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Complex Systems - Jean-Philippe Bouchaud - Lecture 8: Hawkes; Kirman & Moran models (Valentina Ros) — Transcript

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0:01this conference will now be recorded
0:04great
0:06okay so good morning everybody
0:12so today we are gonna have a
0:15longer session
0:17and what I'd like to do is to
0:19uh discuss maybe uh first the end of the
0:24today five or six that we started last
0:26time
0:27so this was about uh melee hoax
0:30processes but as we will see we will
0:32also connect to some of the things that
0:34were discussed in the last lecture about
0:36meta stability and we are going to do
0:38this looking at a very simple model of
0:41financial transactions and exchanges and
0:45that's part one then I think we will
0:47make a break and then we will go to uh
0:50what was supposed to be the S7
0:52and that is about
0:54some simple model of opinion Dynamics if
0:58you want so this is a close again to
1:01what has been discussed in the last
1:03lecture and this is called in the
1:05literature either the kiermann model or
1:07the Moran model so this is also
1:09connected to uh to the homework
1:12number seven
1:15and uh so we will try to do this in
1:19detail and then if there is time I will
1:21give you just the idea of the third part
1:24of the seven slash eight which is uh
1:28going back a little bit to statistics
1:30and discussing using idea of brown and
1:33motion to discuss tests about or what
1:36are called goodness of fit test so how
1:39do you check that assumptions that you
1:42make on on a given set of data and on
1:45their underlying distribution are
1:46actually good or or whether you have to
1:49discard them so this is in case
1:51we managed to arrive them there but
1:54let's say the first two parts are the
1:57most important ones uh today so let's
2:00start with uh with the first one
2:03that is in the first uh today uh after
2:09the exercise that we discussed last time
2:12and I just wanted to start by recalling
2:14something that you saw in the lecture so
2:17this was
2:21from the last lecture
2:24and the idea is to recall a little bit
2:26this concept of meta stability
2:35and activated Dynamics because this is
2:38what we are going to discuss in more
2:40detail later on
2:42so in in the last lecture you were
2:44discussing things related to these
2:47models of choice where you have many
2:50agents that need to choose between
2:53different options and at a certain point
2:56you introduced
2:59some Dynamics so s was one of the
3:03possible binary choices of the agent
3:05label by I and you were writing
3:08something like this
3:10so the choice of the agent I at time T
3:13was related to the one of
3:15all of the other agents by uh by the
3:19following Expressions so this was the
3:20sign of some external field which I
3:23think was called Capital H then there
3:26was some uh let's see
3:28term which which was specific and
3:31typically randomly distributed and
3:33specific for agent I and then you add an
3:37interaction term
3:38that I write already in the meme field
3:41Expressions that was jij that I here
3:45write as j0 Over N so the normalization
3:48comes from the fact that you have all to
3:51all interactions between the agents so
3:53you have to divide by a factor of n in
3:55order for this to be of the same order
3:57of magnitude of your external field and
4:00then you have
4:01the choice of the agent J at the
4:04previous time T minus 1.
4:08and in order to discuss this model at a
4:11certain point uh we we or you made the
4:14assumption that this uh different random
4:17Fields were set equal to zero and then
4:19the model can be uh discussed in terms
4:23of a global meanful like variable which
4:27you can think of as the magnetization if
4:30you want in in the language of spins
4:32which was just the sum overall of your
4:35agents
4:36of the corresponding variable at time t
4:40and this quantity satisfies a dynamical
4:43equation that is essentially launch
4:44events so there was
4:47DM so this is a total derivative
4:50over DT was of the following form so the
4:54first term
4:56it was convenient to write it
4:59as a derivative of some potential which
5:02depend on the particular value of
5:04magnetization that you have at time T
5:06and then there was a noise term which
5:09contains some coefficients so the
5:11coefficient was set to one in the
5:13lecture I think but let me call it
5:15capital sigma
5:17divided by n and then there is the usual
5:21White Noise term
5:23and so as you see your Dynamics is given
5:26by as usual you have
5:29some noisy fluctuations
5:31and then the first term is what people
5:34usually call some gradient descent
5:38theorem
5:45so what you are saying is that you have
5:47a Dynamics in a given potential the
5:49Dynamics wants to push you downhill uh
5:53at the points which are or towards the
5:56points that are essentially Minima for
5:59for this potential or in general
6:00stationary points where the gradient of
6:02the potential is equal to zero and if
6:05you had no noise once you reach a
6:07stationary Point as you see the Dynamics
6:09this is equal to zero so your Dynamics
6:11would become stationary and you just
6:13staying there forever but if you have a
6:16little bit of noise then you can have a
6:18more richer Dynamics so you can use the
6:21noise to escape from these stationary
6:24points and this is what we are going to
6:26see today
6:27so in particular the shape
6:29of the potential in the lecture
6:33was or maybe I can draw it here
6:42so if you remember
6:50your potential was looking like the
6:54usual potential in using like problems
6:56so this was V beta
6:58of M when your field h capital H in here
7:03is different from zero
7:05it had a shape
7:08that was more or less like this with two
7:12Minima one of which is the global
7:14minimum in here and the other one is
7:16instead a local minimum which is
7:19sub-optimal with respect to uh to the
7:21global one
7:23and now what we're going to discuss
7:25today is how to interpret this local
7:28minimum so in particular
7:30in our language this will be
7:33a metastable state
7:38so what metastable state means is
7:41roughly the following so if you think
7:43about these Dynamics and you imagine
7:44that you start
7:46somewhere within the Basin of Attraction
7:49of this minimum so let's say that my
7:52Dynamics starts now maybe blue is not a
7:54good color
7:56starts somewhere in here than what you
7:58expect
7:59your system to do under these Dynamics
8:01if the noise is small
8:04then the dominant term will be this
8:06gradient term so this gradient term
8:08pushes you towards this point in here
8:11where indeed the right hand side is
8:14equal to zero so you would expect that
8:15you have some noisy dynamics that
8:17somehow goes down into this landscape
8:20and then it relaxes into this metastable
8:23state and maybe it performs a little bit
8:25of fluctuations within the Basin of this
8:28local minimum because you have the noise
8:32and if you're looking at your system in
8:35the limit in which n is exactly equal to
8:38Infinity so in general we think it n as
8:41being a large parameter but in the limit
8:44in which n is strictly Infinity which is
8:46the mean limit then that's it so you
8:49would converge to into this minimum and
8:52you will stay there essentially forever
8:54but if n is finite if you remember the
8:57comments that were made in the last
8:59lecture something more trivial and more
9:01or less trivial actually can happen
9:04namely you will have a certain instance
9:07of times which are very rare where your
9:11system
9:12which is confined in this region uses
9:15the noise to escape from this local
9:17minimum and to eventually converge
9:20to the global minimum of your potential
9:22in here
9:23but doing this is very hard with this
9:26Dynamics if your noise is small because
9:29as you see so if your noise is small
9:31then typically this is the term that
9:33wins and this term as I said pushes you
9:37down so in order to overcome this
9:39barrier so this is what was called B in
9:43the lecture is the difference in
9:45potential between let's say one of the
9:47Minima and the local maximum so to
9:50overcome this barrier you need very
9:52large fluctuations of your noise that
9:56are typically very rare So Rare means
9:59that you have to wait a lot of time to
10:01see this process occur and you can
10:04estimate what is the time so this was
10:05given uh in the lecture as well
10:09so let me call Tau the typical time
10:12scales for this type of jumps
10:16then
10:18what explicit calculations tell you
10:24is that this Tau will depend
10:27on the parameter that you have
10:30in the following way so it will be
10:32exponentially large in the barrier so
10:36what I call B here or actually to be
10:40more
10:41intuitive let me call it DV so this is
10:44the difference in potential between the
10:47local minimum and the local maximum and
10:50then in front you have this factor of n
10:52which is nothing but so it would be a
10:56factor of 1 over a sigma Square where
10:58Sigma is the variance of your noise
11:01and our unusual applications whenever
11:03you look at launch event what people do
11:06is to choose the variance of the noise
11:08to be proportional to temperature so
11:11noise is a measure of thermal
11:13fluctuations so this square root is in
11:17typical in the usual launch events
11:19formalism is of the order of square root
11:22of temperature so you see that this
11:24expression here you can write it as e to
11:27the beta
11:29times DV
11:31and this is uh what people call in the
11:35literature the the so-called Arena's
11:41Arrhenius Behavior so it tells you that
11:44the time that you need to wait in order
11:46to jump this barrier grows exponentially
11:49in the barrier itself but also in the
11:52temperature or in the inverse
11:54temperature or if you want in the
11:56inverse
11:57of the variance of your noise and in
12:00here because you're looking at models
12:02that are main field or fully connected
12:05you have this factor of n which makes so
12:07when n is large this makes your jump
12:09processes very very rare meaning that in
12:13the limit when n goes to Infinity these
12:15jumps will never occur because Tau goes
12:17to Infinity but if you have a finite
12:19size system even if fully connected you
12:23do have some finite time scales over
12:25which you will see this jump occurring
12:27and these are called activated processes
12:32or activated jumps
12:36I hope that you can see
12:44that are something that is studied even
12:48currently in situation a little bit more
12:50complicated than these for example in
12:52systems that are Glacier which have very
12:55many metastable states
12:57so what we are going to see today with
12:59the exercise number one is something
13:01quite similar so we will also estimate
13:04times for jumps over barriers of a
13:08potential but in this time so this time
13:11this let's say JUMP time will be also
13:14large in the model that we are going to
13:16see but it will be large in a parameter
13:18that is not the number of agent capital
13:21N but which is another parameter that we
13:25will call Epsilon or one over Epsilon
13:28which is related to non-linearities in
13:31your model so the setting is a little
13:33bit different but this was just to
13:35somehow remind you that in essence the
13:39type of processes that we want to
13:40describe are exactly of the form of what
13:43you saw in the previous lecture
13:45and let's keep in mind this expression
13:48because
13:50at the end of the exercise we will
13:51compare with uh with uh with the case of
13:56easing-like potentials
13:59okay so having say this let's go to the
14:02exercise and let me take the text
14:07and check where can I write
14:11[Music]
14:12um
14:14here is fine no here it's probably fine
14:17okay
14:18so this is the exercise number two I
14:21think uh of the today
14:23and it is a very simple model of uh
14:27Financial exchange or if you want
14:30transactions
14:32and you may look at the reference so
14:35this exercise is taken from this paper
14:44that you find in the archive
14:501 9 12.
14:54zero zero three five nine so there are
14:57more details in here that you can you
15:00can easily look at
15:02so let's introduce the model dance
15:07okay so as I say this
15:10is a model for
15:15Financial transactions
15:23so what does it mean well what it means
15:25is that as usual you have many agents
15:29or some agents
15:33and these agents want to exchange a
15:35given Financial products so they want to
15:38buy
15:40or to sell
15:42some financial asset
15:49it doesn't matter what this is so for us
15:51it will be something that they want to
15:53either buy or uh or or sell so this can
15:57be stocks
15:58bonds
16:02whatever you can think of
16:07and how do they do this well they do not
16:10exchange directly by the way they trade
16:13is by placing uh what people call an
16:16order so they
16:20place
16:22orders
16:26and an order is basically an an
16:29intention to buy or to sell something so
16:31an order is made of three things in
16:34general
16:35so one is what I will call the
16:38directions so you have to say
16:41whether you want to buy or whether you
16:43want to sell the given Financial
16:45products
16:49okay then you have to specify what is
16:52the size of your order so what is the
16:56amount of product that you want to bound
16:58to buy or sell
17:02and then you want to specify the price
17:05so
17:07in general
17:10what you say is I want to buy I don't
17:12know six units of my given financial
17:16asset at a given fixed price so you you
17:19place this order into the market and all
17:22of these orders are collected into what
17:24people call an order book
17:29okay
17:31which is just a collection of the
17:33intentions of uh of all of the agents uh
17:37as we specified in here so let me give
17:40just a sketch so this is you don't need
17:42to write this down it's just to uh to
17:45give an example of how we may think
17:46about this uh order book
17:54so let me denote the agents
17:58with I don't know symbols so there will
18:00be an agent that is a square an agent
18:03which is a circle an agent which is a
18:06triangle and so on
18:08and then let me take an axis which is
18:11the price
18:15and I divide this axis into slots
18:20okay
18:22and for each of these slots I put a
18:25symbol whenever the agent wants to buy
18:28or to sell the amount of quantity that
18:31we are thinking about at the given price
18:34which corresponds to the slot so suppose
18:36that the agent denoted with the square
18:39wants to buy something wants to buy
18:42three amounts
18:44of this asset at this price in here so I
18:48will
18:50put it here with the three squares then
18:53there is another agent that wants to buy
18:59two amounts of this thing at a lower
19:01price and then so so as you can imagine
19:05when the price is lower these are the
19:07people that want to buy
19:10they typically want to buy it at a lower
19:12price and then you will have
19:14at a higher price the people that want
19:17to sell so let's say that the guy which
19:20is the triangle wants to sell
19:24three units
19:25at this price and then there will be
19:27another one I don't know stars that
19:30wants to sell in this at this price and
19:32so on and so forth
19:34and as you see there might be a gap
19:36between
19:37best price that we
19:40've got that we saw the higher price at
19:42which people are available to buy and
19:44the best sell price so let me denote to
19:47this
19:47uh price here at a given instant of time
19:51with BT and then with the know this with
19:5480. so there might be a gap between
19:58these two quantity and this Gap is what
20:00is called uh the spread
20:02so the spread
20:06is
20:07s of T at any instant of time and it is
20:10just the difference a T minus v t
20:18okay
20:20so just one last thing about the
20:24terminology just to understand the model
20:25so this is uh discussed again in here
20:29but uh so what I'm what I've been
20:32talking about so far uh are actually
20:34called limit orders so I should add
20:39a limiting here so a limit order is when
20:42you state indeed what is your intention
20:45so you say as I said in here that you
20:47want to bound to buy a given amount at a
20:50fixed price
20:51but now there might be people that just
20:53want to bow to buy or to sell a given
20:56amount at whatever price the market
20:58chooses uh for them so there will be
21:01people that that say I really want to
21:04sell as fast as possible three units of
21:08this uh of this financial asset at the
21:13best price uh which I can at which I
21:16find people to buy and these type of
21:20orders which are executed immediately
21:22are called
21:24uh in in this financial language Market
21:30orders
21:37so this means when you place a market
21:39order what you're saying is that you
21:40want to buy or sell
21:42a given amount
21:46X
21:48immediately
21:55at any price
21:58foreign
22:02to give you the terminology of the more
22:05General models of dynamics of order
22:08books
22:09and now what we are going to do to do
