stationery
2:45:13X
2:45:15is equal to this constant that now I
2:45:17bring this on the other side which is
2:45:20already into the minus
2:45:23zero over d i
2:45:29which means that might be
2:45:33stationary next
2:45:37East
2:45:40what
2:45:42B you see that I can eliminate it here
2:45:48I divide everything by this in it
2:45:52and my uh this stationary will be
2:45:55j0 over v0
2:45:581 minus E to the minus is zero over Z
2:46:10of course under the assumption that X is
2:46:13smaller than Theta
2:46:16foreign
2:46:22of our solution
2:46:25and now what do we have to do well now
2:46:27we have to look at what is the V over
2:46:29when X is larger than Theta and when X
2:46:33larger than Theta then we have an extra
2:46:35contribution to our integrated soccer
2:46:39plant equation which comes from the
2:46:40Delta so now I assume that X is larger
2:46:43than did I play the same game that I did
2:46:46before
2:46:49so I say 0 equals to be integral from x
2:46:530 equal to 0 up to this x of the right
2:46:57side
2:47:00and what I get is the following so let
2:47:02me write
2:47:08shorter than well I get to zero equal to
2:47:12one contribution from
2:47:15as before from b0 so b0 stationary
2:47:20of x
2:47:22then I have the contribution from the
2:47:24diffusion
2:47:26and this was
2:47:32B
2:47:35the derivative attacks
2:47:38minus D times the derivative at 0 which
2:47:41we say was equal to J zero
2:47:45and then we have the final contribution
2:47:48from Delta function which is just Phi
2:47:51one line
2:47:59in this remember these two fellowships
2:48:06okay and now we should remember
2:48:10something
2:48:13so do you recognize any consolation
2:48:17IRAs
2:48:19all of the formula but remember
2:48:21that when we impose a stationarity and
2:48:25we equated the flux the equation
2:48:28was precisely the following so the flags
2:48:30of firms which were dying has to be
2:48:33equal to the slacks of those which were
2:48:36injected which was five one minus
2:48:39the different five years
2:48:42so this was because
2:48:45missionaries so we can use this in here
2:48:48and we see that these two terms
2:48:50constant
2:48:54so the equation is now particularly
2:48:56simple it's just uh it's telling me that
2:48:59my stationary
2:49:03distribution for X larger than Theta has
2:49:05an exponential form so it will be a
2:49:07constant
2:49:09times e to these
2:49:12this minus zero
2:49:15over D times h
2:49:18or X largest
2:49:24right
2:49:27thank you
2:49:30okay
2:49:34so now there is a final little step to
2:49:37do so this would one piece of the
2:49:39solution the other one is up here
2:49:45small in effect it is for large enough X
2:49:51and you see that I have some
2:49:52undetermined constant a in here
2:49:58so what is a clever way to match uh
2:50:03so it is a clever way to uh determine
2:50:07the value of the constant well uh what
2:50:09you can do is you you ask that
2:50:12the division is
2:50:15so it's derivative will not be
2:50:17continuous because you have the data
2:50:18fund but the distribution itself
2:50:20is continuous so you have to equate the
2:50:23expression that you have for smaller
2:50:25values of x to the expression that you
2:50:28have for larger values of X when
2:50:30computed exactly at Theta
2:50:37so this
2:50:38is that
2:50:44which is perfect
2:50:47for almost the last step
2:50:59which is to fix my constant a
2:51:03using continuity
2:51:08of B
2:51:17B
2:51:19okay so for X smaller than Theta we add
2:51:23the expression that U perhaps to get
2:51:26more
2:51:27but let me write it so if x is smaller
2:51:30or equal
2:51:33s Theta
2:51:35we have an expression and then I compute
2:51:37this expression exactly as at sometimes
2:51:40it gives me j0 over b0
2:51:451 minus E to the minus b0 over D times
2:51:49Theta
2:51:52and then I equate it to the expression
2:51:55that I get for X largely Theta that is
2:51:58just the exponential
2:52:01a e to the minus
2:52:04P0 over it B times Theta
2:52:09and this allows me to fix
2:52:12the value of a so a will just be equal
2:52:16j0 over P0 e to be
2:52:200 over d
2:52:22Theta minus one
2:52:24by multiplying each side of the equation
2:52:26by e to the BC over B
2:52:31okay and once I have this here the full
2:52:34solution for
2:52:35almost the full solution for my
2:52:39for my stationary space
2:52:43should I rewrited
2:52:48foreign
2:52:56X
2:53:00or X
2:53:02foreign
2:53:04and it was
2:53:0680.
