13 - SN Partial Differential Equations and Applications Webinars
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Join Pierre-Emmanuel Jabin of Penn State as he discusses large stochastic many-particle or multi-agent systems. Don’t want to miss another webinar? Register here: https://go.sn.pub/lXgEaxHave an article on stochastic systems? Submit Your Article: https://go.sn.pub/GT525VLearn more about publishing with SN Partial Differential Equations and Applications: https://go.sn.pub/UFulhi
00:05
uh
00:06
of today uh pierre manuel jaban
00:09
professor of mathematics at the
00:11
university of
00:12
maryland um sorry so so so i moved i
00:16
i should have seen that i moved this
00:18
summer i mean
00:20
oh thank you thank you for letting me
00:22
know where where are you now uh penn
00:24
state
00:25
in penn state uh okay okay no not very
00:27
far away
00:28
not very far away okay okay so he's now
00:31
at the universe penn state university
00:34
awesome about no further ado i passed
00:37
the floor to you
00:38
thank you again for joining us today
00:42
thank you very much uh
00:45
and thank you so much for the for the
00:48
kind invitation
00:50
um so what i want to do today
00:54
is i want to talk about a relatively
00:58
recent result
01:00
that we obtained with what we do at the
01:03
beginning
01:04
with dj brash and my former student
01:06
zenfone
01:08
and which consists in the the derivation
01:11
of the mean field limit for the keller
01:13
segal
01:13
uh system uh
01:17
now the actual result is of course
01:19
relatively technical
01:21
so what i i will do is
01:24
give a very broad overview and a broad
01:26
introduction
01:28
of uh what all this is about
01:31
um why we're interested in the mean
01:33
field limit
01:34
and and things like that and then i'll
01:37
give you
01:37
some uh technical hints of what's going
01:41
on
01:42
but of course relatively limited so that
01:44
i i really want to keep the the top
01:46
broadly accessible
01:48
um
01:52
so system of of interacting particles um
01:56
nowadays show up in a lot of different
01:59
settings
02:00
and i've just tried to gather a few of
02:02
them here i'm not going to go through
02:04
the list in details
02:06
uh just to give you the sort of idea of
02:10
uh the various things that that
02:12
particles can
02:13
can mean and you know you can start from
02:16
the
02:17
um classical systems and and plasma is
02:20
the
02:20
first one that comes to mind because
02:23
this
02:23
was uh the setting where the midfield
02:26
limit was first derived
02:27
formally by blazoff in the 30s um
02:32
but of course still in physics
02:34
astrophysics is another very classical
02:36
example
02:37
um newton's equations
02:41
for points uh masses probably
02:44
the first system uh of interacting ode
02:47
is ever to be written
02:50
and it's still used nowadays because for
02:54
the study of galaxy formations for
02:55
instance at large scale
02:58
the universe is mostly flat and when you
03:00
study concentration of dark matter
03:03
dark matter particles are supposed to be
03:05
interacting essentially only through
03:06
gravitation
03:08
so it's a very common model
03:11
of course you have other examples in
03:13
physics as in fluid and i will actually
03:16
go back to that later um but as i was
03:20
saying
03:20
um nowadays this has really
03:23
spread in in various other fields
03:27
from the biosciences you know aerosols
03:31
microorganisms onto economics social
03:34
sciences
03:36
you have opinion dynamics consensus
03:37
formation
03:39
swarming flocking um
03:42
all all things like that
03:46
and just to uh
03:49
give two extreme examples of such
03:51
systems so this is
03:53
uh an actual recent simulation you know
03:56
formation of large scale in the in the
03:58
universe um
04:00
and this is uh the dynamics of galaxies
04:05
and the color indicate the density with
04:08
as as usual
04:09
red indicating a higher density of blue
04:12
a lower one and the small yellow
04:17
lines indicate the velocity of the
04:19
galaxies
04:23
and at the other um end of the spectrum
04:26
both in terms of applications so it's
04:28
biology instead of physics
04:30
and it's very small scale instead of
04:32
very large scales
04:34
you have neural networks
04:39
and so here what uh
04:42
what you see is the cytoskeleton of the
04:45
neuron
04:46
but this allows to to observe relatively
04:49
well
04:50
on both the the the center of the neuron
04:52
and then the axons that connect them
05:00
now before going into the mean field
05:01
limit this is a slide that if you've
05:03
seen
05:04
me give a talk on infield limits this is
05:06
the one slide that is shared by sort of
05:08
all my talks
05:10
[Music]
05:12
because it gives a scale of what we mean
05:14
by a limit
05:17
so basically the mean for limit
05:20
consistent saying what happens when the
05:22
number of particles let's call that
05:24
n becomes infinite
05:28
but that's a bit of a mathematical
05:30
formulation because in practice of
05:32
course you
05:32
have a fixed number which can be very
05:36
very large
05:37
but how large is it and so how close are
05:40
we to infinity
05:42
in in various problems
05:45
and of course if you're thinking of
05:47
plasma for example then
05:50
order of magnitude is the avogadro
05:52
number so
05:54
around 10 to the 24 right so branches
05:57
10 to the 20 if it's very dilute to 10
06:00
to the 25
06:01
if it's more dense as a plasma
06:04
uh and as an order of magnitude for
06:06
physical setting it's it's
06:09
not uh it gives you the right of
06:12
the right figures um but again as i said
06:16
this is not the only use
06:18
and if you're now thinking microorganism
06:21
you're much lower um
06:26
so how large is a bacterial population
06:30
and for example and and here it's useful
06:35
to think of it between 10 to the 6 10 to
06:37
12 right if you have a bacterial
06:39
infection in your body for
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um for instance my usual joke is that 10
06:45
to the 6 you
06:47
cannot detect the infection uh by taking
06:49
blood sample for example
06:52