22:11the exercise we are going to simplify uh
22:14the model uh quite a lot so we want to
22:18simplify it in such a way that the only
22:20thing that we care about to describe the
22:24Dynamics of the system is the spread so
22:27it is just the gap between the best sell
22:30price and the best price at which people
22:32want to buy
22:33this this particular asset so how do we
22:37simplify it so can I use that part of
22:40the Blackboard yes
22:44so we will make the following
22:46assumptions
22:52which Define our model
22:55so the first one is that
22:58the size of each order we fix it to be
23:01equal to one so in this drawing this
23:04means that each of these different agent
23:07can only buy or sell a fixed amount of
23:10of this quantity so we will have
23:13to adjust the drawing in such a way that
23:16you have only one
23:17symbol for each agent
23:19okay so please stop me if if this is not
23:23clear
23:26in the meantime I will write it so size
23:29is fixed to one
23:34then we make a second assumption that is
23:37that once one is locked in this price
23:42axis is filled and you are a new agent
23:44that wants to place another order on top
23:47of what it is already contained in the
23:49book
23:50you can only place it in front so at a
23:54higher price if you want to buy or at a
23:57lower price if you want to sell so in
23:59front of those price slots which are
24:02already occupied
24:05so
24:11once the slot is filled
24:16Place orders
24:19at only
24:24in the let's say in the
24:27let me run it in higher
24:32or lower slot
24:37okay so what this means is that the
24:40people that come first will occupy so
24:42now I ran out of symbols but let me
24:45repeat a little bit the symbols but the
24:47idea is that the people who come first
24:49will occupy already all of the slots
24:52which are
24:54lower or higher so this will be all
24:57occupied
24:59and therefore you get a model in which
25:01you essentially have no holes so you can
25:03think of this in physics language like
25:06uh like fermions so you can put only one
25:09agent at each slot they cannot pile up
25:13at the same slot and if you want to add
25:15something you have to add something
25:17either in here or in here
25:20okay
25:22and then the third assumption has to do
25:24with this Market orders that I just
25:27introduced
25:28so a market order so whenever somebody
25:32places a market order this gets executed
25:35immediately so he says or she says I
25:38want to buy at this price at this price
25:40at this price slot there is a guy that
25:43wants to sell so whenever this happens
25:45this guy sells the amount and therefore
25:47this order disappears
25:51so this means
25:52that
25:54orders
25:57are executed
26:00so you actually buy or sell
26:03only at the boundary
26:13so if you have a situation like this
26:16the idea is that you cannot buy at this
26:18price whenever you really buy and
26:21execute the order you do it at the best
26:23possible sell price
26:25so why are we introducing this uh these
26:28assumptions well because once you have
26:30this as you can easily realize the only
26:33thing that can happen so you have such a
26:35situation at a given time t
26:38you assume that now there is somebody
26:40who wants to place an order and the only
26:42thing that he or she can do is either to
26:45place an order inside the so-called
26:48spread or to cancel one of the orders
26:52which are at the boundary of the spread
26:54which means that all of the Dynamics of
26:58this type of order book is encoded into
27:00the evolution of the spread of this
27:02difference
27:04okay so I hope this was more or less
27:07clear and now we are going to write
27:09formulas so maybe it will become even
27:13more clear but if you have questions
27:14don't hesitate to stop me
27:18foreign
27:20so I think this gives a little bit of
27:22context but in essence the only thing
27:24that we have to keep in mind is that we
27:26have a model of order books where there
27:28are no holes in this price axis and
27:31everything is encoded in the Dynamics of
27:34this quantity in here
27:36okay so now that we have this assumption
27:38let's write down the model
27:41so writing on the model means that now
27:44we have to specify what is the frequency
27:47at which
27:48uh we or the agents
27:52actually Place their orders in the order
27:56books
27:59and this is where
28:01we introduce these hoax processes that
28:03we have discussed
28:07in the previous exercise
28:24okay first of all a general question on
28:26the model so we say that everything
28:28depends or is encoded in the Dynamics of
28:31the spread
28:33so now I basically saved it already in
28:35words but what are the events that
28:38decrease the spread and what are the
28:40events that increase the spread
28:43uh in the smaller
28:46and this is question one
28:49so anybody has a guess
28:58okay so we basically have just two type
29:01of events so if you execute
29:07or cancel you can even cancel
29:10an order
29:14then does the spread increase or does it
29:17decrease
29:24any guess
29:29okay so if you go back to this picture
29:31executing an order means
29:33that now I come into the order book and
29:35I say that I want to buy to buy one unit
29:37at whatever is the available price so I
29:40will buy
29:41one unit of my product that's this
29:43particular price which means that this
29:45order will be executed and if this order
29:47is executed you are enlarging as you see
29:50your spread
29:56okay
29:59whereas if you place
30:02what I call the both a limit order
30:10then as I said you have to place it in
30:13front
30:13of those which are already in there so
30:16if I say that I want to buy I will have
30:18to place myself in this
30:20slot in here and therefore I will
30:22decrease
30:24the spread
30:34okay
30:37so we have only these two type of
30:39possible events and now what we have to
30:42say is what is the frequency at which
30:45this type of events or these type of
30:48events occur
30:50and this as I said is modeled via these
30:53hoax processes so
30:55there will be a rate for the events
30:58which increase uh the spread and I will
31:00call it
31:02Lambda Plus
31:06and this will be given by
31:08a hoax process so the hoax process as as
31:13you remember
31:14gives you the intensity in terms of a
31:17self-excited kernel so there is first a
31:19term which is the so-called background
31:21intensity so this we call this mu before
31:25but now let me stick to the notation of
31:26the paper so this is a constant Lambda 0
31:29Plus
31:30and then you have the term which
31:32contains a
31:34kernel
31:36that I will write in general like this
31:39so this is a quantity which depends on
31:42time so we have Alpha
31:45so now here we go again back to this
31:47issue of notations so
31:51in the lecture Alpha was called beta and
31:53we are going to call beta what was
31:54called Alpha so forgive me for that but
31:58I think this is good because if you go
32:00back to the paper at least you have the
32:02same notation
32:03so we have this term in here where X of
32:07T well I will specify it later so this
32:10would be the linear hoax process that we
32:13have discussed already and then what we
32:14do
32:15is to add some small non-linearity which
32:19is important in here so we add a term
32:21which is proportional to Epsilon and
32:24which goes like
32:25X of t to the power of 2
32:28where what is X of T well X of T is the
32:32usual self-excited exciting kernel so
32:36this will be the integral
32:37from 0 to T
32:40in the Tau
32:42of a function Phi
32:45of T minus Tau so in the previous
32:48exercise and also later on we are going
32:50to choose this to be an exponential
32:54and then you have so let me call the S
32:58the differential Associated to this
33:01process
33:03so what d s of Tau tells you is
33:06what is the number of events that occur
33:10in a given small interval D Tau
33:13of time
33:15and I put a plus in here and what I mean
33:18is that this
33:21yes plus
33:23of tau is
33:26the maximum between the s
33:30of Tau and zero
33:32so what do I mean by this well what I
33:35mean is that the self-exciting part of
33:38the kernel is only or gets on the
33:41contribution from those events that
33:44occurs at previous times that increased
33:47the spread so if you have somebody that
33:50wants to execute an order and therefore
33:52increase this difference in here
33:56whenever this effect occur this will
33:59influence the total rate of your plus
34:04spread increasing events but if you have
34:06an event that decreases the spread so if
34:08somebody places a limit order then this
34:10has no effect into this kernel
34:14okay so this is for Lambda plus and then
34:17what about uh the others well the others
34:20in the model
34:21so the rate for uh limit orders is just
34:25taken to be a constant
34:28so it's like a it's a poisson process if
34:30you
34:31uh
34:35remember from the lecture
34:38and from
34:40homework 5.
34:44okay so this is the model in full
34:48detail
34:50the model of the paper
34:57now what are we interested in well we
35:00want to understand
35:01whether this models can gives give rise
35:05to crisis
35:07and how can we understand how this
35:10Crisis occur
35:12so that's point
35:14so this was point two of the exercise
35:19and point threes
35:26is how do we describe
35:30a crisis
35:34in this model
35:38well a crisis in this setting means that
35:42you are in a situation in which
35:44dynamically you see that your spread
35:46starts increasing and it it increases
35:49without bounds and if the spread
35:52increases
35:53this means that this difference becomes
35:55larger and larger which means that
35:56essentially you have no longer exchanges
35:59in your economy in the sense that the
36:01people that want to buy want to buy at a
36:04price which is much lower than those uh
36:06that want to sell and they people keep
36:09canceling their order or executing their
36:12order and therefore there are no longer
36:14exchanges and and this is how you uh
36:17interpret that Uprising is occurring so
36:20more formally what this means is that X
36:22of t
36:24becomes
36:29infinite
36:31infinite time
36:42which means basically that you start
36:44having an explosion of the events that
36:46are increasing the spread
36:49okay now in the usual linear case so let
36:54me just add this comment when Epsilon is
36:57equal to zero we know one way
36:59to generate a crisis
37:01so let's forget about Lambda minus so if
37:04we have just a hoax process we know what
37:07is the limit in which the number of
37:08events is exploding
37:11and this is
37:13the case in which if you remember
37:16this parameter Alpha was going to one
37:25so the problem is well defined whenever
37:27Alpha is smaller than one if Phi is
37:30normalized to one
37:31and as you increase Alpha and you get
37:34closer and closer to one the process
37:35becomes unbounded and this is encoded in
37:38the fact that X of T explodes so this is
37:42something that of course already in the
37:44linear case and now in here we are going
37:46to see another way in which a crisis can
37:48occur which is instead due to the
37:50presence of this non-linearity and which
37:53is related to this idea of metastability
37:58okay so let's do this so this was a
38:02little bit an introduction to the model
38:04and how to actually uh do the proper
38:08calculation
38:09let's simplify it further
38:13so what we do from now on is forget
38:15about the events that decrease the
38:19spread so I will put this Lambda 0 minus
38:23exactly equal to zero so after all we
38:26are interested into this scenario of
38:27Crisis so we only want to care about the
38:30events which increase the spread so I
38:33just simplify the Dynamics I set this to
38:35zero and then I will denote the Lambda
38:38plus simply by Lambda so I forget about
38:42so the Lambda t plus
38:45let me simply call it Lambda t
38:47so in the end we are going to focus on
38:50this non-linear hoax process and
38:54see what's going on
38:58yes
39:12yes
39:16so the question is that in here I should
39:19add so the idea is that you can
39:22only place orders if you have a spread
39:27which is non-zero
39:29so if in this drawing
39:32you are in a situation in which these
39:34two somehow collapse then there is no
39:36space to place any further order in
39:38there
39:44yes so you're asking why I don't put one
39:47if we yes okay so that depends on
39:50whether
39:52so what since they cannot overlap so I
39:55execute
39:56the order if I have a guy in here that
39:58wants to buy and the guy in here wants
39:59to sell them this they match I consider
40:02it in this way so the question for those
40:04who uh didn't listen
40:07is the following so in the text indeed I
40:09forgot that here you should put an
40:10indicator function
40:12that asks you that at the given time
40:15the spread is larger or equal to two
40:18because if the spread is equal to 1 so
40:21this means in our drawing
40:25that these are the people that want to
40:27buy and I have a guy in here these are
40:29the people that want to sell
40:31and I have a guy in here so yes okay
40:37so in here in principle
40:40yeah this is a little bit of a choice so
40:42in principle you could place an order in
40:44here
40:45and then execute this and this will give
40:48you immediately
40:49an event which is opening so we want to
40:52forget about this so we just ask
40:55that you have at least two slots so that
40:58if the guy puts an event in here you're
41:01really decreasing the spread and you're
41:03not executing directly
41:05so there is this a little caveat in here
41:09thanks for
41:19yes exactly so so the question is
41:22why do we include
41:25this self-exciting Dynamics in here well
41:28I think that one way to think about this
41:30is that
41:32uh if you have some stocks or bonds and
41:36you start so there is some sort of uh
41:39they say Dynamics which depends on what
41:41the other people are doing which is
41:43encoded in here so the idea would be
41:45that if you see that many people uh are
41:48selling the stocks that they have
41:51because there is no longer Trust on
41:54on the company or on whatever is
41:56associated with the stocks then you are
41:58also pushed to sell and this is why uh
42:00or to execute very fast your orders and
42:03this is why you self-excite these type
42:06of processes
42:08okay
42:10people online please also stop me
42:13if something is not clear
42:16okay
42:19okay so this was all basically
42:20introduction to the model and now let's
42:22go to the actual mass
42:25well it's not enough
42:28and this is exercise three I think
42:56foreign
43:08so now we have let me rewrite the
43:11process without the Plus
43:13we have this nonlinear box process
43:17D plus Epsilon XF Square
43:22X of t
43:27is my kernel
43:30the answer file so that's the
43:33differential of the process and as usual
43:36we are going to make the
43:38Simple Choice for our kernel which is
43:40the exponential so we choose V of T to
43:43be
43:44beta e to the minus
43:47beta t as we did in the linear case
43:50in the exercise of last time
43:53okay so the first point
44:00of these exercises to write down
44:02an equation for our
44:05dynamical variable X of t
44:08and the starting point is as follows so
44:11we are going to assume so this is
44:14something that you can do rigorously but
44:16I'm not going to do it uh in here but
44:19what we are going to assume is that beta
44:21which controls the decay of your kernel
44:24is small
44:28so whenever you say small you have to
44:30specify with respect to what so in our
44:32case it would be for instance small with
44:35respect to
44:37lambda zero
44:39okay which in words means that your
44:43kernel is slowly varying in time so if
44:47if beta is small at least for for the
44:50smaller times or for a window of time
44:52which becomes larger this will be equal
44:55to a constant or roughly equal to a
44:57constant beta and whenever you are in
44:59this situation what you can show is that
45:01you can simplify the Dynamics of your
45:05process so let me write it you can
45:08assume that the Dynamics of your process
45:10is given by adrift that is simply given
45:14by lambdarti and a variance or a noise
45:17which has a strength which is of the
45:19order of square root of Lambda T so I
45:21will write it and then I will comment
45:24so the assumption is that
45:27you can use that
45:29your Dynamics
45:32vs of t
45:35looks like this so you have a Lambda t
45:38DT
45:40and then you have a nice term
45:43foreign
45:50T I always put it as a subscript in here
45:53but
45:54it's just a notation so ETA of T DT
45:56where ETA is the usual white noise
46:00so what this means so if you if you
46:02imagine that Lambda is constant then you
46:04will have a poisson process and for the
46:06poisson process you know that this would
46:09be true so this is telling you that the
46:11average of your process is Lambda and
46:14the variance of your process is Lambda
46:16again these are precisely properties of
46:19your poisson process so this is like a
46:22poisson
46:24it has the same structure
46:29of a poisson process
46:31but now you have still this dependence
46:34on time in your uh in your rate Lambda