2:53:09X for x
2:53:11larger equals 3 then it discontinues to
2:53:13Theta and a is given here
2:53:19okay so this is
2:53:21almost the solution
2:53:26why do I say almost because now we have
2:53:29to remember that we did some assumptions
2:53:31when we
2:53:32point
2:53:34and we have to check that these
2:53:36assumptions are actually self-consistent
2:53:38with the solution that we found
2:53:42and this is a step which substitutes a
2:53:44little bit of C in the user solution for
2:53:47this additional state which is the
2:53:48normalization
2:53:54so in here we don't we don't want to
2:53:57check the normalization in fully
2:54:00interval but remember that we
2:54:02had this parameters 5
2:54:07by 0
2:54:09which was which we assumed to be fixed
2:54:13and this was the fraction of terms which
2:54:17are active
2:54:19that was defined as the integral from
2:54:22zero to Infinity index
2:54:24of
2:54:27this case stationary solution
2:54:30to stationary of x
2:54:37and now what I have to do is I plug my
2:54:40solution for the stationary I compute
2:54:42this Instagram I'm not going to do this
2:54:45this is just integrated explanation you
2:54:48just have to split into the different
2:54:50regime from zero to P time from C to
2:54:52Infinity
2:54:53and you get out
2:54:55simple constants
2:54:58that is j0
2:55:01Theta
2:55:03divided by
2:55:05b0
2:55:09so this was the last point to see
2:55:13and this
2:55:14is nothing but a self-conception
2:55:23question
2:55:28for
2:55:30five zero
2:55:34why is it just consistent well because
2:55:37you have in here this constant v0
2:55:41but then you have to remember what would
2:55:43be zero so this zero
2:55:45uh here
2:55:48and you have to remember what is j0
2:55:52so remember that J
2:55:54zero was the Flux Of or was equal in the
2:55:58stationary states to the flags of uh I
2:56:01mean Burns for this we both could be
2:56:04equal to one minus by zero
2:56:08so inside this j0 there is a Phi zero
2:56:12which appears and also inside of b0 so
2:56:16if you remember this was B
2:56:18Plus
2:56:20beta Sita
2:56:22itself
2:56:24and therefore this is
2:56:26B plus beta Theta Phi
2:56:301 minus five zero
2:56:34so if you plug
2:56:36these two expressions
2:56:38inside this equation you get
2:56:41an equation for your five zero which you
2:56:43need to solve in order to complete this
2:56:46uh the solution of your mother
2:56:50because spicero is not a parameters
2:56:54it's part of the solution of the model
2:56:55so let me rewrite the equation and then
2:56:58you see that it is simple
2:57:02so Phi 0 is beta times j0 so it is a
2:57:06Theta times pi 1 minus Pi zero and I
2:57:10will divide both numerator and
2:57:12denominator by Theta times Phi so what
2:57:15I'm left with is 1 minus by zero in the
2:57:18numerator
2:57:19and then I have in the denominator
2:57:22B divided by
2:57:24C plus five
2:57:27Plus
2:57:30beta
2:57:33and then I have Theta Phi 1 minus Pi
2:57:36zero but I divided by so theorem left
2:57:38with one minus zero
2:57:45and the reason why I wrote it in this
2:57:47form is just that you recognize you hear
2:57:50something that we defined at the
2:57:53beginning so this was the ratio between
2:57:54time scales that we called said
2:58:00so we can rewrite this as a quadratic
2:58:03equation for sine zeros to this by zero
2:58:06Z Plus beta
2:58:091 minus by zero is equal to 1 minus 5 0.