and and 10 to 12 you're likely dead or
06:55
dying
06:55
so that's sort of a range
07:00
this is also roughly uh the
07:04
range of number of particles that you
07:07
can
07:07
actually put in the computer in the
07:10
numerical solder
07:11
all right so if i want to calculate the
07:14
solutions to a couple system of ods
07:17
um how many particles can i
07:20
can i use well if you do things poorly
07:23
uh probably a lot less like that but if
07:25
you know a little bit what you're doing
07:28
uh you're using some fast multiple
07:30
methods for example then you can ramp
07:32
things up quite a bit
07:34
um the best simulation i have seen so
07:38
far
07:39
was using 10 to the 14 but that's weeks
07:42
of calculations on the on the super
07:44
calculator
07:46
and in quite a few other settings the
07:49
numbers can even be lower
07:53
if it's collective dynamics or swarming
07:56
i mean how many animals are we talking
07:58
about
07:58
maybe a few thousand you know probably
08:01
not that much more
08:04
so when when you look at that ranch
08:07
right
08:07
um it's relatively clear that there is a
08:11
bit of a danger
08:12
in talking about the limit n tends to
08:14
infinity
08:18
because why we believe that
08:21
relatively well if there's 10 to 25
08:24
right
08:25
it does raise the question well
08:28
how true is it for 10 to 10
08:32
how true is it for 10 to before and to
08:35
firm
08:39
um and so the big point here
08:42
is to say that whenever you can
08:46
sometimes you cannot but whenever you
08:48
can having quantitative convergence
08:50
result is so much better
08:52
because then for a fixed end it will
08:54
tell you precisely how close you are to
08:57
the limit
08:58
and then it's your decision to say uh
09:01
whether the error
09:02
is acceptable or if it's not
09:09
um so what kind of couple
09:12
um od or uh sd
09:15
as uh we will see in a few slide systems
09:18
i'm talking about here
09:20
um so let's
09:23
take n particles
09:26
identical uh or
09:29
exchangeable uh as as probabilist would
09:32
say
09:33
uh now this is very important um
09:37
assumption it doesn't mean that the
09:38
trajectory of each particle is going to
09:40
be
09:40
the same because they may start from
09:42
different places
09:43
but it means that if i swap them
09:46
then the system remains the same so they
09:50
have the exact same properties okay if
09:53
i'm considering plasmas then
09:55
all the ions in the plasmas have the
09:57
same mass and the same charges for
09:59
example
10:01
you can do multi species but i'm not
10:04
going to go into that during this talk
10:07
um so i will represent all i need to
10:11
know about each particle by a given
10:13
vector in finite dimension and
10:15
i'm going to call that x i so the kind
10:18
of things that we put into excise the
10:20
position of the particles sometimes the
10:22
velocity
10:23
sometimes you add some internal state
10:25
right could be
10:26
a rotational velocity for example some
10:29
internal energy
10:30
there are lots of systems and
10:34
if the particles interact two by two
10:36
then what you get
10:37
is is pretty much the equation that is
10:39
written on that slide
10:43
um so you can have self interactions
10:45
that's with this term capital f
10:47
of x i and then you have a sum of two by
10:50
two interactions that is described by
10:52
some kernel k
10:55
um and of course the whole modeling
10:57
question is how we would choose fk
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the initial data things like that
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um but one important thing uh during all
11:07
my talk
11:08
uh that is the second thing i want to
11:11
insist on
11:11
in this slide is this one over n that
11:14
you see in front of the sum
11:16
and this is what is called the mean
11:18
field scale
11:19
now its effect is very simple because uh
11:22
i have a sum of n terms right
11:24
and i'm divided by n so formally what
11:28
that makes it
11:29
makes my sum of order one
11:32
and i usually describe that
11:36
as uh the simplest interesting scaling
11:40
in such a system it is not the only
11:44
interesting scaling right there is no
11:46
particular reason that once you put the
11:48
physical constant and everything in
11:50
uh and your scale time and maybe your
11:52
scale position velocities
11:54
there is no particular reason why you
11:56
should end up with a one over n
11:57
in front of that sum you will end up
12:00
with some constant
12:02
um but if that constant
12:05
is a lot less than one over n
12:10
then you don't get something that is
12:12
very interesting
12:14
because as n becomes larger then you
12:17
expect the interaction to go to zero
12:20
and that's not a lot of fun on the other
12:23
hand
12:24
if the constant in front of that is much
12:25
larger than one over n
12:27
then you probably have something
12:29
relatively difficult to deal with
12:33
because now the sum of priority could be
12:35
very very large
12:37
and so what that means is there are
12:41
probably some pretty drastic constraints
12:45
going on in the system so that you know
12:47
you have the right configuration of
12:49
positions
12:50
position velocities things like that
12:53
so that the uh the interactions have
12:56
cancellations
12:58
and this is a lot harder
13:03
and of course you know
13:06
another example of a multi-agent system
13:09
many particle system
13:11
that has a much harder scaling is the
13:14
one that leads to boltzmann and
13:16
collisional models
13:19
oh so here i will focus on that one
13:21
which again is
13:23
far simpler but still gives an
13:25
interesting limit
13:28
um so let's go to the first classical
13:30
example right which i had mentioned
13:32
which is the the
13:34
newton dynamics of point masses so i
13:37
consider now n
13:38
uh point masses so point means that i
13:41
neglect the size of each mass right
13:43
in gravitational interaction so what
13:46
what do i need to track i need to track
13:48
your position qi
13:49
and their velocity vi
13:54
and i just apply newton's law so on
13:57
time derivative the position is the
13:59