46:37of t
46:38and this is something that as I said you
46:41can argue more rigorously using the fact
46:44that beta is sufficiently small so that
46:46your kernel varies does not very very uh
46:50very fast
46:52okay so once we have this then we can
46:54try to compute an equation for
46:58the changes in time of our variable X of
47:01T so remember that to characterize
47:03crisis we want to check whether X can go
47:05to Infinity so let's look at what is the
47:08Dynamics of X of t
47:11so let me write the differential of X of
47:14T So to compute the differential I have
47:15to take the derivative of this object
47:18with respect to time
47:20so time appears twice it appears as an
47:24extreme
47:25of the integration and it appears inside
47:27of the kernel so if I first take the
47:30derivative with respect to the extremum
47:33this means that I have to compute
47:35whatever is inside precisely at the
47:39value Tau equal to T
47:41and if I do this my kernel simplifies
47:44because the exponential of Tau minus tau
47:47is is equal to one so this first term
47:50will give me a contribution which is
47:52beta
47:53times
47:54BS of t
47:59and then I have a second contribution
48:00which comes from deriving inside the the
48:04Integra so I have to take the derivative
48:06of the kernel and this will bring me
48:07down a factor of minus beta
48:10so I will have
48:13from here minus beta times
48:16again the integral
48:19from 0 to T
48:21of
48:22uh now I write it
48:26so okay
48:28e to the minus beta
48:31T minus Tau d s of Tau
48:33okay
48:39and if I
48:41look at my definition of X of T then
48:43what I recognize is that this is
48:45precisely
48:46wait I forgot a DT because now I'm
48:49looking
48:50at the differential
48:53so I have a DT in here
48:56and this closes the parenthesis
48:59so again this is the derivative with
49:00respect to the extremum and this is the
49:02derivative inside the integration and
49:04what is this quantity in here well this
49:07is just
49:08my original X of tip
49:12foreign
49:15because of how we defined it
49:19okay
49:20and in here I have this the S but yes I
49:24just told you
49:27what is the form of the differential for
49:29the S so what I can do is to plug this
49:32expression inside here and use the
49:35explicit expression for Lambda which
49:37appears here and here
49:39so
49:43let's do it in here
49:45and I hope that you can see
49:48yes
49:49where does the two less Expressions come
49:51from
49:52this one
49:55yes and the one above
49:57okay so this one is something that we
50:00are not proving
50:02it is let's say an assumption and if you
50:05really want to prove this it takes a
50:07little bit of work but what you have to
50:08use is that you're you're assuming that
50:11the parameter beta which controls how
50:14fast your kernel is decaying you're
50:16assuming that this is small so that the
50:19kernel decays uh slower and under this
50:23assumption you can argue that the
50:26equation for your process so this is
50:29what gives you the number of events that
50:31you expect to occur in a given interval
50:33DP
50:34this can be written in the following way
50:37so it has a drift term which is
50:39proportional to your excitation rate
50:42which is Lambda plus and then it has
50:45some noise which is proportional to uh
50:47to the square root of this excitation
50:49rate and the way you can motivate this
50:51is having in mind a person process so
50:54the person process would be the case in
50:56which Alpha is equal to zero and Epsilon
50:58is equal to zero
50:59and in that case you know that the
51:01number of events in a small interval of
51:03time DT is poisson distributed so being
51:07poisson distributed means that your
51:09average is equal to Lambda 0 and your
51:13variance so the sigma square is again
51:16equal to Lambda and this is what is
51:19encoded in here so in our case we do not
51:21have really a person process but we are
51:23somehow assuming that this Lambda of T
51:27changes very slowly so we can kind of
51:29think at it as a constant and this is
51:31why eventually you get out this
51:34expression in here but this way we we
51:36are not proving this let's say this is
51:38uh what we are assuming and once we
51:42assume this then we can write down an
51:45equation for the X
51:47so where does this come from now so you
51:50I want to write a differential for this
51:52quantity X of T is given in here so what
51:55I have to do so I could I could write
51:58directly the derivative
52:01over time and now I'm writing it in the
52:03differential form but you see that if I
52:05take the derivative over time of this
52:06expression
52:08I have that the time appears both in
52:11here and appears inside the kernel so
52:13when I derive I have two contributions
52:16so the first contribution comes from
52:17evaluated the integrand the into the the
52:20quantity which is inside the integral at
52:22the time t
52:24and this is this quantity in here right
52:26because if you evaluate the kernel at
52:29the time Tau equal to T then this is
52:33like evaluating Phi at 0 so you get a
52:36factor of beta and then this DS of time
52:39of Tau becomes the S of T okay
52:43and then the second contribution to the
52:45derivative I keep the integral as it is
52:47and I take the derivative of the kernel
52:50itself with respect to T
52:53and this is the expression that you see
52:55in here
52:58so if I derive this with respect to P I
53:01get a factor of minus beta that I am
53:03collecting outside with a minus sign and
53:06then the rest I recognize that it is
53:08just X of T is it here
53:11yes
53:13okay great
53:16okay so now
53:18let's try to write this in a way which
53:21is cleaner so I have this beta
53:26then I plug the expression for the uh
53:28the S of T and I plug the expression for
53:31Lambda itself so let me try to collect
53:34everything so I have a Lambda 0
53:38DT
53:41let's do this I have a Lambda 0 plus
53:43alpha x t
53:46DT
53:50which comes from here then I have the
53:52noise so the the ah sorry I forgot the
53:54Epsilon
53:57plus Epsilon
53:59XP Square Times VT the noise I will
54:03write it as a last contribution let me
54:06write this so then I have minus beta
54:10sorry the beta minus X of t
54:13times DT
54:15and then I have the noise which is
54:18encoded in here which was square root of
54:20Lambda so I will write it as plus square
54:24root of Lambda t
54:26e times T DT
54:31okay
54:35and now that I have this so now you see
54:37that this is uh
54:39now a closed equation for x
54:42so let me try to rewrite it in the way
54:45we wrote the language of an equation
54:48before
54:50so
54:56let me try to write it
54:58now I divide by the T if you want so I I
55:01run I want DX
55:04T over DT
55:09so let's
55:11collect everything so I have a Lambda 0
55:14beta
55:17foreign
55:22term
55:24so there is a factor of beta always in
55:27front and the linear term is Alpha minus
55:30one so this is minus
55:331 minus Alpha times beta X of t
55:37then I have the non-linearity beta
55:39Epsilon X of T Square
55:43and then I add the noise
55:46plus beta square root of Lambda t e
55:50times t
55:51okay
55:54and now what I want to do
55:56as before
55:58is to write this quantity in here
56:02as the derivative of sample tension so
56:05as minus
56:07the derivative of some potential that I
56:09call V
56:11which depends on x
56:13derivative with respect to X evaluated
56:17to x equal to my dynamical variable X of
56:21t
56:22so this will be the gradient descent
56:25term and then we will have a nice term
56:26as it was done in the case of the
56:29magnetization in the lecture
56:32so what is the potential well what I
56:34have to do to recover
56:36the potential is so this is minus its
56:39derivative so I have to integrate this
56:42expression with respect to X to X
56:45all right so I will have
56:47well a minus sign in front
56:52integrated with respect to X will give
56:54me Lambda 0 beta X
56:58then I have something that is linear so
57:02the integral
57:03will be x square over 2.
57:07and then I have the quadratic term which
57:10will give me
57:13x cubed over 3.
57:17so if I now take minus the derivative of
57:20this object I recover what I have in the
57:22right hand side
57:39uh you mean that ah yes because I killed
57:43all the events which
57:46decrease the spread by setting Lambda 0
57:49minus to zero so the question is uh why
57:53don't I put in here the constraint or it
57:56was actually inside the x that BS is
57:59actually the S Plus so it counts only
58:01the
58:02spread increasing events and the reason
58:05is that to simplify the model I kill the
58:07rate at which the spread decreasing
58:09event support so I set Lambda 0 minus to
58:13zero
58:14so all of the events which occur are
58:17those that increase the spread
58:19so my DS coincides with
58:22with Ds Plus
58:24for uh
58:26pointing out
58:28okay so now this is our potential which
58:32depends on our non-linearity Epsilon
58:36and now to try to understand the
58:39Dynamics what you have to do is to study
58:40what we have to do is to study this
58:42potential so let me rewrite it
58:46a little bit so let me write this as
58:50beta
58:521 minus Alpha over two
58:55and then I want to
58:57compose these two terms in such a way it
58:59was Global Square so I will write it as
59:03x minus
59:05Lambda 0 over
59:07one minus Alpha Square
59:11so if I take x square I recover this
59:14if I take the double product I recover
59:17the linear term
59:19and then I have a console a global
59:22constant which I have added that I will
59:24need to subtract so false
59:29oops
59:30the cubic term and then let me subtract
59:34the remaining term which I just added so
59:36this will be Lambda 0 square beta
59:39uh divided by two one minus Alpha but
59:43this we don't really care about it
59:44because it's just a global shift the
59:46concept
59:48that will not
59:50matter when we look at the shape of the
59:52potential
59:57very good
1:00:00okay
1:00:04okay so let's study this
1:00:09and let's start from
1:00:11Epsilon equal to zero where we go back
1:00:14to our linear hoax
1:00:29so I will forget about the
1:00:32oops the constant term
1:00:37that doesn't really matter so in the
1:00:38case of sine equal to zero then the
1:00:40shape of our potential is simple
1:00:42because I'm killing the cubic term and I
1:00:46just have essentially a parabola so if I
1:00:49plot
1:00:51the Epsilon equal to zero
1:00:54of x
1:00:57plus the constant
1:00:59Lambda 0 square beta over 2 1 minus
1:01:02Alpha
1:01:04then I have a parabola and the center of
1:01:07the parabolize a quantity that you might
1:01:09recognize
1:01:12it is Lambda 0 divided by y minus Alpha
1:01:16so my potential will look like this
1:01:19okay
1:01:21and so
1:01:24is there anybody who recognizes what was
1:01:26this expression
1:01:28for the linear hoax process so I'm
1:01:31changing the notation so maybe that's a
1:01:33bit
1:01:34so that was in the exercise of last time
1:01:37that was mu
1:01:39over one minus beta
1:01:42but anyway if if you go back to that
1:01:45notation you will recognize that this is
1:01:47the average intensity of uh the uh of
1:01:52the linear hoax process so this was uh
1:01:56if you want the average
1:02:00sorry
1:02:04in this notation
1:02:07it was the average of the S of T
1:02:09actually
1:02:19okay
1:02:22which was also the stationary values so
1:02:24there were if you recall if you remember
1:02:27there was a discussion about the process
1:02:29being stationary for Alpha smaller than
1:02:31one so the value at which this rate
1:02:34eventually converges was given precisely
1:02:39or was related to this expression in
1:02:42here and this is this is good because
1:02:45what we recover with this potential
1:02:48description is precisely this so in here
1:02:50if you think at the process in terms of
1:02:53this language Dynamics then what do we
1:02:55expect to happen well you have just one
1:02:57minimum
1:02:57so your Dynamics will have some noise
1:02:59will descend because of the gradient
1:03:03term and eventually it will relax into
1:03:06this unique uh this unique minimum that
1:03:10is what will govern your your stationary
1:03:14state so if you ask what will be the
1:03:16stationary value of my X of t
1:03:20then this because of uh
1:03:24ergodicity if you want this will be
1:03:27given by
1:03:29so I'm just
1:03:33this is just a definition then remember
1:03:35that you have here this vs of Tau so at
1:03:39very large time you expect that this
1:03:41converges to to the average of this
1:03:44quantity and if you take the average
1:03:47what you have to average is this DS
1:03:54and if this average of the S converges
1:03:57to this quantity in here using then you
1:04:01the this is independent of time you
1:04:03bring it out of the integral the kernel
1:04:05is normalized so what you get is
1:04:07precisely that the long time limit of
1:04:09your value of x coincides with with this
1:04:12expression in here that we had already
1:04:13computed so this is consistent with our
1:04:17previous analysis of the linear hoax
1:04:20case
1:04:21but now if you add uh Epsilon which is
1:04:24positive of course this picture will
1:04:26change
1:04:28and it will change because if you add
1:04:30some non-linearity there are
1:04:33more stationary points of your potential
1:04:36which appear so let's compute them
1:04:39so now let's
1:04:42look at Epsilon larger than zero
1:04:47so how does one compute the stationary
1:04:49points of the potential well what you
1:04:51have to do so stationary points are the
1:04:53Minima or or the maximum
1:04:55so to compute them what you have to do
1:04:57is to take the derivative and set it to
1:05:00zero so the derivative we have it in
1:05:02here
1:05:04up to the minus sign so we have to
1:05:07compute those values of X where this
1:05:09expression is equal to zero so this is a
1:05:12quadratic equation for x
1:05:14that we can solve so let me just give
1:05:17you
1:05:18being it quadratic it will have two
1:05:20solutions
1:05:22so let me give I have a two solution
1:05:26directly so this will be 1 minus Alpha
1:05:30plus minus
1:05:32square root of 1 minus Alpha Square so
1:05:35the beta simplifies
1:05:38and you get something like this
1:05:44okay
1:05:48so these points will be stationary
1:05:51meaning that they are either Maxima or
1:05:54or Minima
1:05:56and now what should you do to understand
1:05:59which one is the maximum and which is
1:06:01the minimum
1:06:02and yes
1:06:11so in general the way in which you
1:06:14distinguish whether you are in a minimum
1:06:16or in a maximum is to look at the second
1:06:18derivative right so if the second
1:06:20derivative is positive you are in a
1:06:21minimum if the second derivative is
1:06:23negative you are in a maximum
1:06:26now here you can compute the second
1:06:27derivative and check the sign when you
1:06:30evaluate it at this point but let's do a
1:06:33shortcut so what is the shortcut well is
1:06:37to understand
1:06:41is to use what we already know
1:06:44from Epsilon equal to zero
1:06:49and to guess what is then the shape of
1:06:51the potential for Epsilon different from
1:06:53zero
1:06:55so this will be now v e of X Epsilon
1:06:59plus the constant
1:07:01so before this was a minimum now we
1:07:04switch on Epsilon so first of all
1:07:08as you see if you take X which is very
1:07:11very large
1:07:13the term which dominates in this
1:07:15expression will be the cubic term that
1:07:17has quite importantly the minus sign in
1:07:21front so this is telling you is that 4X
1:07:24very very large this potential will
1:07:27Decay like minus X cubed
1:07:31in here
1:07:33and then you know that you have two
1:07:35stationary points so it is quite easy to
1:07:38expect if you study the behavior at x
1:07:40equal to zero you will find something
1:07:42like this so it is easy to expect that
1:07:44your potential will have some shape like
1:07:46this
1:07:48with a minimum and then a maximum that
1:07:51allows you then to Decay to minus
1:07:53infinity
1:07:55so we have to understand which one is x
1:07:57minus and which one is X Plus
1:07:59and a fast way to do this is to
1:08:04try to look at the behavior of this
1:08:08quantity when Epsilon is small because
1:08:09we know that when Epsilon is zero the
1:08:12minimum will collapse to lambda zero
1:08:15divided by 1 minus Alpha so let's expand
1:08:19this two terms
1:08:21these two points for small Epsilon
1:08:26so if I look
1:08:28at the term with a plus
1:08:31when Epsilon is zero I can kill this
1:08:34term inside the square root I just have
1:08:361 minus Alpha to the power of 2 it's
1:08:40square root so this quantity is positive
1:08:42so it's just 1 minus Alpha I have a plus
1:08:45so I have twice 1 minus Alpha divided by
1:08:48Epsilon so this will go
1:08:50when Epsilon is very small
1:08:53like one minus Alpha
1:08:56over Epsilon
1:08:59and this is X Plus
1:09:02whereas if I look at x minus and I do
1:09:05the expansion
1:09:08I have to be a little bit more careful
1:09:10because
1:09:11then the order one terms will cancel
1:09:14because I have a minus so I have to keep
1:09:17also the expansion toward the Epsilon
1:09:19that comes from expanding the square