2:58:15and then we can solve this
2:58:22uh where
2:58:25after
2:58:41so we just saw this second order
2:58:44equation and we have a now decide how to
2:58:47choose the sign in front of the square
2:58:49root
2:58:54in general we get two solutions plus or
2:58:57minus which will be of the following
2:58:59form I have one or two beta
2:59:05Z Plus beta plus one
2:59:09lash line square roots
2:59:13that plus beta plus one where
2:59:18minus 4
2:59:21.
2:59:22okay
2:59:26and here comes
2:59:31here comes the comment that we made at
2:59:34the beginning so when we introduced uh
2:59:36we said so now we have to choose
2:59:39what is the meaningful solution between
2:59:41plus and minus and we can use this
2:59:43equation that we have when introducing
2:59:45that so this idea that
2:59:47when Z goes to zero
2:59:50and always
2:59:52besides the should be small
2:59:54then you should expect that whenever
2:59:57somebody dies it gets immediately
2:59:59reinjected with a time space which is
3:00:01much much faster than the one of that
3:00:04and therefore if you are in this
3:00:06situation what do we expect
3:00:09in this limit for the value of 5
3:00:12well if whoever dies gets immediately
3:00:15rejected then we expect that all of the
3:00:18firms in our model will be in the
3:00:20interval of active first because as soon
3:00:23as they go out I put them back
3:00:25immediately or very fast into the model
3:00:28at x equals Theta and therefore we
3:00:31should expect that in this limit
3:00:33i0 goes to one
3:00:37so this is just to say that using this
3:00:39we can select what is the good solution
3:00:43so we just have to look at the limit
3:00:45instead going to zero for this equation
3:00:47and see if we get 1 with a with either
3:00:51with Plus or with minus
3:00:54and what you find is that the good
3:00:56solution if you do this is actually the
3:00:58one with the mind
3:01:00so the good price zero tools
3:01:02as a minus sign
3:01:04in front
3:01:13well why because when Z is equal to zero
3:01:15here you just have to there is let's
3:01:17tell you this thing let me point it out
3:01:19so when Z is equal to zero you can yes B
3:01:22plus one square minus four B you can
3:01:24rewrite it as
3:01:25Theta minus one square
3:01:29and then you you have to remember that
3:01:33parameters data that we introduced was
3:01:36the ratio of time scales when we can
3:01:39neglect
3:01:40uh data so we we always have this
3:01:42assumption that beta is small
3:01:44and if beta is small then this value is
3:01:47negative so when you take the square
3:01:49root of the square you have an absolute
3:01:50value which Clips one sign and this is
3:01:53why if you do the math you find that
3:01:56minus is a good solution just just keep
3:01:58in mind that this uh
3:02:00what you have inside the square is
3:02:02typically negative if the issue that we
3:02:04have
3:02:06that we are where you should expect if
3:02:07you're going to want
3:02:11okay so with this
3:02:13uh we uh
3:02:15conclude the first part
3:02:19with super late well
3:02:22okay
3:02:23so let me give you an idea so what this
3:02:25shows is that you can find a solution
3:02:29for your stationary State and actually
3:02:31you can argue that you always have a
3:02:34solution to this self-consistent
3:02:36equation uh which always lives in a good
3:02:39regime so in some sense
3:02:41[Music]
3:02:43so you may wonder when does when do I
3:02:45have to throw away this solution well
3:02:46one thing that you could expect which
3:02:48would be a trivial thing is that at a
3:02:51certain point you find values of Phi
3:02:53zero such that you get that one of these
3:02:57parameters flows and these two signals
3:02:59that what you're doing is not good any
3:03:02longer and you have to look for another
3:03:04suit now this is not what happens in
3:03:06this smallness so in this model you
3:03:08always find that Phi 0 is a solution
3:03:12that is admissible
3:03:16but what you have to check is and
3:03:19therefore
3:03:20stationary that we found
3:03:23is an admissible solution but what you
3:03:26have to check is really as I said before
3:03:27the stability so what changes
3:03:35is its stability
3:03:40so let me go very fast forward or
3:03:42through the second exercise where you
3:03:45and give you the idea of how you check
3:03:48actually the stability and determine
3:03:51when it breaks down
3:03:59an idea used to use precisely this
3:04:02picture of preserving a little bit and