velocity time derivative of the
14:01
is the acceleration which here is the
14:03
sum of the
14:04
gravitational interaction between
14:08
between all particles and this is the
14:11
model that was solved in the first
14:13
picture that i showed you for example
14:17
uh it's also a model for which we still
14:20
have no idea
14:21
of what whether the mean field limit
14:23
reverse little holes almost so
14:25
it's kind of still very very important
14:29
um but in this talk uh what i will
14:32
typically consider
14:34
are a couple system of stochastic
14:36
differential equations
14:38
and the logic here is that i add a noise
14:42
to the dynamics right and now
14:45
it's like that so i have this plus a
14:48
square root of two sigma dwi
14:51
i put square root of two sigma because i
14:53
will move
14:55
relatively fast to pd formulation and
14:58
i'd rather have sigma than sigma square
15:00
over two with my
15:02
in my equations and in my formula this
15:04
is just a question of notation
15:06
uh and the wi are just
15:09
grown in motions or linear processes
15:12
that are independent
15:14
and there are several cases where you
15:18
want to
15:18
add noise like this um
15:22
it can be because there's uh some
15:25
well some sort of actually random
15:28
effects in in your model
15:30
and that can be random collisions for
15:33
instance
15:34
uh in the gravitational settings um
15:37
this could be to try to model some sort
15:40
of random
15:41
behavior of microorganism the run and
15:44
tumble
15:44
of bacteria for example it's not a great
15:47
model for that but
15:49
it's at least something that tries to do
15:51
the job
15:53
it can also represent some um
15:56
uncertainty in the system and so because
15:59
you're not very sure of what's going on
16:01
you just add noise in general
16:03
uh to represent this sort of
16:06
uh type of other behaviors
16:09
that can happen now there are many ways
16:12
of adding noise you can multiply it
16:14
multiplicative noise uh you can use
16:17
other type of noise
16:18
and then um that growing in motion
16:22
uh again here it's a little bit the
16:24
simplest case
16:27
uh note as well that here i'm i'm fixing
16:30
the size of the diffusion
16:31
it's also possible and you have quite a
16:33
few applications when you want to
16:35
rescale the diffusion with a number of
16:37
particles
16:39
and for simplicity i'm going to look at
16:42
the noise that is non-degenerate
16:44
um in some models it is degenerate and i
16:48
i mentioned random collisions on
16:50
gravitational particles
16:52
random collisions only change the
16:54
velocity right so then you would only
16:56
put the noise in the velocity
17:02
um before going to the mean field limit
17:05
uh the the first um
17:09
question i want to discuss here a little
17:11
bit briefly is which kind
17:14
of initial distribution we want to put
17:17
on the x i zero
17:20
now in in many cases when you have a
17:23
time evolution system
17:25
uh there's an easy way to to
17:30
to not have to make any decision on the
17:32
initial data which is to say well
17:36
if we are trying to solve for an
17:38
experiment then
17:39
the initial data is whatever the
17:41
experiment provides
17:44
um and unfortunately unfortunately here
17:48
that's not going to to work
17:51
because there's simply no way that
17:54
anyone is going to give you
17:57
the positions of of 10 to 20 particles
18:00
with
18:01
any kind of reasonable precision
18:07
so this is out of a question there's no
18:09
way to measure it
18:11
even very roughly there is way to
18:13
measure it precisely on top of that
18:16
so this is doomed to fail
18:19
so we do have to make a decision on
18:22
the the initial distribution
18:25
and what people typically do then as
18:28
they suppose that the initial
18:29
distributions
18:30
are iid or very close to being iid
18:36
and this is called chaos and this
18:38
follows the notion of molecular chaos
18:40
that was introduced by boseman
18:42
in the mid 19th century
18:46
um the logic to this
18:49
is not only that we are making our life
18:52
simpler
18:52
by doing something like that um
18:55
[Music]
18:57
a way to justify that for example is to
19:00
say that those initial distributions if
19:02
you're looking at an experiment
19:04
are themselves the result of a long-time
19:07
evolution
19:08
of some other dynamics
19:11
the dynamics is whatever was used to
19:13
prepare the experiment
19:16
and if you assume ergodicity
19:20
in that um first dynamics for example
19:24
then yes the longtime
19:27
gibbs equilibrium will look
19:30
somewhat like something that is iid or
19:33
close to being my id
19:41
now the idea of the mutual approximation
19:43
is
19:44
on paper relatively simple uh you have
19:47
this discrete
19:48
force field which is a sum of two by two
19:50
interaction
19:52
and we are going to replace that by
19:55
its expectation and then
19:58
instead of the original uh coupled
20:01
system of sde
20:03
now i have a different coupled system
20:04
which is called the machine lazov
20:06
process and which is the second equation
20:10
written here
20:12
the difference of course as you see when
20:14
i write it like that is just that as
20:15
i have this expectation that is taken
20:18
before all the interactions
20:21
and it's a conditional expectation right
20:23
it's a conditional expectation
20:25
knowing the position of the particles
20:28
whose trajectory i'm trying to calculate
20:32
now this is not a couple system of sde
20:34
anymore because if i want to solve that
20:37
i need to find the um
20:41
joint law of uh
20:44
particle one particle two and so on till
20:47
particle
20:47
n knowing
20:51
particle i
20:54
so um again this is a different type of
20:57
system
21:01
now the first question is why something
21:03
like that would be true
21:05
and the second question is why would it
21:07
be useful
21:11
um i'm replacing a
21:15
discrete sum of many many terms by
21:18
its average so of course i'm thinking
21:21
low
21:21
large numbers and it would be exactly
21:26
those large number if i knew that the x
21:28
i
21:29
are independent