1:09:21root
1:09:22okay and if I do this so this is
1:09:27you you can do it for money so this will
1:09:30be one minus Alpha I just look at the
1:09:32numerator now
1:09:33plus minus
1:09:361 minus Alpha so I bring 1 minus Alpha
1:09:39outside the square root and then I have
1:09:411 minus 4 Epsilon Lambda 0 divided by
1:09:451 minus Alpha Square
1:09:48and then if Epsilon is small I can
1:09:50expand this square root so this is 1
1:09:53minus x to the power one alpha and the
1:09:55expansion for X being small is so this
1:09:59will be the expansion of this is 1 minus
1:10:01one alpha times x
1:10:04as you can check so if you do this
1:10:07in the limit
1:10:09Epsilon being small you find that this
1:10:12is precisely
1:10:13Lambda 0 over 1 minus Alpha plus terms
1:10:17which are of order of Epsilon whereas in
1:10:20here you have plus terms which are
1:10:22ordered one
1:10:25so this is to say that by continuity in
1:10:28epsilon we expect that the point which
1:10:30is the minimum is x minus so this here
1:10:34will be x minus
1:10:37that when we switch Epsilon to zero
1:10:40we'll collapse to this point in here
1:10:43whereas the new stationary point that we
1:10:46have
1:10:47is X Plus
1:10:49which as you see is is very far away if
1:10:52Epsilon is small because it is of the
1:10:54order of 1 minus
1:10:56Alpha divided by Epsilon
1:11:00okay
1:11:03so this tells you that in the limit
1:11:06Epsilon going to zero this goes to
1:11:09infinity and you recover this picture
1:11:11with a unique uh with a unique minimum
1:11:16okay so the last thing that we need uh
1:11:19to to understand the Dynamics is then to
1:11:21compute the barrier
1:11:23and the battery is a measure so you what
1:11:26we can do is to compute then the value
1:11:28of the potential at this
1:11:31local maximum that tells you uh so if
1:11:34you shift the potential this is actually
1:11:36zero so it tells you what is the barrier
1:11:38B what is uh the difference in potential
1:11:40that you have to overcome with this type
1:11:42of jumps or activated processes
1:11:46so this means that we want to compute V
1:11:51evaluated at X Plus
1:11:54so this is algebra it's a you you can do
1:11:57it so let me go a little bit fast I will
1:11:59expand it so this is
1:12:02no I don't expand it this is exact
1:12:07ly if you do the math you will find that
1:12:09this barrier goes like
1:12:11one over Epsilon Square
1:12:17so when Epsilon is small this point is
1:12:19very very uh
1:12:21up this is very very large and very very
1:12:24far away
1:12:25so this barrier here
1:12:28Delta V is of the order of one over
1:12:30Epsilon Square
1:12:33and so again what happens you take
1:12:35Epsilon going to zero this X Plus goes
1:12:38to Infinity the barrier shoots up and
1:12:41you recover as you see the the nice
1:12:43Parabola that we had before
1:12:47okay so this is the potential and now
1:12:49that we have uh the potential we can ask
1:12:53this question about Activation so in a
1:12:56similar way as for the double well of of
1:12:59the magnetization of the easing we can
1:13:02imagine that the Dynamics if you start
1:13:04from somewhere within the Basin of
1:13:05Attraction of x minus will relax
1:13:08in this metastable state but then there
1:13:10will be very rare activated
1:13:15processes that kick you up across this
1:13:20barrier and once you go across this
1:13:23barrier then you start rolling down
1:13:26because from where on the potential goes
1:13:28to minus infinity and therefore if you
1:13:30manage to kick this barrier then you go
1:13:35into this scenario in which you have a
1:13:37crisis of your economy or of of your
1:13:40financial exchanges that if you remember
1:13:42was defined by X exploding and uh and
1:13:48going to Infinity
1:13:51so how often does this happen
1:13:55well
1:13:57we can use
1:13:59results
1:14:01in the literature to compute what is the
1:14:03average time that is required to see
1:14:06this type of jump events
1:14:12and there are explicit expressions for
1:14:14this
1:14:18and
1:14:19for this particular example
1:14:23the expression for the time is given
1:14:28in full detail
1:14:30in the text of the today
1:14:37so this is now 0.3
1:14:43okay so if you if you look at there
1:14:47what you see is that you can write down
1:14:49what is the average time for this
1:14:51activity jumps
1:14:53which will be given by a pre-factor that
1:14:55I just call a I'm not going to rewrite
1:14:57it so that is the square root that you
1:14:59see in the text
1:15:01and then you have the exponential
1:15:04that is what we want to compute
1:15:08of what of an integral so the integral
1:15:10goes from so in the text I wrote X
1:15:14equilibrium and X star so what I mean by
1:15:16this
1:15:18X equilibrium is x minus is our
1:15:20metastable minimum and F star is is the
1:15:23point where you at the barrier so
1:15:26let me call it x minus
1:15:29NX Plus in DX
1:15:33and then you have a ratio so you have
1:15:35the ratio between the derivative
1:15:37of your potential V Prime of x
1:15:40and another quantity that is called d of
1:15:43x
1:15:45that if you look at it you will
1:15:47recognize that it is basically
1:15:51the variance of the noise of our lunch
1:15:53event process so D of X is
1:15:57B squared over 2
1:16:01Lambda 0 plus alpha x square
1:16:04sorry X
1:16:06Plus
1:16:08Epsilon s x square
1:16:12so this was basically Lambda T and if
1:16:15you look at your language
1:16:18in here
1:16:21it is basically the square of this
1:16:24expression so it is the variance of the
1:16:25noise
1:16:27foreign
1:16:30this time what we have to do is to
1:16:32compute the this integral which is at
1:16:36the exponent so
1:16:38this is a little bit lengthy so I will
1:16:40not do it in detail but you find it in
1:16:43the solution maybe I will just give you
1:16:45uh
1:16:48an idea so and the idea so it's lengthy
1:16:51because
1:16:52well
1:16:53you have to integrate on a finite
1:16:55interval the ratio of two quantity which
1:16:57are quadratic so what's the fastest way
1:17:00to do this
1:17:01well in I think that the fastest way is
1:17:04first of all to look at the denominator
1:17:06and try to write it as a sum of two
1:17:10terms which are only linear in in x
1:17:13so to do this what you can do is you
1:17:15compute The Roots
1:17:17now I will sketch just how to do the
1:17:20computation so you compute the roots of
1:17:22this expression let me call it
1:17:25X1 and X2 so you you set this equal to
1:17:28zero and you will get two values for x
1:17:31and then you can show that you can write
1:17:331 over DX
1:17:35as
1:17:37so if you compute the root then you know
1:17:39that the X
1:17:41is essentially equal to a constant so DX
1:17:45will be
1:17:46some constant
1:17:48x minus X1 x minus X2
1:17:51right and the constant is chosen to
1:17:54adjust the coefficient of the term x
1:17:57square
1:17:59so you plug this into the denominator
1:18:02and then you try to split these two
1:18:04terms into a sum of two terms which are
1:18:06only linear in the denominator and you
1:18:08can do this you will find that
1:18:11you will have an expression like this
1:18:15you have a pre-factor and then you have
1:18:161 minus 1 over x minus X1 minus
1:18:211 over x minus X2
1:18:24okay
1:18:25and this helps
1:18:28because now let me call
1:18:32this integral
1:18:34at the exponent
1:18:36curly I
1:18:38so this expression is the coefficient
1:18:39times the exponential of the integral
1:18:44so then you can write your integral as
1:18:47what as the prefactor
1:18:56times the difference of two integrands
1:19:02one which contains
1:19:04V Prime of x
1:19:07minus X1
1:19:09minus the one which contains
1:19:15V Prime of x
1:19:18x minus X2
1:19:22then what you have to check is that X1
1:19:24and X2 are always smaller than x minus
1:19:27so you see you have in principle a
1:19:29singularity but this is outside the
1:19:31domain of integration and then you can
1:19:33this integral is something that you can
1:19:35do quite easily so now you have a
1:19:37quadratic form
1:19:39in the numerator
1:19:41and you can rewrite
1:19:44so that's just the hint
1:19:47for the math
1:19:49but you can rewrite
1:19:52V Prime of x
1:19:54as
1:19:57x minus X1 so suppose that we want to do
1:20:00this integral what I have to do is to
1:20:02play a little bit with my V Prime of X
1:20:05and I can rewrite it as a constant
1:20:07times x minus X1 Square
1:20:11Plus
1:20:13another constant times x
1:20:16plus another constant so this is an
1:20:18exercise and once you rewrite it in this
1:20:21way you plug it in here and you see that
1:20:24you have either terms which are linear
1:20:26in X or the only
1:20:28most non-trivial term that you get is of
1:20:31the form gamma over x minus X1 that will
1:20:34give you a logarithm so if you do these
1:20:36tricks you can compute these Expressions
1:20:39quite easily
1:20:40but I'll uh leave it to you
1:20:43and you can of course check the
1:20:44solutions
1:20:46and now let's comment uh what comes out
1:20:48of this
1:20:51so doing all of the math
1:20:58and then doing an expansion
1:21:00for small Epsilon which is
1:21:03what we are interested in
1:21:19the expression that you get is as
1:21:21follows so you get that this e of Tau
1:21:27or actually
1:21:29the integral that you have at the
1:21:31exponential
1:21:33will go like
1:21:35Alpha Square minus one
1:21:38minus two alpha log Alpha
1:21:42divided by
1:21:43Alpha Beta Epsilon
1:21:46plus you have a logarithmic Corrections
1:21:50in epsilon
1:21:53plus terms which are regular
1:21:55when you send epsilons to zero
1:22:01okay
1:22:02and of course I am assuming here that
1:22:05Alpha is different from 0 if Alpha is
1:22:07equal to zero
1:22:08you will have to redo the calculation
1:22:11uh with a little bit of hair but you get
1:22:13a very similar result
1:22:16and so the important point and with this
1:22:18we can conclude is that you find then
1:22:21that your average time for your Escape
1:22:24or activated processes
1:22:26goes like e to the some constant
1:22:30times 1 over Epsilon and so again
1:22:34when you take Epsilon to zero you see
1:22:37that you recover the situation in which
1:22:40this time explodes and you cannot have
1:22:43any of these activated processes but as
1:22:46long as Epsilon is finite and you have
1:22:48some non-linearity in your process
1:22:51you go back to this meta stable
1:22:53situation so you you have to wait a very
1:22:56large time but eventually you have
1:22:58situations in which the noise kicks you
1:23:00up this very large barrier and if this
1:23:03happens then after that you are bound to
1:23:06roll down uh to minus infinity and
1:23:09therefore you have a crisis for your for
1:23:12your economy and this is a scenario that
1:23:14is different with respect to
1:23:18so let me add two comments
1:23:24one on the content which is the last
1:23:26point of the exercise
1:23:28so this is a scenario for the crisis
1:23:37that is different
1:23:41with respect to what we have already
1:23:43discussed
1:23:44that was the case in which
1:23:47Alpha is very close to one
1:23:50that is what was called in the
1:23:52literature as self-organized
1:23:55criticality
1:24:05so it is different because in this
1:24:07scenario here you have an economy or
1:24:09whatever you are describing that is
1:24:11always at the verge of being critical so
1:24:13any small fluctuations will pull you
1:24:17into the unstable phase where Alpha
1:24:21reaches one and then you have an
1:24:22explosion in the number of events
1:24:25whereas in here you have something very
1:24:27different because you have a state a
1:24:29local minimum that actually looks quite
1:24:32stable and it looks stable for very
1:24:33large times so you are in this economy
1:24:35and you think that everything is going
1:24:38well because you are in your meta stable
1:24:40local minimum and you have no clue so
1:24:43your parameter Alpha is very far away
1:24:44from one so you have no clue that
1:24:46something could go wrong but just
1:24:49because you have this little non-linear
1:24:52term
1:24:53eventually if you wait long enough then
1:24:56your noise will actually bring you up uh
1:25:00uphill in here and will lead you to a
1:25:02crisis which has uh which is very hard
1:25:05to predict a priority if you don't have
1:25:08this picture in here in mind so the
1:25:10scenario for the behavior of the system
1:25:13is really very very different this is a
1:25:15crisis which is due to activated
1:25:17processes whereas if you choose Alpha
1:25:20cross one you go back to this idea of
1:25:23self-organized criticality
1:25:25and the second comment that I wanted to
1:25:27just point out and then the details you
1:25:30can figure them out is that this matches
1:25:34the calculation that we just did so in
1:25:36particular
1:25:37this time going like one over Epsilon
1:25:40matches with
1:25:41the scaling that I wrote at the
1:25:45beginning for the easing like case which
1:25:48was that this time was going like uh the
1:25:51the barrier in your potential divided by
1:25:54the variance of the noise
1:25:56why is it so well because in here so it
1:25:59it might be a bit surprising that you
1:26:01have one over Epsilon because if you
1:26:03remember the height of the barrier
1:26:06was of the order of one over Epsilon
1:26:09Square
1:26:11so you would expect a priori one over X
1:26:13Epsilon squaring here but the point is
1:26:16that you have to be careful about the
1:26:18noise
1:26:18and the noise depends on on the
1:26:21particular point x where you are because
1:26:24the noise in our lunge of an equation
1:26:26was the square root of Lambda t
1:26:30or was going let's say the variance
1:26:33what I call Sigma Square there
1:26:39you know our problem goes like Lambda T
1:26:41and Lambda T goes like Lambda 0 plus
1:26:45Alpha X Plus Epsilon x squared
1:26:49and what you can check is that if you
1:26:51compute this expression at the barrier
1:26:54which was X Plus
1:26:57you see that this goes like one over
1:27:00Epsilon
1:27:04I am not mistaken
1:27:10but this we can check so this should go
1:27:12when computed at the battery it goes
1:27:15like uh one over Epsilon so if you plug
1:27:17it into this expression you will have DV
1:27:20which is one over Epsilon Square
1:27:22and then you have a factor of so you
1:27:26have a sigma Square
1:27:28which is or Sigma Square over n and now
1:27:30becomes
1:27:311 over Epsilon and combining these two
1:27:35scalings you get out precisely this
1:27:38factor of one over Epsilon for the
1:27:40relaxation time now this is not quite
1:27:43the right argument because uh the noise
1:27:47changes all along the potential so to
1:27:50recover the scaling I computed it only
1:27:53at the barrier but what you have to do
1:27:55properly is precisely what I sketched in
1:27:58here
1:28:00so you you really have to compute this
1:28:03integral all over your path so
1:28:06what I'm saying is that if your noise is
1:28:09constant you could bring this out of the
1:28:11integral and then you just have the
1:28:13integral of uh of a derivative which
1:28:16gives you the difference of the
1:28:18potential at the two points so that will
1:28:19be the barrier divided by B if D was
1:28:22constant now here it is known constants
1:28:24so you have to go through all of the
1:28:25integration which depends on X but you
1:28:28somehow recover the scaling if you just
1:28:30compute the uh precisely at the point
1:28:33which corresponds to your barrier and so
1:28:35you see that you have an expression
1:28:38which matches with what we know from the
1:28:41lecture
1:28:42okay so I think now
1:28:45we can maybe
1:28:47[Music]
1:28:48so this was it for this exercise on meta
1:28:51stability are there questions
1:29:01okay if there are no questions I think
1:29:03we can
1:29:04maybe make a break now and then go to
1:29:06the second part
1:29:08if you agree
1:29:11so now is 10 40 so we kind of 15 minutes
1:29:19of break and then we go to uh to part
1:29:23two
1:29:25okay
1:29:46this conference will now be recorded
1:29:49okay good
1:29:52good so let's start again with part two
1:29:56and the text is in the the seven slash
1:29:59eight
1:30:02and what we are gonna discuss in here is
1:30:05uh something which goes under the name
1:30:07of uh imitation models or models for
1:30:10herding
1:30:11uh
1:30:13uh or even
1:30:15well let's say imitation models
1:30:17and in particular we are looking at a
1:30:19particular model that was introduced by
1:30:21Kirman in this paper of 1993 and the
1:30:26motivation of human so this is called
1:30:27the ants model so once like the animals
1:30:30and his motivation was to understand
1:30:32indeed behavior of animals so the idea
1:30:36is that if you have this collection of
1:30:39uh of animals and then you put two
1:30:42sources of food what he was realizing is
1:30:45that these animals always somehow choose
1:30:49to go to it either in the source a or in
1:30:52the source B but somehow sometimes there
1:30:55are some very abrupt changes where all
1:30:58of the animals suddenly want to go to
1:31:00the source a or all of the animals want
1:31:02to go to the source speed and so the
1:31:04idea was to write down a simple model to