3:04:05seeing to get back to your original
3:04:08solution
3:04:09except that now what we have to preserve
3:04:11well we have to preserve a full function
3:04:13which is our solution for uh for the
3:04:16plant equation
3:04:18so how do we preserve a function where
3:04:20we introduce
3:04:22foreign
3:04:32plus some small perturbation in
3:04:34functional space which I call B1
3:04:39that now being a preservation we can
3:04:41assume is no longer stationary but it
3:04:43will depend
3:04:44on time and this is small because I'm
3:04:47putting
3:04:48a factor of a child in front
3:04:52so we make this answers for our equation
3:04:55then of course if we perturb our
3:04:57distribution we are also perturbing all
3:05:00of the parameters which implicitly
3:05:01depend
3:05:02on the distribution so we have to assume
3:05:05that beta
3:05:07goes to some
3:05:09sorry B goes to be zero plus F times P1
3:05:11J goes to j0 Plus
3:05:16a one and what else Phi
3:05:25foreign
3:05:31all of these
3:05:36into our soccer plank equation
3:05:41and we will get the term
3:05:43which does not depend on Epsilon which
3:05:46counts as because
3:05:48precisely so if we select b00 I zero
3:05:52empty stationary yet before
3:05:54the soccer blank equation is satisfied
3:05:56and it is stationary so that term
3:05:57resunction
3:05:59and then we will get a correction which
3:06:01is uh of the order of Epsilon
3:06:05so I will just
3:06:07we've gone for like five minutes don't
3:06:09worry I will just
3:06:12write
3:06:15what is the equation that you get
3:06:19and how do you study it
3:06:21so if you plug into the blank
3:06:23equation you do the math you isolate the
3:06:25term which is a word that Epsilon you
3:06:27get an equation for this
3:06:29correction P1 of accent if
3:06:32that they write in this form so you have
3:06:34a b over DT now I collect all the terms
3:06:38which depend on P1 on the left side
3:06:42you have a diffusion term
3:06:45and then you also have a Drifter
3:06:52all of this applied P1 respective
3:06:57and on the right hand side you will get
3:07:00terms which depend on the correction
3:07:03on our family Bank quantities
3:07:06and if you do this properly you should
3:07:08find this V1 of t
3:07:10times our stationary
3:07:15so this is over the zero and afternoon
3:07:16that this was over there absolute so
3:07:18that's why you get a distribution and
3:07:20then you also get a contribution from
3:07:22the source which depends on this
3:07:24correction file
3:07:29okay
3:07:36and now you can so now you have to solve
3:07:39this equation to get T1
3:07:43is a function of your time dependent
3:07:46quantities
3:07:48and this is not something that we are
3:07:50going to do so you can look at the paper
3:07:52but what is the idea so the idea is that
3:07:54what you have in here
3:07:55is an operator
3:08:01that I call
3:08:03G to the minus one
3:08:06and so you have what you have to do to
3:08:08get P1 is just to invert this operator
3:08:11and this operator so you see it's an
3:08:13operator which depends
3:08:16derivative but you know how to write the
3:08:19inverse so G is
3:08:22doing function for those
3:08:24who have this terminology mind but
3:08:26anyway you can show that this equation
3:08:28can be inverted
3:08:30well formally
3:08:33I call this function
3:08:36f of x and t
3:08:39my P1 will just be my operator G applied
3:08:43to the function f
3:08:46and this will itself give a function
3:08:48which will dependency
3:08:51and you can show so this is
3:08:54similar to what you do when you want to
3:08:55sort of keep the equation which is
3:08:58basically the equation that we are
3:09:00looking at we can show that the inverse
3:09:02so G which is the inverse of this
3:09:05operator
3:09:06we know how to solve for for this we
3:09:08know uh that this is an integral
3:09:11operation with a kernel which is
3:09:14that we can compute so there will be
3:09:18uh kernel that is given in the paper
3:09:21explicitly and I can write this right
3:09:25side as
3:09:27the kernel evaluated myself
3:09:32times my function that's why now
3:09:34integrated in d y
3:09:39and this kernel is
3:09:42is essentially a gaussian
3:09:47is the usual kernel that you get when
3:09:50you look at problems of diffusion
3:09:52so if you want details you can look at
3:09:54the paper
3:10:02but somehow The crucial point is that we
3:10:04can solve for P1 and we have P1 as a
3:10:08function of C like this D1 and C1 and
3:10:11Phi 1.