21:32
not necessarily id but at least
21:34
independent
21:36
um but you know just
21:40
two slides before i chose them initially
21:43
to be independent
21:46
of course the problem is even if they
21:48
are actually rigorously independent at
21:51
the initial time there is no way that
21:53
they remain independent
21:54
as time goes by
21:58
and that's because the dynamics itself
22:00
will correlate with positions
22:03
there's nothing you can do about that
22:04
that's what it is
22:07
so this leads to a relatively subtle
22:10
notion
22:11
which is usually called propagation of
22:13
chaos
22:15
and propagation of chaos is the
22:17
conjecture that if
22:18
n is large enough then they might not be
22:21
strictly independent but they are almost
22:24
independent in some sense
22:30
this is very fuzzy as i formulated like
22:33
that
22:34
but for a good reason there are various
22:36
definitions of propagation of chaos
22:38
depending how you want to measure
22:41
this independence and
22:44
i'll show you in the second part of the
22:45
talk i'll show you the notion that we
22:48
are using to get the result
22:52
um
22:56
second question is why is it useful
23:00
well it's useful because it turns out
23:01
that this making blastoff process
23:03
can be greatly reduced
23:07
and if we show propagation of chaos so
23:11
if we can justify this
23:12
right and we get to that uh
23:16
but now uh if i take the same initial
23:18
data for the
23:20
x i bar zero as i took for the x i
23:23
zero then there are iid or close to b
23:26
guy id
23:27
and it turns out that this system
23:30
this making lasso system propagates
23:33
independence
23:35
so they remain iid
23:38
therefore instead of solving for n
23:40
particles now i just have to solve for
23:42
two
23:43
and i get the second equation
23:47
below for x bar one and then i would get
23:50
the symmetric equation for x bar two
23:54
and well because i'm i'm doing pd
23:58
of course i'm greatly tempted to first
24:00
look at the low
24:01
right of uh each x bar i
24:05
so here's the type of x bar i at the
24:08
at the bottom and well if i'm using
24:11
nito's formula
24:13
uh i get an addiction transport equation
24:16
for the law
24:17
which is the machine glass of pde and
24:20
that is this equation on robot
24:23
now i have an n minus 1 divided by n
24:26
because uh
24:28
here i'm neglecting the the self
24:30
interaction so there's no k
24:32
of um
24:36
k of x i minus x i
24:39
just goes away right so maybe i should
24:42
every scale by 1 over n
24:44
minus 1 but i didn't want to to make it
24:46
too cumbersome at the beginning
24:48
of course as n tends to infinity this
24:51
goes to one
24:52
and so you get a pde that is completely
24:54
independent of n
24:57
so what the mean tail limit does is it
24:59
replaces
25:00
at the macroscopic scale right so if
25:02
you're looking at
25:05
the distribution of particles
25:09
okay robot at the macroscopic scale
25:12
it replaces a potentially very complex
25:15
very difficult to solve
25:17
coupled system of 10 to 20 sde
25:20
by one nonlinear rpd
25:24
and that's the whole point
25:30
now let me go into some of the models
25:32
that that uh
25:34
i'm i'm interested to uh to solve and uh
25:38
the first one is it corresponds to
25:40
actually a paper that
25:42
we've done with my student a few years
25:44
ago but that gave the first idea of the
25:46
method
25:47
and it contains the the dynamics of
25:50
point orders
25:51
in the fluid so here is another
25:54
experiment this is a soap film
25:57
which is um used a lot
26:00
for images of that uh because the soft
26:04
sofian is sort of naturally 2d and
26:08
and then of course it's relatively easy
26:10
to see
26:11
the velocity of the fluid uh in the film
26:15
uh by looking at it because it's so
26:19
which is exactly uh you know the game
26:21
that kids play with this whole bubble
26:23
and then the nice color and all that
26:25
uh and here what you see are
26:29
some relatively well-defined vortices
26:33
which i found is very very nice
26:37
so mathematically what this corresponds
26:39
to is is
26:41
essentially the biosavalo right which
26:44
tells you that
26:45
uh if you're in an incompressible fluid
26:48
and you look at one point vortex then
26:51
this creates a velocity in the fluid
26:53
that is given by
26:54
by this expression and here i'm using
26:57
the
26:58
symbol to to represent the rotation by
27:01
pi over 2
27:02
the trigonometric sense
27:06
and the dynamics of point borders is now
27:08
just consistent saying i'm going to put
27:11
several search vertices in the fluid
27:14
and each of the center of the vortex
27:17
is going to move according to the total
27:20
velocities
27:21
created by all the others so the dynamic
27:25
of x i is given by the formula below and
27:28
you just
27:28
sum right over the velocity field
27:32
created by
27:32
all the other vertices across noise
27:39
now here i'm looking at a relatively
27:41
simple example
27:42
where each vortices has the same voltage
27:45
and in particular the same sign of the
27:47
vorticity
27:49
and there are various issues in
27:50
particular if you're able to change
27:52
signs so
27:53
but i'm going to skip but i do make my
27:56
life simpler
27:58
by by doing that
28:02
now if you don't put noise this is a
28:04
reformulation of to the other
28:07
and if you put noise this is not a
28:09
reformulation of 2d nanostrokes
28:11
but in this conjecture it was
28:13
conjectured to converge to it
28:18
and this is actually a very old
28:21
conjecture because it dates from the
28:23
19th century again
28:26
by elmo's and
28:31
elmo's id was to use the system of point
28:34
orders
28:34
as a first way to approach
28:37
um turbulence influence
28:42
and again mathematically the formula are
28:45
relatively simple if i take the mean
28:47
field limit that i had written so if i
28:50
take the system before
28:52
and i just introduced the poisson kernel
28:54
in dimension two
28:56
then this machine of equation exactly
28:58
corresponds to the two dna structures
29:02
written on the verticity so on the curl
29:04
of you you being the velocity