1:31:06uh to try to understand this type of if
1:31:10you want Collective behavior of or
1:31:12behavior of a large number of agents and
1:31:16we can generalize it a little bit so we
1:31:18can think of this as being a description
1:31:20also of human behavior so whenever you
1:31:23have a choice between two binary choice
1:31:25between two uh options
1:31:29and there is a huge amount of people
1:31:32that have to make choices sometimes you
1:31:34see that you have this abrupt switches
1:31:37between one choice or the other and we
1:31:40would like to understand why this
1:31:42happens and what is a good description
1:31:44for this so this could also be
1:31:46in economic Trends where some you have
1:31:50some reversal of of the behavior of
1:31:53people you can also so the
1:31:55it was mentioned in the lecture this
1:31:57example of using cell phones
1:31:59so people have to choose whether buying
1:32:02or not a cell phone and this can change
1:32:04massively over time so these are all
1:32:07things that we can think of in terms of
1:32:10this very simple models so the kiermann
1:32:13model that we are discussing today you
1:32:15also find something about that in the
1:32:19lecture notes so I uploaded yesterday
1:32:21night a new chapter which is a chapter
1:32:24which is about random field using model
1:32:26and which contains something about what
1:32:30we are going to discuss today
1:32:31and the Amazon which is similar is the
1:32:35Moran model that if you have time you
1:32:38can look at in the homework number seven
1:32:41and again I will upload the solutions uh
1:32:45after the day today
1:32:47so let's describe the models so what are
1:32:49the rules of the game so the idea is
1:32:51that you have again many uh agents
1:32:54that
1:32:56interact with each other and which are
1:32:58of two types so
1:33:05rules of the game so you have two
1:33:07species if you want
1:33:12A and B
1:33:14so this can denote uh I don't know the
1:33:17people uh who want to buy a cell phone
1:33:20and the people who don't or the people
1:33:21who want to vote for something and the
1:33:23people who want to vote for something
1:33:25else and and whatever you want to think
1:33:28of the ants which want to go to the
1:33:30source of food on the right versus those
1:33:33which want to go to the source of food
1:33:34on the left
1:33:36and uh all of these individuals interact
1:33:40in the following way so there is they
1:33:43meet so let's choose a typical time
1:33:46scale so an interval of time
1:33:53DT
1:33:56and what happens is that we assume that
1:33:59uh well these people
1:34:02or these agents can meet uh with each
1:34:06other in within this interval of time in
1:34:08pairs
1:34:09and they meet with a rate so the rate
1:34:14of let's say
1:34:18meetings
1:34:23we denote it with gamma
1:34:26and whenever they meet if they are of a
1:34:29different type some discussion takes
1:34:32place there is some influence of uh one
1:34:35uh with uh on the other and there can be
1:34:38some conversion so you can have
1:34:41let's say some conversion
1:34:45so the conversion means that you have
1:34:48two individuals one of type A and one of
1:34:51type B that meet together with the rate
1:34:54gamma and as a result
1:34:57either the individual of type A is able
1:35:01to recruit the individual of type B in
1:35:04its team or the vice versa kind of core
1:35:07so you have a process in which you start
1:35:09from this configuration and you end up
1:35:10in either one of these two
1:35:12configurations and in in this model now
1:35:14we assume that this happens
1:35:17with equal probability
1:35:20whereas if you look at the model
1:35:23uh well you have some Fitness so if you
1:35:25look at the homework the situation is a
1:35:27little bit different but let's stick to
1:35:29this example for the moment
1:35:32so you have some conversion which can
1:35:35occur with rate gamma and you also have
1:35:38something else so you can have that some
1:35:40individual decides to change uh its team
1:35:44spontaneously and this also happens with
1:35:46a fixed rate which I call in here
1:35:49Epsilon so this will be uh
1:35:55so let me put a slash in here you have a
1:35:58rate
1:35:59of
1:36:01let's say self
1:36:03okay
1:36:06I'll write it properly
1:36:13so you have conversion and you have
1:36:16what I can call self-switching
1:36:23self-switching means that without
1:36:26meeting anybody else an individual of
1:36:28type A can decide to become of type B
1:36:31and vice versa
1:36:33and this happens with a rate that I call
1:36:37Epsilon and I call it Epsilon for a
1:36:39reason that we will discover
1:36:41afterwards
1:36:44okay so these are the rules and now uh
1:36:47well before going to the exercise let me
1:36:49stress something so we are going to
1:36:51solve this modus and from the technical
1:36:54point of view
1:36:55what will be important in here is what I
1:36:58will discuss in point two of the
1:37:00exercise and the idea will be to go from
1:37:03a discrete setting which is the one that
1:37:06I'm sketching in here to some sort of
1:37:09continuous limit which in essence means
1:37:11that we are going to start from some
1:37:13master equation with transition rates
1:37:15and then we will try to derive from that
1:37:17a kind of focal Planck equation for uh
1:37:21for the process taking a continuous
1:37:22limit
1:37:23and and this I will do a little bit in
1:37:25detail and I think it is uh important uh
1:37:28for you to uh to have a look at how you
1:37:31can do this properly but this is point
1:37:34two now let me start from point
1:37:36one
1:37:41so point one which allows to understand
1:37:44a little bit better what I wrote in here
1:37:46so the idea is
1:37:48now to describe this process in terms of
1:37:51transition rates so let me fix one type
1:37:55so let's choose type A and let me
1:37:59introduce a variable case Okay will be
1:38:01just the number
1:38:03of individuals
1:38:11of type A
1:38:14out of n so let's say that we have
1:38:18a total of n individuals so as you see
1:38:21with this type of Dynamics and remains
1:38:24constant because every time
1:38:26two individuals meet
1:38:28they can change their opinion but you
1:38:31end up with two individuals at the end
1:38:33so n remains constant and K is a number
1:38:36is the variable which changes uh with
1:38:38time which denotes uh if you want the
1:38:41number of individuals of a given type
1:38:43and then we write down transition rates
1:38:46which tell you what is the probability
1:38:48that you increase or decrease this value
1:38:51of K so
1:38:54I will call them W as it was done in the
1:38:57lecture so w will be the transition rate
1:39:00to go from K for example to K plus 1
1:39:06okay
1:39:07and what the rate means is that this is
1:39:09a probability divided by time so if I
1:39:12write it as W Times
1:39:16my small interval DT then this will be a
1:39:19full probability so this will be the
1:39:21probability that over a small interval
1:39:24of time DT I go from having K
1:39:27individuals of type A to K plus 1
1:39:29individuals of of type A
1:39:32and if you uh if you look at the model
1:39:35what we are always assuming is that the
1:39:38interval is chosen in such a way that
1:39:40you have at most one encounter between
1:39:43two individuals at each interval DT
1:39:47so what is this rate now this is given
1:39:49already in the text so let me write it
1:39:54in a slightly different way and let me
1:39:56comment
1:39:57so this is one half times gamma
1:40:01and then you have Phi so Phi is defined
1:40:04as a fraction so I would write it as K
1:40:07Over capital n
1:40:101 minus K Over capital n
1:40:14Plus
1:40:16Epsilon 1 over K Over capital n
1:40:21so indeed what this is describing is uh
1:40:25is what we just saved when describing
1:40:28the model so the idea is that in this
1:40:30small interval of time
1:40:32two individuals can meet and they will
1:40:34meet with the rate so let me
1:40:38erase the DT here
1:40:40so two individuals we will meet with the
1:40:42rate gamma and if they meet a conversion
1:40:47can occur only if they are of different
1:40:49species so you need that one of those uh
1:40:52is of the species a and this will happen
1:40:55with a probability that is equal to the
1:40:57fraction of individuals of the species a
1:40:59so it's K Over capital N the other one
1:41:01has to be of species B so this is the
1:41:04corresponding probability
1:41:06and then when they meet a conversion
1:41:08will occur and the conversion can be so
1:41:11you have an equal probability that the
1:41:13conversion is good for the individual of
1:41:15type A so that the type B converts and
1:41:19becomes type A and therefore you
1:41:22increase your population or the
1:41:24conversion can go uh on the other side
1:41:26but the probability for the good
1:41:29conversion to occur is encoded in this
1:41:31Factor one alpha in here
1:41:34so the first term is uh describing when
1:41:37you increase your population because of
1:41:39this recruitment or interaction effect
1:41:42and then as I say do you have this extra
1:41:44contribution of the self-switching and
1:41:47uh and this of course has to be
1:41:49proportional so this will occur with the
1:41:51rate Epsilon and it has to be
1:41:54proportional to the population of type B
1:41:57because only those of type B will
1:41:59convert to type a and increase your
1:42:02population
1:42:04so this is essentially encoding just the
1:42:07rules of the game that we wrote uh
1:42:09upstairs in words and of course you have
1:42:12uh the other way
1:42:16so you have the rate of decrease of your
1:42:19population
1:42:20and now as you can easily guess the
1:42:22first term uh will be exactly the same
1:42:24so
1:42:26you have the same probability that two
1:42:29individuals so core and now you have
1:42:33the conversion of both to type B which
1:42:36decreases your population whereas in
1:42:39here
1:42:40to decrease your population you need
1:42:42that an individual of type A converts to
1:42:45type B and this happens with a rate
1:42:47which is Epsilon and with a probability
1:42:49that is equal to the fraction of
1:42:52individual of type A
1:42:54foreign
1:43:00of your model in a small interval DP
1:43:05and now from here
1:43:09what we want to do
1:43:11is uh to so if you have these rates you
1:43:14can write down some generic Master
1:43:17equations for the behavior of your
1:43:22probabilities so let me call it
1:43:26P of K at time T so this is the
1:43:30probability of course to have
1:43:39of having K individuals of type A
1:43:46at time t
1:43:48and the master equation will contain the
1:43:51transition rate so if you want we can
1:43:53write
1:43:54with this type of notation we can
1:43:57interpret
1:44:00our transition rates times the small
1:44:04interval DT as
1:44:06the conditional probability that you are
1:44:08at K plus 1
1:44:10time t plus VT given that you were at K
1:44:14at time t
1:44:19okay now
1:44:21I will not write down well yes I will do
1:44:24it uh in a minute or actually let me do
1:44:28it so let me write down
1:44:30a discrete Master equation for this
1:44:33process
1:44:34foreign
1:44:39I think so
1:44:43so I ask what is the probability that
1:44:46the time P plus delta T I have K
1:44:50individuals
1:44:56so this will be equal to the following
1:44:59so
1:45:01I have a certain probability that at the
1:45:03previous time I already have K
1:45:05individuals and then I am I'm asking
1:45:08that nothing happens in the interval DT
1:45:10so that the situation remains unchanged
1:45:14and the probability that nothing happens
1:45:16is 1 minus the probability that
1:45:18something happens
1:45:20so this is 1 minus the probability that
1:45:24two individuals meet and and sweet and
1:45:28one of those switches and this is
1:45:32gamma K Over N times 1 minus K Over N
1:45:42and then I also have to subtract the
1:45:45probability that one of those switches
1:45:47by itself
1:45:49thank you
1:45:51so I remember I add the DT because gamma
1:45:54is always a rate so it has a dimension
1:45:56of one over time and the probability
1:45:58that one of those switches is just
1:46:00Epsilon DT
1:46:08okay so this is the contribution when
1:46:11nothing happens and then of course you
1:46:13have contributions to this probability
1:46:16coming from a conversion which occurs in
1:46:20the small interval DT
1:46:22so you will have here A plus the
1:46:25probability that you were at K minus 1
1:46:28at time t
1:46:31times the probability that somebody
1:46:33converts
1:46:35to the type A during the small interval
1:46:38DT and this is precisely given by the
1:46:41rates that we wrote before so I will
1:46:43write them compactly so this is w
1:46:47to go from K minus 1 to K
1:46:51times DT
1:46:55okay
1:46:57and then you have
1:46:59the term coming from above if you want
1:47:01so you have to add
1:47:04the probability that you were at K plus
1:47:061
1:47:07at time T and then somebody goes
1:47:12to type B
1:47:15which is the other rate
1:47:20okay so all of this is in the
1:47:23description setting
1:47:25and now to do exercise two
1:47:28so I'm modifying it a little bit with
1:47:30respect to the text
1:47:31but the idea is that I just want to show
1:47:34you a little bit
1:47:36how one goes from the discrete setting
1:47:39to The Continuous plank like
1:47:41equation
1:47:44so we will do it in two steps
1:47:48so I will first sketch how you do this
1:47:50properly and then I will sketch a
1:47:53shortcut
1:47:56that uses some of the things that we
1:47:58discussed in pedia 3 and also very
1:48:01briefly
1:48:02at the end of the last today
1:48:08okay so the point too is go to the
1:48:11continue
1:48:17which means
1:48:19that we want to take the number of uh
1:48:22animals or or people or agents going to
1:48:26Infinity
1:48:27and write down an equation which holds
1:48:30uh in this limit
1:48:32and to do this we have to introduce some
1:48:35continuous quantity which are kind of
1:48:38what you introduce whenever you look at
1:48:41this and going to Infinity limit so you
1:48:43introduce densities and you try to
1:48:45develop a description in terms of
1:48:47density like the magnetization as we did
1:48:49for the spins
1:48:51so let me introduce this quantity five
1:48:54so if I will
1:48:56just be the fraction K Over N
1:49:01and let me also introduce something that
1:49:03hopefully uh
1:49:06will become clear in a minute
1:49:08so I will also think at the fact that if
1:49:11I want to develop a continuous
1:49:13description I not only I have to rescale
1:49:16k
1:49:17to be able to send n to Infinity but as
1:49:21we will see I will also have to rescale
1:49:23time to have a properly defined equation
1:49:27so I will write it down here and then we
1:49:30will
1:49:31see this in action concretely but the
1:49:34idea is that I will need to rescale my
1:49:37time variable with a given power of n
1:49:39which for the moment uh that I will have
1:49:43to choose to get a well-defined limit
1:49:45when n goes to Infinity so for the
1:49:47moment let me write down a generic power
1:49:49and to the alpha
1:49:52and then what I want to do is that I
1:49:54want to show
1:49:58uh that
1:50:03I get
1:50:05a well-defined
1:50:08equation
1:50:10which will be of the type soccer
1:50:12plank type
1:50:14or a new probability which I it's now a
1:50:19function of this continuous variable now
1:50:22Phi end of my rescale time
1:50:26and which I can think
1:50:28as the limit
1:50:31when n goes to Infinity
1:50:33of the probability of my discrete
1:50:36process
1:50:37evaluated at K which is nothing but
1:50:42n times Phi so this is a bit formal but
1:50:44then we will see what I mean
1:50:48and at the previous time T which is
1:50:52very scaled with respect to
1:50:55to the one of the continuous process
1:50:59so I will start from my master equation
1:51:02for this capital P and then I will try
1:51:05to write it down in terms as an equation
1:51:07of for a function which is always a
1:51:09function of only a function of Phi and
1:51:11of summary scale time
1:51:14and I will see if uh doing that I can
1:51:17reach uh I can take then the limit and
1:51:20go into infinity and find a closed
1:51:22equation for this quantity in here
1:51:24so this is all in words but I think it's
1:51:26much clearer if one does the calculation
1:51:30directly
1:51:36so to do the calculation let me start
1:51:38from
1:51:40the master equation observed
1:51:43and maybe I go
1:51:46I go down here
1:51:52okay so you see that you have P of k t
1:51:55plus DT and then on the right hand side
1:51:56you have P of k t
1:51:59plus something which depends on DT so
1:52:01the idea is that I want to get on the
1:52:03left hand side some derivative so I
1:52:06rewrite
1:52:07that as P of k p plus DT minus
1:52:13P of KT divided by DT
1:52:18and then I start manipulating what
1:52:21remains on the right hand side
1:52:26and I started doing that so since
1:52:28eventually I want to introduce this
1:52:30variable file let me try to introduce uh
1:52:32to rewrite what I have on the right in
1:52:35terms of five
1:52:37so let me try not to do mistakes
1:52:41so the first term which I get is a minus
1:52:45gamma
1:52:47then I have an instruction which is
1:52:49precisely 5.