3:10:12but now B1 and Phi 1 are also
3:10:16somehow unknown they are the perforation
3:10:19that we used in our functional space
3:10:22so once we have P1 we then have to
3:10:24impose
3:10:25some self-consistency
3:10:30okay
3:10:33so you have to impose
3:10:37as consistency
3:10:44so yes P1 as a function of Pi one but
3:10:47then remember what was Phi 1 well Pi one
3:10:49is
3:10:51to order Epsilon is the integral of C1
3:10:55it says the first that consistent
3:10:57equation that you have is that Phi 1 has
3:11:00to be equal to the integral
3:11:02text of your P1 electricity
3:11:09and then you have a self-consistent
3:11:11equation for B1
3:11:15which is related to the derivative of uh
3:11:18of one
3:11:22so B1
3:11:25should be equal to the following three
3:11:33okay so these are the two circumstances
3:11:36equations that we are supposed once you
3:11:37have uh this solution for B1
3:11:42and now our last
3:11:44comments
3:11:46how do you
3:11:48check and try to find a solution to this
3:11:50equation
3:11:57well one thing that you can do is to
3:11:59make an Anzac for the form of this Phi 1
3:12:02and T1 and this is what they do in the
3:12:04paper
3:12:10and you make an answer which
3:12:13Heat
3:12:16to the presence of
3:12:18population but
3:12:21if you understand from the full solution
3:12:23of the model but anyway you make the
3:12:25following answer
3:12:29assume that you can write Phi 1 as some
3:12:32constant High Times
3:12:34some exponential will where Alpha is
3:12:38General complex
3:12:43and B1
3:12:45you can write it as some other constant
3:12:47times the same
3:12:49extension
3:12:51you plug this into the self-consistent
3:12:53equation you do all of the algebra and
3:12:56what you get out of this are
3:12:58the equation then the paper
3:13:06so now we have three parameters to fix
3:13:09five b and Alpha and you see that the
3:13:12equation 10 in the paper that maybe you
3:13:14also looked at
3:13:16can be homework have the following form
3:13:19so you can
3:13:20rewrite the process consistent equation
3:13:22in in the form of a matrix which depends
3:13:26explicitly on Alpha
3:13:29acting on the vector
3:13:31constants
3:13:33being equal to zero
3:13:40now you ask how can I find a solution to
3:13:43this equation which is non-trivial of
3:13:45course a previous Solution that's sine
3:13:46Theta zero but we have no previous
3:13:48solution you need this Matrix
3:13:50it's not The Interpreter which is
3:13:52objective so that you have no zero
3:13:54vectors
3:13:55which belong to the kernel of the Matrix
3:13:59M so such that if I apply M to them I
3:14:02guess zero and so what you have to ask
3:14:04is that the determinant
3:14:07Alpha
3:14:08is equal to zero because this tells you
3:14:11that the kernel of Matrix
3:14:14so the set of vectors where they
3:14:16actually be zero is not just given by
3:14:18the zero sir
3:14:21and in this way you get
3:14:24an equation
3:14:28that you try to solve
3:14:30you have different regimes so here you
3:14:32have played solve this numerically play
3:14:35a little bit remember that Alpha the
3:14:39answers that we are making complex so we
3:14:42will get the equation for the real
3:14:43participation imaginary part
3:14:46and you will find when this is the
3:14:48final point
3:14:50you will find three type of solution so
3:14:53three regimes
3:14:56depending on the parameter you can find
3:14:58three type of solutions
3:15:00for your Alpha
3:15:05so in the first case you find that
3:15:09your real part of alpha
3:15:11is smaller than zero and imaginary parts
3:15:13of alpha is zero
3:15:19and this is precisely
3:15:22the regime
3:15:25where you can claim
3:15:29that your solution is stable because if
3:15:31we go back
3:15:33in here
3:15:35so remember if I want it was the