29:07
and you have a very well known system
29:10
here
29:14
uh now another example which of course
29:16
is the one that i'm
29:18
then going to
29:21
focus on uh are models of chemotaxis
29:27
and chemotaxis is simply the the
29:30
dynamics
29:31
according to chemical gradients
29:36
so this is uh again an actual uh
29:40
experiment uh
29:44
that shows the the dynamics of cells
29:47
when you put some some
29:48
chemoattractant and something that you
29:51
observe
29:52
in this experiment is that the cells do
29:54
tend to congregate
29:55
right so you use clusters of cells
30:03
we can do a very elementary basic model
30:06
for
30:06
chemotaxis in the following manner
30:12
let's just consider one chemicals and
30:14
denote by
30:15
c its concentration
30:18
and then the simplest way of doing this
30:20
is to say that the dynamics
30:22
of one bacteria again we are going to
30:25
neglect its size
30:26
so i'm just looking at its center the
30:29
center will move in the direction of
30:31
higher concentrations
30:32
of c so it will move according to the
30:36
gradient of c
30:36
at point x i and then i'm adding noise
30:40
which is
30:41
a very rough um way of of modeling
30:45
run and tumble type type of behavior of
30:49
it's not a good way by any means but
30:52
it's simple
30:55
um and if i do that right
30:59
i also have to calculate the evolution
31:01
of the concentration
31:04
and the interesting case is when you
31:05
have a coupling and that coupling
31:07
for instance can um proceed
31:10
from the fact that bacteria
31:13
microorganism produce the chemical
31:16
themselves
31:18
so if they produce the chemicals
31:20
themselves i should have a diffusion
31:21
equation
31:22
on c with source terms centered at
31:26
each xi and if i'm making one favorite
31:30
assumption which is
31:31
that the chemicals diffuses much faster
31:35
than the microorganisms are moving
31:39
then i can look at the stationary
31:40
diffusion equation
31:42
and again if i'm using the the brain
31:45
function
31:48
which is 2d log i get the formula on c
31:51
that is written below
31:53
now there could be outside external
31:55
sources and other stuff but
31:58
that's pretty much the basic one
32:02
and um
32:05
this is of course completely a toy model
32:08
right it's not a very serious chemo
32:10
texas
32:12
model in that regard on
32:16
it has one uh well it has two virtues
32:19
one it actually probably correctly gives
32:23
the scaling of the interactions between
32:25
uh between microorganisms
32:28
so in a more complicated model we should
32:30
still expect that when
32:32
microorganisms are close for instance
32:34
their interaction
32:36
should scale in a manner that is
32:38
relatively well
32:40
um relatively close to what we have here
32:46
in the second virtues it's very simple
32:48
to study
32:49
the actual limit and and it's
32:53
a super well-known equation right it's
32:56
uh
32:56
what we nowadays usually call the path
33:00
killer signal system
33:02
um so the modeling id that i showed in
33:05
the previous slide was actually
33:08
already done by batlack in 53 but he
33:10
didn't write the system that was done
33:13
independently later by keller and zeigel
33:15
in 70.
33:18
and again here if i'm just writing down
33:20
the machine glass of equation
33:21
corresponding to that system this is
33:23
what i get
33:27
there's also a fun fact that is uh
33:31
this is actually also connected to
33:33
astrophysics because it
33:34
looks very much like the so-called
33:37
poison equation
33:38
that had been derived by chandrasekhar
33:40
in the in the 40s
33:44
um all right the two bit of differences
33:47
here
33:48
in the way i'm writing the way i'm
33:50
writing it
33:51
in comparison to the way we usually
33:53
write it in pds
33:55
um row bar for me is a probability
33:58
density
33:59
so it has total mass one
34:02
and second uh the way i
34:06
i use lambda uh you see here uh
34:09
i put the two pi which comes from the
34:12
green function explicitly in front of it
34:17
now as i said this is a a nice system
34:21
and in particular we can
34:25
pretty much determine its behavior and
34:28
that's interesting
34:30
because obviously there is a possible
34:33
self-reinforced loop in the dynamics
34:37
because if you start having a little bit
34:39
too many
34:40
microorganism at the point right then
34:43
they are going to produce more chemicals
34:45
at that point and this in turn will
34:48
attract more organism which will produce
34:50
even more chemicals and so on and so
34:52
forth
34:53
so there is a very natural question
34:55
which is um
34:57
does this system blow up
35:01
and because it has a very simple
35:04
structure we can give
35:05
a very precise answer um
35:09
and here i'm i'm quoting double per time
35:12
there have been a lot of works on this i
35:14
just don't want to go over all the
35:16
literature for this
35:18
uh so this is sort of the nice summary
35:20
that has
35:21
the the clear-cut distinction so
35:25
um it turns out that you have global
35:27
existence or you have blow up
35:30
independently of the of the initial data
35:32
you just have to look
35:34
whether the diffusion is dominant or
35:37
the advection is dominant so you have to
35:40
compare
35:41
lambda with sigma and here
35:44
with the rescaling that i have done the
35:47
comparison is between lambda
35:49
and for sigma uh
35:52
which actually corresponds to the
35:54
critical mass that is eight pi
35:56
uh if you do another rescaling where
36:00
now robar doesn't necessarily have to do
36:02
mass one
36:03
but you've rescaled the strength of the
36:05
interaction and the diffusion
36:08
so that's the same critical version and
36:11
if a diffusion dominates so if lambda is
36:12
less or equal than four sigma you have
36:14
global existence
36:15
and if your um addiction dominates to
36:18
lambda is strictly larger than four
36:19
sigma then you
36:20
always have blow up by night time
36:26
uh now i should add that if i look at
36:29
the same system
36:30
in multi d uh
36:33
still with a log of interaction so
36:36
that's not poisson anymore right