1:52:51then I have 1 minus 5.
1:52:57and then I have I can also put
1:53:01plus Epsilon in here
1:53:05and everything is multiplied by my
1:53:08probability of k and t
1:53:11and using now for this type of relation
1:53:15for for finite and what I can say is
1:53:18that
1:53:20in in the limit okay so let me
1:53:23I will write it down and then so the
1:53:25idea is that you have in mind that you
1:53:27will take the limit and go into Infinity
1:53:28of what you have on the right hand side
1:53:30but if you take the limit I'm going to
1:53:32Infinity this P you expect it to
1:53:35converge uh to a function which is a
1:53:38function of Phi and Tau for
1:53:42and so
1:53:44I will write it here so this will be my
1:53:47P of Phi and Tau in terms of T since I
1:53:52have t still t on the left hand side let
1:53:54me keep T so this will be
1:53:57e n to the minus Alpha
1:54:05so if you see p of KT is precisely this
1:54:07then I am assuming that this has a limit
1:54:10when n goes to Infinity so I replace
1:54:13uh I take the limit on the right hand
1:54:15side and I replace with my limiting
1:54:18function curly p
1:54:20but then if I look at what I have below
1:54:23things are a little bit more complicated
1:54:26so now
1:54:28we have to remember which were
1:54:31the rates in there
1:54:34but are a little bit more complicated
1:54:36because so what is what was w
1:54:41so this was W going from K minus 1 to K
1:54:44so you have to look at this expression
1:54:47in here and you have to replace K with K
1:54:50minus 1.
1:54:52so I will write just the first terms so
1:54:55I will have in here Plus
1:54:58what is the rate from K minus 1 to K so
1:55:00it's the constant terms so is one half
1:55:04gamma
1:55:06and then since I'm starting from K minus
1:55:081 instead of K Over N I will have K
1:55:11minus 1
1:55:12Over N so K minus 1 over n is Phi
1:55:17minus
1:55:18uh yes
1:55:20is 5 minus 1 over n
1:55:24do you agree because K Over N is is just
1:55:26Phi and then i s
1:55:29a 1 over n
1:55:34then I have 1 minus K minus 1 over n and
1:55:37this I can write as 1 minus 5 plus
1:55:411 over n
1:55:46and then the same thing for the term
1:55:48with Epsilon so this will be plus
1:55:51Epsilon
1:55:54yeah sorry for the spacing but
1:55:57you have the same thing you have 1 minus
1:55:59five plus one over n
1:56:02When I close the parenthesis I don't
1:56:05know if you see this but all of this
1:56:07thing in parenthesis then is multiplied
1:56:11by
1:56:13by what well in principle I have capital
1:56:16p
1:56:17of K minus 1 t
1:56:20so let me rewrite it so let me take
1:56:24assume that n is large and this I can
1:56:26rewrite
1:56:28basically as my curly p
1:56:36evaluated at what is case Okay is
1:56:41uh
1:56:43so it's my so this is
1:56:46if you want n over Phi
1:56:49minus 1
1:56:51so I can rewrite it as n times P minus 1
1:56:55over n
1:56:57and then I know that my probability p
1:57:00as a function of n times something will
1:57:03map into my function curly P evaluated
1:57:07at the something so let me write it down
1:57:10so this will be Phi minus 1 over n
1:57:16and then I have to rescale time as above
1:57:20foreign
1:57:37and now I have done the other terms so
1:57:39the other term is is quite similar
1:57:41except that this rate in here is
1:57:45changing but if you do this
1:57:47along the same line of reasoning you
1:57:50have
1:57:51foreign
1:57:53the first part of the rate is unchanged
1:58:03and the second part was just multiplying
1:58:09so now you go
1:58:12no it was not unchanged because I have a
1:58:15plus somewhere
1:58:25okay so to compute the second term you
1:58:28have to evaluate the rate now which goes
1:58:30from K plus 1 to K
1:58:33so if I use the expression that I wrote
1:58:35before and I do the same expansion you
1:58:37realize that I have a plus in front of
1:58:39the correction because it's K plus 1 and
1:58:42not K minus 1 so you should find
1:58:44something like this please stop me if
1:58:47if this is not immediate
1:58:50then I will do it explicitly and then
1:58:52the term in here is proportional to K
1:58:55plus 1 over n which I will write
1:58:58I will rewrite as V plus 1 over n
1:59:07times
1:59:09my curly p
1:59:11which now is evaluated at K plus 1 which
1:59:15in my new variable is V plus 1 over n
1:59:19and then the rescale the time
1:59:26okay
1:59:42yes
1:59:45this this is minus yes because I have a
1:59:48minus in front thanks
1:59:51exactly
1:59:56okay so now I'm doing things a little
1:59:59bit it's loppy so as you see I'm
2:00:00translating from one notation to another
2:00:02and taking first one limit uh before the
2:00:06other but let's
2:00:08so you can do things more properly but
2:00:11what I just want to emphasize is the
2:00:13step
2:00:14that comes now
2:00:17and the step that comes now is that you
2:00:20now have
2:00:22so you rewrite things in such a way that
2:00:24on the right hand side it appears the
2:00:27function uh of which you want to compute
2:00:29or for which you want to compute an
2:00:31equation
2:00:32the left hand side we have to work on
2:00:34that but we will do it last but the
2:00:37point now is that this function is now
2:00:40computed at Phi which is good but then
2:00:42you have this correction of 1 over n
2:00:45and so what we can think of is
2:00:48now to do an expansion of uh of what
2:00:52appears in the right hand side in powers
2:00:54of 1 over n which is uh which is our
2:00:58small parameter when n goes to Infinity
2:01:05so I'll just give the idea
2:01:19and then we will not do all of the
2:01:21calculations because it's a bit tedious
2:01:24but
2:01:26foreign
2:01:29give the idea so now you assume that you
2:01:33can expand your functions so whenever
2:01:35you have P of
2:01:36Phi minus 1 over n
2:01:40and
2:01:42my T to the N minus Alpha will be what I
2:01:46will later on call Tau
2:01:49this you will write as P of Phi Tau
2:01:53and you're happy with that
2:01:55then you have the first derivative
2:02:02over Phi times minus 1 over n so let me
2:02:06put
2:02:08and this is evaluated at 3 and Tau times
2:02:111 over n
2:02:14and then it turns out for reason that
2:02:16you realize when you do the calculation
2:02:17that you have to go
2:02:19to second order here
2:02:23so you will have a second derivative
2:02:28to keep into account
2:02:34within one knife in front
2:02:39one over n Square
2:02:43and of course you do the same whenever
2:02:45you have a p evaluated at five plus one
2:02:48over n
2:02:51and now what you have to do is to plug
2:02:54these expansions on the right hand side
2:02:59combine all of the terms so you see that
2:03:01also the rates will have a term which is
2:03:04of order one which just depend on Phi
2:03:06and then we'll have terms of order one
2:03:08over n in terms of order 1 over n Square
2:03:12and you have to collect everything to
2:03:14expand your right hand side in in powers
2:03:18of 1 over n
2:03:20now if you do this this is a little bit
2:03:22lengthy but what you realize is that
2:03:24many terms will cancel so in particular
2:03:27the term of order one of order 0 and the
2:03:29term of order one over n will exactly
2:03:31cancel from this expression
2:03:34and so in the end
2:03:36you end up with the following things so
2:03:38let me keep the left hand side
2:03:54which was this
2:03:57and then on the right hand side so if
2:03:59you trust me
2:04:02you will see that you reduce everything
2:04:04to the following so you have a one over
2:04:06n Square
2:04:09times the second derivative
2:04:15of your
2:04:18original rates times now your function
2:04:22evaluated at 3 Tau
2:04:29minus
2:04:33minus
2:04:37nope
2:04:39a term which looks like this so there is
2:04:41a term 1 over n
2:04:46Cylon
2:04:50D over d c
2:04:54one minus 2 Phi
2:04:57times your p
2:05:03okay
2:05:08okay so you have something that on the
2:05:10right hand side
2:05:12has some scaling within which remains
2:05:14non-zero but the scaling has some
2:05:17mismatch so you see that you have a 1
2:05:19over n Square Times what you would like
2:05:21to have quantities of order one in the
2:05:24limit I'm going to Infinity minus 1 over
2:05:26n instead times quantities of order one
2:05:30so in order to get a meaningful right
2:05:32hand side uh limit of this equation
2:05:36I have to do an extra step which
2:05:37explains why I choose this notation
2:05:40Epsilon
2:05:41so I have to assume that Epsilon also
2:05:44scales whenever I look at the
2:05:48discontinuous limit
2:05:50I have to assume that if Cylon also
2:05:52scales like 1 over n so I
2:05:55say that Epsilon is equal to some
2:05:57constant is zero
2:06:01divided by n and in this way e0 is a
2:06:05border one and I match the scaling
2:06:07between these two quantities
2:06:09so why do you do do I do this well if
2:06:12you think about the usual
2:06:14mean field or fully connected things
2:06:17you realize that this is actually the
2:06:20meaningful thing to do
2:06:22and the reason is that uh so usually
2:06:25whenever you have this uh interaction
2:06:28terms
2:06:29and and you have a scaling within you uh
2:06:32you have to rescale with the interaction
2:06:34to have something which remains uh of
2:06:36order one uh in the limit angle is going
2:06:39to Infinity but anyway in here you see
2:06:41it directly from from the calculation
2:06:43that in order for this to term uh to
2:06:45match
2:06:46you uh you assume that Epsilon
2:06:49when n goes to Infinity scales as one
2:06:53over capital n
2:06:55and if I do this then I can replace
2:06:59here so let me erase this
2:07:03I put an upside on and I put an end
2:07:05Square
2:07:08and I have a global factor of 1 over n
2:07:12Square on the right hand side that I
2:07:15simply bring on the left hand side
2:07:17so I will have a BT divided by n Square
2:07:22in here
2:07:24so of course this can be done properly
2:07:27and rigorously but let me just
2:07:29uh give you roughly the idea
2:07:32and so you see that now my right hand
2:07:35side looks like
2:07:37uh a properly well defined Hawker blank
2:07:41equation with a drift term which is a
2:07:43order one and with a diffusion term and
2:07:45my Tau
2:07:48I Define it as some rescaled time with
2:07:52an unknown power
2:07:54and to the minus Alpha
2:07:57and in order to make sense of this
2:08:00equation what I want from the left hand
2:08:02side is that I can rewrite this
2:08:05as the derivative of my P
2:08:12with respect to some to the same Tau
2:08:15variable
2:08:19so the dependence on K is fine you have
2:08:22K on both of these terms so this once
2:08:26every scale will become just a function
2:08:29of Phi I don't have factors of plus
2:08:31minus 1 over n the two care about
2:08:34you see that what I have to do in order
2:08:36to uh
2:08:38for this mapping to make sense is to
2:08:42choose my power
2:08:44Alpha in here
2:08:48in such a way that I can rewrite the
2:08:51derivative over T if you want
2:08:55which is related to the derivative of
2:08:57our Tau by 1 over n to the uh to the
2:09:02alpha I have to choose that I find such
2:09:04a way that I cancel this factor and
2:09:07square
2:09:08in the left hand side of my equation so
2:09:10this is to say if I rescale time
2:09:13or if I introduce a variable of time Tau
2:09:17which is rescaled
2:09:19as t to the N minus 2.