3:15:36preservation of Phi so we are assuming
3:15:38that Phi is equal to 5 0 plus some
3:15:41perturbation that we are imposing we are
3:15:44assuming that it has this form but then
3:15:46you see that this exponential will be e
3:15:49to the real part of alpha times p and
3:15:51then you have two sine of imaginary
3:15:54parts of alpha times T Plus either sine
3:15:57simply
3:15:59so if the imaginary part is zero
3:16:02of alpha this constant this is equal to
3:16:04one and if the real part is negative you
3:16:07you see that you have an exponential
3:16:08which indicates very fast to zero and
3:16:11therefore in your preserved Dynamic you
3:16:13go back to a value of Phi of T which is
3:16:16precisely the size zero that you have to
3:16:18determined before
3:16:19and so this corresponds to stability of
3:16:21your equation
3:16:23and then you find another regime where
3:16:26your imaginary part is different from
3:16:29zero but the real part of alpha is still
3:16:31uh more than zero so this is still
3:16:34stable
3:16:37but you see that if you have an anterior
3:16:39imaginary part this somehow suggests
3:16:42ventilation so that you converge back
3:16:45let me do a little drawing
3:16:47you preserve your fine you will have
3:16:50some populations and then eventually you
3:16:52converge to the stationary valued by
3:16:54zero
3:16:56and finally as you expect you have a
3:16:59regime of parameter where actually you
3:17:01find that the real part
3:17:03of alpha is larger than zero and the
3:17:06imaginary parts do whatever
3:17:09and this is really so the first value of
3:17:12parameters
3:17:13at which this happens is really what is
3:17:16telling you that you are developing an
3:17:18ability
3:17:24and therefore that you should throw away
3:17:26the solution that we have we alluded in
3:17:28the first exercise because as soon as
3:17:31you preserve a little bit from that you
3:17:33flow somewhere else and to determine how
3:17:36you flow in this regime you then have to
3:17:38solve for the full time dependent soccer
3:17:42plan problem in this generally you don't
3:17:44know how to do so there are in the paper
3:17:46used to marriage uh
3:17:49for simulation of the Dynamics but let's
3:17:52say General this
3:17:54going into this unstable phase and
3:17:56finding the solution is very hard
3:17:58problem
3:18:00that there is no discussed analytically
3:18:03in the paper
3:18:05okay so that's it sorry it is very late
3:18:09and so for uh for the last exercise on
3:18:14the formograph
3:18:15so you have it
3:18:17and the solutions of the today and of
3:18:19course if you want we can discuss it on
3:18:23Wednesday or if the in the question and
3:18:25the answer file and if you want to write
3:18:28questions before Wednesday in there uh
3:18:31feel free to do it and we will address
3:18:32them
3:18:34either there or any discussion
3:18:38questions
3:18:39no questions yeah
3:18:51yes so just to make it clear there is a
3:18:54I will write an email because I think it
3:18:56was not clear so there was a question on
3:18:58the exam the exam will be uh written I
3:19:02think
3:19:03unless she was maternity today today or
3:19:06tomorrow but I think it will be written
3:19:07and it's gonna be a paper to read and to
3:19:11discuss so there are some technical
3:19:13questions and then many other questions
3:19:15about the interpretation uh of the paper
3:19:18so it is uh there is not the choice
3:19:21between let's say a standard exam and a
3:19:23paper exam but it is only the paper and
3:19:25you can look at it the exams from last
3:19:29years which are in the folders who have
3:19:32an idea
3:19:33okay
3:19:35okay so if there are no questions
3:19:40have a nice study week

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