36:38
stalling freely and higher
36:40
so it's not this equation it's this
36:43
system and the mean field limit of this
36:45
system then i keep the same behavior
36:48
and now the critical parameter is two dc
36:50
bar instead of force um
36:54
[Music]
36:57
at the level of a multi-agent system
37:00
this raises so many questions
37:03
starting with do we even have existence
37:10
so does this couple system of sde
37:13
even make sense and that's hard
37:16
because now you have singular attractive
37:19
interactions between microorganisms
37:24
so if they start getting close right you
37:27
might have problems
37:29
and the other issue is if you're
37:32
thinking diffusion helps
37:33
well at the level of a multi-agent it
37:36
probably does not
37:37
or not so much uh because the
37:40
bad consequences of of the brownian
37:43
motion is that it brings you
37:45
near zero with probability one
37:49
so it's pretty much guaranteed that you
37:51
will have some pairs of particles in
37:53
this evolution that are going to end up
37:55
being very very close
37:57
and what happens then is
38:02
delicate
38:05
so this was a big um open
38:08
conjecture doing the main field limited
38:10
for such a system
38:13
and now we'll finish this broad
38:16
introduction by
38:17
on just quoting some of them on some of
38:20
the main results
38:22
um now there is very extensive
38:25
literature
38:26
so i i hope you will forgive me if i i
38:29
didn't quote your favorite paper here
38:34
just trying to keep it reasonably short
38:40
i want to emphasize two things first of
38:42
all the deterministic cases
38:44
now much better known than the
38:47
stochastic piece
38:50
in the deterministic case for example uh
38:53
point vortices the midfield limit was
38:55
already done
38:56
in the early 90s so we
39:00
fill them into 2d incompressible euler
39:03
and you have results by goodman who love
39:06
and grub and then
39:07
result by shoche with more general
39:09
initial data
39:12
if i am looking at strong repulsive
39:17
interactions then there is a very nice
39:20
result
39:21
by by certainty and endurance and
39:23
certainty
39:25
for risk kernels
39:28
there is no result for
39:33
singular attractive
39:36
potentials of the type of keller ziggler
39:40
because without diffusion
39:41
that is most certainly imposed
39:47
the case that is not as well understood
39:49
as the case of second order system
39:52
uh even though there have been on some
39:54
great recent results
39:56
uh by pico in in particular
40:00
and and again i want to emphasize that
40:03
collisional
40:03
models are are a lot harder than all of
40:06
this
40:08
um the stochastic case
40:11
what we know is so much more limited
40:16
uh for two dynamic stocks for instance
40:18
there was a result by ozada
40:20
in the mid-80s but this was only
40:22
qualitative convergence
40:24
plus a smallness assumption the smallest
40:26
assumption
40:27
was only removed very recently by phony
40:30
and michelle and
40:34
the first quantitative convergence was
40:36
the paper with with
40:38
my students and foo two years ago
40:42
and specifically for the keller's legal
40:43
system uh
40:45
well you see a lot of attempts
40:48
but very very few results the closest
40:51
one
40:52
uh is probably a recent paper by phony
40:54
and jordan
40:57
and they show two things they show well
40:59
pose and nest and they show some sort of
41:01
mean field limit
41:03
it's in the strongly diffusion dominant
41:07
regime there's another typo here it
41:08
should be lambda strictly less than
41:10
sigma
41:12
so it's not four sigma it's sigma so
41:14
this is not the critical mass
41:16
this is a regime where the diffusion is
41:18
even larger or the interaction is even
41:21
smaller
41:22
and diffusion is super dominant but even
41:25
this
41:25
regime does not give you propagation of
41:28
chaos
41:29
because the notion of limit that they
41:31
get is so weak that
41:33
there's we don't have any idea of
41:36
whether
41:36
there's uniqueness of solution at the
41:38
limit on that
41:43
all right so this is um
41:46
the end of this introduction now what i
41:49
want to do
41:50
is uh i want to go to the tablet because
41:53
i don't want to lose you in the
41:54
technical stuff
41:56
and i want to write down
42:00
uh what what we are doing
42:14
so
42:19
let me first write down again the system
42:35
right so this is the object we have
42:52
and typically here k is
42:58
a gradient flow
43:04
and v is
43:08
an attractive potential right so log x
43:11
if x is small
43:12
uh well
43:17
turns out that that log x is uh
43:21
is negative not positive that's going to
43:24
play a very significant
43:26
role uh in the proof i'm not going to
43:28
show you that but that plays a very
43:30
significant role in the proof
43:32
so um the first step
43:35
for this is we have no idea what's going
43:38
what what is going on at that level
43:41
and so the first thing is we're changing
43:44
the point of view
44:00
and so what we are doing is we are
44:01
looking at the joint law
44:04
row n of x one x n at time t
44:13
now something that i really have to
44:16
emphasize
44:17
right away because it's
44:20
critical this is
44:24
not the whole information in the system
44:27
as a matter of fact this misses
44:31
um
44:31
[Music]
44:35
every sort of time correlation
44:39
in the trajectory
44:46
so this is losing information on the
44:49
trajectory
45:00
which in some sense is the whole point
45:02
of the method because now we are looking
45:04
at a statistical distribution
45:07
and not at
45:11
trajectories
45:15
and well if i'm just using at least
45:19
formally ito's formula
45:22
right uh it solves the the louville
45:25
or uh forward commodore mogarf equation
45:33
uh that i'm going to write down in an
45:35
abstract manner
45:36
in the following in the following way
45:39
right
45:40
so it's a linear equation in large
45:43
dimension
45:44
with an operator ln and ln of
45:48
root n is just
45:51
this sum over i the divergence of x
45:54
i one over n
45:58
the sum over j different from i
46:02
k of x i minus x g
46:05
n uh
46:09