2:09:25which means that my Alpha
2:09:27is equal to 2
2:09:30then all of my scalings work correctly
2:09:34because I add the right hand side which
2:09:35had this factor of n to the two
2:09:39globally so I bring it on the left hand
2:09:42side and I have this new factor of P
2:09:45Over N Square which I just defined it as
2:09:48Tau and then I reproduce on the left
2:09:50hand side the derivative with respect to
2:09:53Tau
2:09:54foreign
2:10:02just to give an idea of how uh one
2:10:05should do the things uh properly so I
2:10:07think that what one has to remember
2:10:11is that you want to
2:10:14assume that you can rescale the
2:10:17arguments of your function and then take
2:10:19the limit and go into infinity and in
2:10:21such a limit you recover a closed
2:10:24equation for for a function which is
2:10:26just a function of your rescaled
2:10:27variables
2:10:28now there is scaling for the density is
2:10:30is very natural so the factor one over n
2:10:33is what we always expect because of
2:10:36extensivity of K and in here there is
2:10:38the you also have to rescale time
2:10:41in a way that you get a well-defined
2:10:43equation and this is something that if
2:10:45you think about it
2:10:47uh it is quite natural because we are
2:10:51assuming that
2:10:52over a time scale DT
2:10:55you have only one event which switch
2:10:58your K by plus or minus one
2:11:02but the equation that you're writing
2:11:04down is actually an equation for
2:11:06variations in Phi
2:11:09that you can think of as K divided by by
2:11:13a factor of n so to have a very small
2:11:15variation of order 1 over n you have to
2:11:18look at times which are which are much
2:11:20much smaller than the ones of your
2:11:22original discrete process so at least
2:11:25you understand why your tau is is uh has
2:11:29to be rescaled by a power which is
2:11:31negative in in capital n and the fact
2:11:33that this is equal to 2 well in this
2:11:35case you
2:11:36it just comes out from the explicit
2:11:40expansion
2:11:41and your calculation
2:11:43okay so anyway this is just to give an
2:11:46idea of how one takes a continuous limit
2:11:49by doing expansions in terms of uh
2:11:53powers of 1 over n
2:11:55and as you see you get out
2:11:58a Planck equation
2:12:00uh that now we are gonna
2:12:03interpret and solve for the stationary
2:12:06state
2:12:07and uh and this is a focal Planck
2:12:10equation that is that you could guess
2:12:14a priori
2:12:17remembering how one derives Factor
2:12:21Planck equation from uh from the
2:12:23language one equation so now let's keep
2:12:25this I will rewrite this expression
2:12:28for the equation and then we
2:12:33actually try to see how you could guess
2:12:35it
2:12:37without going through all of this
2:12:39lengthy expansion
2:12:43and this is actually point two of the
2:12:45exercise
2:12:48so this is very active in time so the
2:12:51drift term
2:12:52you have this uh D over D Phi of
2:12:58Epsilon 0 1 minus two Phi
2:13:04your Phi Tau and then you have the
2:13:06diffusion term
2:13:20foreign
2:13:24which looks like this
2:13:27okay
2:13:48now let's forget about this calculation
2:13:51that I just did and let's assume that I
2:13:54give you
2:13:56this blank equation and I ask you
2:13:58to justify it
2:14:00to justify why it looks like this given
2:14:03what you know about the process in
2:14:05discrete space and in discrete time
2:14:10so this is 0.2
2:14:13and the way you can justify it is
2:14:18by recalling
2:14:20something very that I that I
2:14:24sketched very fast at the LA the end of
2:14:26the last day but which you find so
2:14:28recall
2:14:32the solution of State A3
2:14:35where you have a derivation of focal
2:14:37Planck equation from the lunge of an
2:14:39equation
2:14:40and The crucial point in the derivation
2:14:42is just to remember that you essentially
2:14:45need to compute two quantities which are
2:14:48an average and a variance that are what
2:14:51entering here of your incremental step
2:14:55in an interval DT so these are the
2:14:58quantities that I defined
2:14:59last time as E1 and E2
2:15:04this is if you have a if you have in
2:15:06mind an underlying large of an equation
2:15:08they looked like this
2:15:10so you add an average of your
2:15:12incremental step
2:15:18in the interval DT
2:15:22and that is what I call D1 and then you
2:15:24have the the same thing but with the
2:15:26square
2:15:28which was E2
2:15:35and now this is the transition
2:15:37probability coming from your underlying
2:15:39lunge event
2:15:40equation
2:15:47and if you have these two moments
2:15:49then
2:15:51your focal Planck equation takes the
2:15:55general form so your DP in this case it
2:15:58would be of x and t
2:16:01in our case X is equal to Phi but okay
2:16:07equals what so it was minus D over DX of
2:16:11your first moment E1 of x
2:16:17P of x t and then you add the second
2:16:19derivative
2:16:37and this is something you you find uh if
2:16:40you do the derivation properly
2:16:44and I also mentioned last time that so
2:16:47this form is generic whatever is uh is
2:16:50the prescription for the noise that you
2:16:52take
2:16:53what changes and that will eventually
2:16:55give you different forms of your
2:16:57language of your focal Planck equation
2:17:00is
2:17:01uh is the computation of this i1 so this
2:17:05quantity will be different whether you
2:17:07choose ether versus strathana which but
2:17:10this is not important in here but this
2:17:11is just to connect with uh with what we
2:17:14saw last time
2:17:16so given this intuition that we know
2:17:19what we want to do is essentially to
2:17:22recognize that what you have in here is
2:17:25an average increment and that what you
2:17:26have in here is the average of the
2:17:29square increment and therefore with this
2:17:31argument we can justify uh the form of
2:17:34our soccer plank
2:17:36so how can we do this well we start from
2:17:39our discrete process in terms of K
2:17:45and we use the fact that in the discrete
2:17:48setting
2:17:49the probability
2:17:52that you end up in so you can have only
2:17:55increments by one unit
2:17:58with a probability that
2:18:01was just given by our rates
2:18:12okay
2:18:14and so now the analog of these averages
2:18:17will be averages over this now
2:18:21conditional transition probability
2:18:25and the possible values that your
2:18:27increments can take in the discrete
2:18:29setting are always plus or minus one
2:18:33so this means that so let me first
2:18:35compute what is the average
2:18:39increment Delta k
2:18:43so this is equal to Delta k equals to
2:18:46plus 1 times the probability the Delta K
2:18:49is equal to plus one
2:18:51minus Delta k equals to minus 1 times
2:18:54the probability the Delta K is equal to
2:18:57-1
2:18:59and therefore if you just plug this
2:19:02expression you see that this is just
2:19:05BT times
2:19:09the rate
2:19:11of plus 1 plus the rate
2:19:18minus one
2:19:22now we plug in
2:19:23our expression for the rate
2:19:28and if you do a the algebra this you
2:19:31recover that this is
2:19:33sorry this is yes precisely
2:19:38given by this
2:19:46and I forgot the minus
2:19:52so this comes with a minus because uh
2:19:56should be Delta k equals to -1
2:19:59times the corresponding probability
2:20:04so the term which contains gamma cancels
2:20:07and you just have to sum the terms which
2:20:09contains Epsilon and you get this
2:20:11and if you do the variance in terms of K
2:20:16in this K case this will be just the sum
2:20:19of the two transition rates so instead
2:20:22of a minus you would have a plus
2:20:24and you find that this is equal to DT
2:20:28gamma Phi
2:20:301 minus 5 plus Epsilon
2:20:37so we are almost there
2:20:40this looks almost like what we want
2:20:43uh for our focal plank
2:20:46except one thing
2:20:49meaning that we are now reasoning
2:20:52with respect to K but the increment that
2:20:55we care about are actually with respect
2:20:57to Phi
2:20:59so we have to rescale everything by a
2:21:02factor of n
2:21:05so let me go it
2:21:07do it up here
2:21:17so what would be i1
2:21:22foreign
2:21:27in my notation of before is what is
2:21:32the incrementing K divided by n
2:21:35so this is simply
2:21:38Epsilon 1 minus 2 Phi
2:21:42DT divided by n
2:21:45whereas I2
2:21:53is just this
2:21:56and so I have DT
2:22:00Over N Square
2:22:04gamma Phi
2:22:071 minus 5 plus Epsilon
2:22:11and now again you see that you have
2:22:12these factors of n which appear and in
2:22:15order
2:22:16to have a well-defined process where you
2:22:18have both drift and diffusion
2:22:21you have to kill these extra factors of
2:22:24n by rescaling both Epsilon in here and
2:22:27rescaling the time
2:22:29so this means that
2:22:31the time in which your focal Planck
2:22:33Dynamics will occur will be such that
2:22:35this is a order one
2:22:38and therefore you will define t Over N
2:22:40Square
2:22:41to be a rescaled time variable
2:22:45and if you do this
2:22:47this will be of order and
2:22:53and therefore you have to redefine
2:22:56Epsilon to be some constant Epsilon 0
2:23:00divided by n so you recover
2:23:02your your scaling of time
2:23:05from this shortcut and once you do this
2:23:07then this Epsilon will disappear because
2:23:10it is over the one over n so your drift
2:23:12will only contain this term which is
2:23:14what you see in the focal Planck
2:23:16equation up there and sorry your
2:23:18diffusion and your drift will contain
2:23:20Epsilon zero times one minus two Phi
2:23:23which is what you see up there
2:23:26so this is a way that is a little bit
2:23:28shorter to avoid doing the lengthy
2:23:32expansion or at least to justify
2:23:35why your focal Planck equation takes uh
2:23:38takes that form
2:23:41okay so are there questions here
2:23:45that was the most
2:23:47uh let's say technical part
2:23:51and if not then what remains to do is to
2:23:55to to characterize the model is to try
2:23:57to
2:23:58in principle solve the full focal Planck
2:24:01equation so we are not able to do this
2:24:03but we can look at the stationary State
2:24:05yes
2:24:06foreign
2:24:13yes
2:24:23yes I think that whenever you write
2:24:28um
2:24:29when you start from the master equation
2:24:31you are doing some so we're not starting
2:24:34from a launch event
2:24:36so the the problem is not uh let's say
2:24:39defined from the start because we do not
2:24:42have to specify prescriptions for the
2:24:43noise but we always assume Independence
2:24:46uh in in the label time interval and so
2:24:49this will lead you to anito like
2:24:51equation and indeed the focal Planck
2:24:54equation that you get up there so the
2:24:55question sorry for those who are nine is
2:24:57uh we get a focal Planck equation that
2:25:01is in the Ito formal is if you if you
2:25:03remember
2:25:04so why is it so uh from where is it
2:25:08encoded from our derivation and and this
2:25:11is embedded in the fact that we are
2:25:12assuming uh somehow that the processes
2:25:15that occur in the event uh DT are
2:25:18independent and so this corresponds to
2:25:19data
2:25:21but this is a good observation so if you
2:25:23what would be the form uh for uh
2:25:27stratonovich
2:25:28focal Planck equation
2:25:32so if you remember
2:25:34you would just have
2:25:37foreign
2:25:39which you have one derivative than one
2:25:42factors of of G which is now the square
2:25:46root of this and then you have the
2:25:48derivative and then the square root of
2:25:50the times p
2:25:51and this is nice because now we are
2:25:53going to solve for the stationary State
2:25:55and we will find a solution that
2:25:57corresponds indeed to The Ether solution
2:26:00and if you have a different form if you
2:26:02use the satanovic the the form of the
2:26:04stationary state will change so you can
2:26:06try to do this as an exercise
2:26:12okay so let's now try to compute this
2:26:14stationary State and then we conclude
2:26:18uh there was maybe a comment before yes
2:26:28so before or while I raise
2:26:31just one comment on the form of the
2:26:33equation so you see
2:26:36you have the drift term and the noise
2:26:38term and they somehow they
2:26:41push you in different directions
2:26:45so the drift term
2:26:47vanishes when Phi is equal to one half
2:26:51so one alpha means that you have
2:26:53polish population of type A and half
2:26:56relation of time type B
2:27:00so the brief term is somehow push is
2:27:02pushing you towards one out if you are
2:27:05away from one else you have some kind of
2:27:07force that pushes you to reach one half
2:27:11whereas the noise term
2:27:13is pushing you away from one alph
2:27:15because the variance of the noise is
2:27:17maximal at one half so the noise wants
2:27:19you if you are sitting at one half it
2:27:22tends to uh with with this random
2:27:24fluctuations to put you push you away
2:27:28uh from the middle towards the extrema
2:27:32where the noise is zero which are
2:27:34Phi equal to one or two zero which means
2:27:36that all of the population is either in
2:27:38state a
2:27:39or in state B so there is a competition
2:27:42between these two terms and so we expect
2:27:45to see this competition also in the
2:27:48stationary State and indeed we will have
2:27:50a parameter Alpha
2:27:54on which the stationary State depends
2:27:57which is just the ratio of the two
2:27:59with a factor of two so let me Define it
2:28:01properly
2:28:03um
2:28:05yes
2:28:08so depending on whether Alpha is larger
2:28:10or smaller than one you will have
2:28:12different forms for your stationary
2:28:14solution
2:28:16but let's compute them and then we are
2:28:18done so this was 0.3
2:28:22foreign
2:28:23so what it means to compute the
2:28:26stationary distribution so the
2:28:28stationary distribution does not depend
2:28:30on time it satisfies
2:28:32the DP
2:28:35stationary
2:28:38you have to set to zero uh the left hand
2:28:41side
2:28:43so actually whenever this is true you
2:28:45have a
2:28:46stationary distribution which depends on
2:28:49your own fight
2:28:51and now if I set to zero the left hand
2:28:54side let me look at the right hand side
2:28:57so the right hand side contains uh many
2:29:00derivatives and what I can do is
2:29:03so the right hand side
2:29:06I can write it as a global derivative
2:29:11maybe with a minus
2:29:13of something that I can call a current
2:29:16so what is my current so here
2:29:20is 1 minus 2 pi times p
2:29:26minus one half
2:29:28B over DC
2:29:32of gamma Phi
2:29:351 minus 5 times p
2:29:44so if I set the if I set the left hand
2:29:47side to zero then I'm saying that this
2:29:49quantity has to be equal to zero which
2:29:52means that whatever is in parenthesis
2:29:55which just depends on Phi
2:29:59it's derivative with respect to Phi has
2:30:01to be equal to zero so this quantity
2:30:04has to be equal to a constant
2:30:07okay
2:30:09when I evaluate it at the stationary
2:30:12distribution
2:30:17because in that case this derivative is
2:30:19zero
2:30:21now what is this constant well
2:30:25since this constant dip is uh so you
2:30:27have the pin here
2:30:29and you don't want a dependence on Phi
2:30:32so what you can argue is that you should
2:30:34actually choose this constant J to be
2:30:37itself equal to zero
2:30:41and the reason is that if this was not
2:30:43the case then you will uh so you would
2:30:47get a distribution that is uh that is
2:30:49not normalizable so what you want
2:30:52is that P when Phi goes to
2:30:55uh well here we have a bounded intervals
2:30:58but in general if you think about
2:31:00unbounded distributions you want to that
2:31:03pdks such that you are normalized
2:31:06and in order for p to Decay such that
2:31:08you're normalized it must go to zero
2:31:10when the argument goes to Infinity but
2:31:13if it has to be constant that this means
2:31:15that P itself uh or whatever or the
2:31:18combination of p and its derivative has
2:31:20to be equal to uh to zero so you can
2:31:23argue in general that because of
2:31:26normalization you can set your current
2:31:29equal to zero or if you want you just
2:31:32assume that this is zero and you find
2:31:34the stationary solution
2:31:36that satisfies your equation in which by
2:31:39definition uh
2:31:42will be stationary
2:31:44so let's assume that our current is zero
2:31:47so this now gives
2:31:49a linear differential equation for our p
2:31:53did I rewrite and that we can solve in
2:31:56three steps
2:31:59so we have
2:32:010 1 minus two High
2:32:05stationary
2:32:07Pi minus one alpha
2:32:10uh
2:32:12Define
2:32:15gamma Phi 1 minus 5 is stationary of
2:32:20five
2:32:24this is b equal to zero
2:32:29and now we solve this equation in three
2:32:32steps
2:32:34so the first step or at least I find it
2:32:36easier to solve it in three steps so the
2:32:38first step is to do some rescaling
2:32:42so you see I have a linear equation but
2:32:44here I have Phi which appears everywhere
2:32:47so let me Define this
2:32:49foreign
2:32:54so that I have the derivative of P tilde
2:32:56which is easier to handle
2:33:00so I introduce
2:33:02such that
2:33:04this is gamma Phi
2:33:061 minus 5 is stationary
2:33:12and then I write the equation for
2:33:13pitilda so I will have 0 1 minus two Phi
2:33:19the MP stationary is
2:33:23the killer Phi divided by gamma
2:33:27PSI 1 minus Phi
2:33:30and on the right hand side I just have a
2:33:32factor of one half so I bring the two up
2:33:35here
2:33:37and I just have B
2:33:40over the side of my PT love Phi
2:33:45okay
2:33:48and now once I did this rescaling I can
2:33:52do
2:33:53something very easy which always
2:33:56comes out from the structure of this
2:33:59equation which is that I can integrate
2:34:01this equation with separation of
2:34:03variables
2:34:17so in a natural separation of variables
2:34:19the seen examples in the lecture already
2:34:21means that I
2:34:23bring on the right hand side all the
2:34:25terms which depend on pitilda and I live
2:34:28on the left hand side all the terms
2:34:30which depend on Phi or if you want that
2:34:33I rewrite this equation in differential
2:34:35form
2:34:40so if you are sloppy it means that you
2:34:42take this Diffie and you bring it on the
2:34:44other side
2:34:45but in the financial form you see that
2:34:48this is equal to
2:34:502 Epsilon 0 over gamma
2:34:54one minus two five five one minus five
2:34:59times DC
2:35:03so everything you have on the right
2:35:05depends on fee everything you have on
2:35:07the left depends on P tilde
2:35:11and so now what you can do is to
2:35:13integrate
2:35:14both terms
2:35:17so you integrate the right hand side
2:35:19from sum Pi zero to some Phi
2:35:22well let me call it five tilde
2:35:26so that I integrate up to five
2:35:29and the left hand side you integrate
2:35:31from the corresponding value of the
2:35:33distribution at Phi 0 and the one
2:35:37fine
2:35:39okay
2:35:42and now these are integrals that you
2:35:44know how to do so on the left hand side
2:35:46you have a logarithm because you have
2:35:48one over p
2:35:50so you will get log of
2:35:54field of Phi divided by
2:35:5715.50
2:36:01and on the right hand side we use
2:36:04exactly the same trick
2:36:06that I mentioned in the first exercise
2:36:10so you have a quadratic expression
2:36:13inside tilde in the denominator I want
2:36:15to split it as we did
2:36:17with the X1 X2 and here it's easy so
2:36:21these you can write it I think as
2:36:26simply this
2:36:32you see that this matches exactly
2:36:35and then you can integrate so these are
2:36:37just logarithms
2:36:39so if you integrate you have two apps
2:36:42sign on zero gamma
2:36:45and then you have the log of
2:36:49I over Phi 0
2:36:53minus but then the minus is already
2:36:57is inside the derivative so this is the
2:37:00derivative
2:37:01of the log of 1 minus 5
2:37:04so here I have a plus
2:37:06log of 1 minus 5 divided by 1 minus 5 0.
2:37:20and then while Phi 0 is arbitrary
2:37:23so let me rewrite this as 2 Epsilon 0
2:37:26gamma so this was the alpha that I
2:37:28introduced before
2:37:31then you have log of
2:37:33Phi 1 minus 5 divided by Phi 0
2:37:391 minus 5 0.
2:37:41and then I can exponentiate right so if
2:37:44I expunish it I get
2:37:47my petilla
2:38:05so PT La Phi will be some constant which
2:38:08depends on this arbitrary Phi zero
2:38:11which I just called C So eventually all
2:38:13of the constants I will fix them by
2:38:15normalization so I don't need to
2:38:17keep track of what is the constant at
2:38:20this stage and then I have Phi 1 minus
2:38:22Phi
2:38:26to the power of 2 Epsilon 0.
2:38:30over gamma
2:38:34but now I have to remember that this is
2:38:36very scaled so this was tilde
2:38:40so from here what is
2:38:42I rescale back if you want my stationary
2:38:46F5
2:38:48is a is what is 1 over gamma which
2:38:53is into my constant so I have a new
2:38:55constant C Prime
2:38:57and then I have a factor of 1 over 5 1
2:39:00minus 5 in the denominator so this would
2:39:02be
2:39:05I will write it like this 5 1 minus 5
2:39:09to the 1 minus
2:39:11to Epsilon zero
2:39:14over gamma
2:39:18so all of the constants
2:39:20which will not did not depend on Phi I
2:39:23encoded them in C Prime
2:39:26and the reason why I don't need to care
2:39:28about those is that I can fix the global
2:39:30constant Now by simply normalizing my
2:39:34distribution so this is Step C
2:39:41which is normalization
2:39:46and this is a chance for me just to to
2:39:48introduce
2:39:49one identity that maybe is useful to
2:39:52remember which is a gamma function but
2:39:55anyway what I need
2:39:56is that the integral
2:39:59from 0 to 1 of my Phi
2:40:03of my distribution
2:40:05is equal to one and this is C Prime
2:40:07integral from zero to one
2:40:12of one over so actually now I will write
2:40:16it 5.
2:40:181 minus 5 to the
2:40:21two Epsilon over gamma minus one in defy
2:40:30okay so c will be the inverse of this
2:40:32inter integral
2:40:35and the reason why I'm writing it
2:40:37explicitly is just to give you
2:40:40something that you know so
2:40:43this is an integral that is well known
2:40:46in the literature it is related to the
2:40:48so-called beta function
2:40:50which in turns is related to the gamma
2:40:53function so the gamma function
2:40:55is a function which is defined in terms
2:40:58of the following integral representation
2:41:02T of x minus 1 e to the minus t
2:41:07and the beta function so anytime you see
2:41:09Phi x 1 minus X integrated between 0 and
2:41:131.
2:41:14it is almost always related to to this
2:41:19beta function in here
2:41:22which is precisely this t to the x minus
2:41:251 1 minus t to the
2:41:28y minus 1 in DT
2:41:31and this is related to the gamma
2:41:33function so you can show these are
2:41:34identities
2:41:36this is gamma of X gamma of Y divided by
2:41:39gamma
2:41:41of X Plus y
2:41:45so using this you can compute
2:41:48the normalization so you can find that c
2:41:50Prime
2:41:54is nothing but gamma of four
2:41:57Epsilon zero
2:42:00over a small gamma divided by gamma of
2:42:04two Epsilon zero over gamma
2:42:07so typically we don't care about the
2:42:09normalization here I want to write it
2:42:11down because I want to take now a limit
2:42:13and discuss what happens when Epsilon 0
2:42:16goes to zero
2:42:19okay
2:42:22so these are General identities
2:42:25and what we also need is that
2:42:30when X goes to zero your gamma function
2:42:34is is Divergent so you you have
2:42:39um a Divergence that you can compute
2:42:41explicitly so you have an asymptotic
2:42:44expansion when X
2:42:45goes to zero which tells you that you
2:42:47have a pole which goes like 1 over X
2:42:50and then you have something of order one
2:42:56okay now we use this just to comment a
2:42:59little bit on the structure of the
2:43:01solution
2:43:05that we obtain and with this we finish
2:43:08I think
2:43:09foreign
2:43:24so it's maybe two comments that are
2:43:27related to point to the final point
2:43:37so the first one is
2:43:40that as we mentioned you have this
2:43:42competition between Epsilon zero and
2:43:45gamma
2:43:45and you see that this emerges
2:43:48in this combination in this ratio in the
2:43:51distribution that you have so you have
2:43:53Alpha
2:43:55website on zero over gamma
2:43:59and you have two situations for your
2:44:01stationary distribution so
2:44:08this goes from zero to one
2:44:13so if Alpha is larger than one
2:44:18then your distribution
2:44:20is proportional to Phi 1 minus five so
2:44:25let me rewrite it so be stationary
2:44:29equals like Phi 1 minus five to the
2:44:341 minus uh
2:44:36Alpha minus one
2:44:40so you see if Alpha is larger than one
2:44:42the exponent is positive so you will
2:44:44have something which vanishes at zero
2:44:46and one and which is symmetric
2:44:49with respect to one half and as a
2:44:51maximum at one half so you expect
2:44:53something like this
2:44:59which means that if you think about your
2:45:01animals half of those will likely go to
2:45:05the source a half of those will likely
2:45:07go to the source B and they will be more
2:45:10or less evenly distributed but if Alpha
2:45:13becomes smaller than one then your
2:45:15distribution changes because now this
2:45:17product goes into the denominator and so
2:45:20you have a Divergence
2:45:22at zero
2:45:24and at one so your CC metric
2:45:27but what I should use colors Maybe
2:45:33you have something which
2:45:35looks more like this for Alpha smaller
2:45:37than one
2:45:41and so this is the regime where you tend
2:45:44to have uh
2:45:46a very opinionated population if you
2:45:48want so your population is either
2:45:50strongly in favor of of the choice B
2:45:53which corresponds to Phi equal to zero
2:45:56or even strongly in favor of the choice
2:45:58a which correspond to Phi equal to equal
2:46:02to one and this is because the term
2:46:05which corresponds to the noise which
2:46:08pushes you toward this extrema is
2:46:10winning with respect to the term which
2:46:13corresponds to the drift that was given
2:46:16by this spontaneous
2:46:19change of opinions of of your population
2:46:23and with all of the Expressions that we
2:46:26derived you can even ask what happens in
2:46:28the limit so this is the second comment
2:46:35you can now take
2:46:37the limit Epsilon going to zero
2:46:42and in this limit so Alpha is going to
2:46:46uh to zero and you get
2:46:49so as you approach this limit
2:46:54so should I do this
2:46:58yes so with the expansion that I told
2:47:00you up there
2:47:03what you find from the normalization
2:47:05constant is that
2:47:08when Epsilon 0 is much smaller than one
2:47:11this is going like one over Epsilon zero
2:47:14I think
2:47:18sorry this is going like
2:47:21Epsilon 0
2:47:26. so you see from this expression you
2:47:28have a coefficient in front which goes
2:47:30to zero when you send Epsilon 0 to 0 and
2:47:33you have an exponent which is going to
2:47:35zero so you have a term which is one
2:47:37over five one minus pi times something
2:47:39which is going to zero so this means
2:47:41that as long as Phi is different from 0
2:47:441
2:47:45you are regular with this term and so
2:47:48your distribution is going exactly to
2:47:49zero in this limit except at the points
2:47:52where Phi is equal to zero and Phi is
2:47:54equal to one where you have a pole and
2:47:56so your distribution is exploding so in
2:47:58the limit if you picture this if you
2:48:01make Epsilon 0 smaller and smaller you
2:48:04start having distributions which become
2:48:06more and more flat towards zero but then
2:48:09which explode uh at zero one so this
2:48:12means that your Phi stationary
2:48:15in the limit
2:48:16Epsilon going to zero converges to a
2:48:19Delta
2:48:20where all of your population
2:48:24is either
2:48:25in zero or it is either in one
2:48:28with half an hour's probability so if
2:48:31you average over time you will find or
2:48:34over realizations you will find
2:48:36that in half of your realization all of
2:48:38your population chooses a in half of
2:48:40your realizations all of your population
2:48:42chooses B but they are somehow all
2:48:45agreeing and there is not a well-defined
2:48:49stationary distribution as you would get
2:48:51uh
2:48:53for
2:48:54um in in the regime of parameters where
2:48:57Alpha is non-zero
2:48:59okay so this concludes and maybe
2:49:02just the final comment
2:49:05to comment on the first exercise
2:49:09so for this type of model you can even
2:49:11compute when Epsilon is different from
2:49:14zero so this
2:49:17says size three
2:49:20when Epsilon is different from zero but
2:49:21find it you still have let's say that
2:49:24Alpha is smaller than one you still have
2:49:25this distribution which is very
2:49:27polarized
2:49:30concentrating on zero and one
2:49:34but in that case in in the Dynamics you
2:49:36can still have
2:49:38let's say uh
2:49:40it is in this case if you start from
2:49:42zero from one you speak to Zero part one
2:49:45these are absorbing states of your
2:49:49Dynamics in the case Epsilon 0 is is
2:49:52positive you can still go from having
2:49:55all of the population concentrated in
2:49:56zero to having all of that concentrating
2:49:59in one and you can compute what is the
2:50:02switching time between these two uh
2:50:06choices and you find that this is of the
2:50:08order of one over two
2:50:10uh Epsilon zero
2:50:12so this we are not going to compute
2:50:15but this is in a way
2:50:18a little bit similar to this jump
2:50:20calculation of the jump times that we
2:50:23did in the first exercise and the
2:50:25interesting thing is that this depends
2:50:27only on Epsilon zero
2:50:29and it does not depend on gamma
2:50:32so on on the recruitment on the
2:50:34discussion between your agents so in in
2:50:36that situation which Epsilon 0 is small
2:50:39you really have that one of your ants or
2:50:43animals or or people changes its mind
2:50:45and then you trigger a full Avalanche
2:50:48where everybody else follows and you
2:50:50jump from this situation to uh to this
2:50:53situation so the Dynamics is is really
2:50:55abrupt and this is related or similar to
2:50:59avalanches
2:51:03foreign in many other systems that are
2:51:07complex
2:51:09and with many agents
2:51:13okay so I think that's enough
2:51:17and the remaining exercise is as I said
2:51:20it's a little bit different it goes back
2:51:21to statistics but it's a nice way to see
2:51:26an application of brown and motion so if
2:51:28you have time have a look at that and
2:51:30there is the arm work related to that
2:51:32which is homework 8.
2:51:36um and that's it so next time we will do
2:51:39the final today I think
2:51:44so is there any question otherwise
2:51:51I will stop

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