minus of course laplacian in dimension
46:20
so this is a simple but crazy equation
46:27
now i should add that for this equation
46:29
we can actually get existence of
46:31
appropriate solutions
46:32
uh whereas we have no idea again how to
46:34
do that on the uh
46:35
on the original system um
46:39
but i will come to that in one minute
46:43
now the problem with this equation is of
46:45
course that there is no way
46:51
to actually use it at the limit right
46:53
you're not going to solve an equation
46:55
and dimension n and there's not much
46:58
that you can do
47:00
so the question that is raised here is
47:02
how to compare
47:20
now
47:23
the first wave that we had used with
47:25
with zenfoo
47:26
for to the united states is to use the
47:29
entropy
47:30
of the system
47:36
and
47:38
the gibbs entropy here just reads
47:43
something like that and so that gives
47:46
you
47:47
a way to use a relative entropy
47:58
so let me define row bar the solution to
48:00
the mean field limit
48:06
which is just this
48:17
and out of that i can tensorize and i
48:20
can get therefore
48:22
the joint lu of the machine blastoff
48:32
process
48:40
and then i can define my relative
48:42
entropy
48:46
between row n and this row bar
48:49
and something like this
48:56
and now this is a very nice object that
48:59
has nice property
49:00
that allows to compare easily
49:05
and in particular
49:09
there's a fund convex property which is
49:12
that
49:12
if i define the marginal
49:16
rho and k which is the integral of rho n
49:21
the x k plus 1 dx n right
49:25
so those are sometimes called the
49:26
observable in physics because those are
49:28
the quantities you can actually
49:30
measure right so rho n1 is the one
49:32
particle distribution or when two is the
49:34
two particle distribution that's going
49:36
to give you correlations and positions
49:39
things like that
49:42
then
49:46
the relative entropy between rho and k
49:54
and robot tensorized k times
50:00
is actually controlled by h n
50:04
so this is very very nice because h n is
50:07
something that you will look at
50:09
the level of the luvial and if you can
50:12
get
50:13
a good convergence rate for h n so if
50:15
you can show that h n goes to zero
50:18
then you will have convergence of all
50:19
the marginals
50:21
and you're done
50:34
the problem is that for gradient flows
50:47
the time evolution of hn is absolutely
50:50
horrible and
50:54
that because it creates a laplacian of v
50:56
and if you
50:57
try to go that way then then you're
50:59
going to fail
51:01
so what was working for uh
51:04
incompressible navier stokes
51:06
where the flow is incompressible right
51:09
so k is the orthogonal gradient of v
51:12
that means divergence k
51:13
zero what was working there
51:16
absolutely cannot work
51:20
for gradient flows
51:24
and instead what we try to do is we try
51:27
to
51:28
use the free energy
51:52
so the free energy is entropy plus
51:55
energy
51:58
right no no no
52:01
big surprise here
52:05
and
52:11
the energy turns out to be the the so
52:14
what we are going to use here is of
52:16
course the modulated energy
52:20
and this is exactly the object that
52:23
cefati
52:23
is using for
52:27
paris gases so this cn
52:30
is going to end up being the right
52:32
linear combination of our previous
52:34
relative entropy with circuit is
52:36
modulated energy
52:38
and so let me write down what it is
52:41
this is
52:49
and this is a modulated energy so this
52:51
is
52:52
if i'm in the taurus this is something
52:55
like this
53:10
where mu n is
53:19
the empirical measure right so if you
53:22
um if you choose all the x i according
53:25
to row n
53:26
then i can build mu n which is a random
53:28
measure now
53:30
on the taurus and then i use it on the
53:32
modulated energy
53:35
so r uh kn let me repeat that again as
53:38
the expectation
53:39
of certainty's modulated generation
53:43
and there's another way of understanding
53:45
why this is so useful
53:47
which is to look at the gibbs
53:54
equilibrium
53:56
and if i define the gips equilibrium
53:59
like this
54:15
and i defined the equivalent of the
54:18
meaningful limit of the gibbs
54:22
equilibrium
54:27
which is not an equilibrium for the
54:28
limit right because when i'm going to
54:30
replace this discrete sum by a
54:32
convolution
54:34
um well of course i have to use the law
54:37
at the limit which is time dependent
54:44
and when i taking this limit i have to
54:46
be careful there's an additional
54:48
constant that shows up
54:51
which i'm going to write here
55:03
then it turns out that hn is a weighted
55:06
relative entropy using the equilibrium
55:25
so the use of the right physics here is
55:28
really
55:28
important it's the starting point
55:31
writing down the time evolution of hn is
55:34
the first thing you have to do i'm not
55:35
going to show you what it looks like
55:37
it's reasonably simple
55:39
if you trust you have the right physics
55:41
right so you really have to use the
55:43
structure of the equation well otherwise
55:45
you will get lost in the calculations
55:47
and i should mention we were not the
55:49
the only ones to try that and and the
55:52
reason we succeeded is
55:54
we actually used uh uh the the right
55:57
form of uh
55:58
of the calculations here
56:01
and that gives you a very nice hearing
56:07
that i am going to write down just
56:10
uh just now
56:14
so
56:20
we have certain assumptions
56:26
i need the limit to be smooth i'm going
56:29
to be vague here
56:30
and i need it to be strictly positive
56:34
so it cannot vanish
56:38
it turns out that
56:43
only the main assumption is just an
56:46
inequality
56:54
and this is very nice because this
56:56
allows a much wider
56:58
shape of v so we are not stuck to the
57:00
exact precise formula of keller's angle
57:04
there are other assumptions of v which
57:07
is v
57:07
okay
57:10
if x is different from zero that i don't
57:13
want to
57:14
um to write down here
57:26
and then i have an explicit gravel
57:38
estimate
57:45
that gives me polynomial decay in n
57:50
now um so this gives you
57:54
essentially everything you want to know
57:57
um i'm essentially going to stop here i
58:01
do want to make
58:02
one final remark on this
58:05
uh note that i'm using
58:08
the absolute value of k n i'm not using
58:11
a n
58:11
directly k n
58:14
does not have a sign
58:23
we are in the attractive case so you
58:27
actually expect kn to be more negative
58:29
and positive
58:30
this is the whole difficulty the whole
58:33
technical difficulty in the proof
58:58
for this we had to develop new large
59:00
deviation estimates and this is really
59:04
where things are painful
59:07
but it works and it gives you what i
59:09
think is a very nice result
59:11
um so this is the first derivation of
59:15
the first trigger is derivation of the
59:16
categorical system
59:20
it works up to the critical mass
59:24
and it works for a relatively broad set
59:26
of kernels
59:28
so i think this is uh i mean this is a
59:32
promising id
59:33
that should uh one should try to push
59:36
further
59:36
in various other cases
59:39
and there still remains plenty of
59:41
questions which is what happens if you
59:43
don't exactly have a great
59:45
flow what happens if you start
59:48
above the critical mass but
59:52
four times that are before blew up
59:58
and what happens for vanishing viscosity
60:02
there are all uh a whole set of
60:05
other questions after that um
60:08
but i will stop here and and i thank you
60:10
all
60:12
okay thank you so much for this
60:14
beautiful talk
60:16
uh is there any question
60:20
so you can either speak up or write on
60:23
the chat
60:40
okay
60:43
so if no one has a any question uh i i
60:46
do have
60:46
one which is more like a curiosity i
60:48
noticed that
60:49
for those type of problems they always
60:51
use the laplacian as the
60:53
as the fusion which gives the brilliant
60:56
motion
60:57
um what happens if you have coefficients
60:59
with the
61:00
within the diffusion is there physical
61:05
interest in these kind of kind of
61:06
problems
61:14
the reason everyone using the laplacian
61:16
is that um
61:17
we already don't understand very well
61:19
what's going on
61:21
so but uh
61:24
sometimes it doesn't cost much to use
61:26
something else
61:28
i see in in the previous result with my
61:30
students on on
61:32
2d navy stocks you can match you can put
61:35
a diffusion coefficient
61:37
see you can even put as long as it's
61:41
smooth
61:41
you can even put a non-homogeneous
61:43
diffusion coefficient
61:46
uh you can probably put something else
61:48
than burning motion and it
61:50
it's still going to work okay
61:53
uh here it's a lot more complicated
61:56
because
61:57
you we really need the gibbs equilibrium
62:01
and the gibbs equilibrium comes from
62:04
using the laplacian
62:07
so what would happen if now i had a
62:10
sigma that was you know a matrix right
62:14
and here well i would want to i have
62:17
seen my minus one so you can still write
62:19
it down but but
62:20
i've really not checked what what would
62:22
happen with
62:23
with the calculations if you go there uh
62:26
it might be that as long as the matrix
62:28
is not singular then everything is okay
62:31
but i'm not going to
62:35
again i haven't checked okay yeah okay
62:38
now thank you
62:40
i have i have a small question maybe a
62:43
related one
62:44
did you mention like people use mostly
62:47
brown in motion
62:48
and you get in the end a linear
62:50
laplacian so
62:52
do you expect if you put something like
62:54
leaving flight then you get the
62:56
fractional diffusion in the end
63:00
for keller siegel i have no idea
63:02
whatsoever
63:04
because the the the proof uses strongly
63:08
the the diffusion
63:11
to control the the the kn term to
63:14
control the modulated energy
63:17
[Music]
63:18
for our previous paper when when we
63:21
didn't have a modulated energy yeah i
63:23
think this should work
63:28
and maybe another stupid question um
63:32
is this some kind of relation
63:36
of this kind of mean view limit with the
63:39
[Music]
63:40
i don't know what it's called maybe
63:41
scaling also from postman
63:43
systems or postman equation to also
63:46
diffusion processes and
63:47
things like that well
63:50
there are connections because
63:55
the concerns are somehow the same
63:59
you need to have an adequate
64:00
representation of your many particle
64:02
systems
64:04
and you need to have a way to compare
64:05
with the with the midfield limit
64:09
which is not an initial limit in both
64:11
cases with the limit right
64:15
so when you look at it at a broader
64:18
level
64:19
there are obvious connections when you
64:21
start looking down
64:23
[Music]
64:24
into the techniques uh then they they
64:27
can be very very different uh
64:32
and in particular uh here
64:35
what i get is i get a strong propagation
64:38
of kills
64:40
because what i get is that if the
64:44
initial entropy
64:48
initial relative entropy converges then
64:51
it keeps on converging
64:53
and that cannot be true for boltzmann
64:56
because you start from an irreversible
64:58
system so entropy
65:00
is propagated
65:03
and you converge to a dissipative system
65:05
where entropy decreases in time
65:09
this is a math entropy here right so
65:11
they cannot fit you cannot have
65:13
convergence at the level of
65:15
the relative entropy for boltzmann for
65:17
example
65:19
so this method would completely break
65:20
down by by necessity
65:26
but the reason for my question is
65:27
because recently i think
65:29
there are several works of devoted to
65:31
showing in some diffusive
65:33
limit and you get the fractional
65:34
diffusion from boseman
65:36
yeah so it's also related to my question
65:38
before but thank you
65:40
but but uh and again i mean
65:43
if you're looking at a broader level
65:44
than else you're going to be interested
65:46
in
65:47
very similar quantities
65:51
but the big difference in boltzmann
65:52
which is going from reversible to
65:54
irreversible
65:55
is going to break a lot of the structure
65:57
that i'm using here
65:59
thank you
66:04
okay any further question
66:12
so if not let's thank our wonderful
66:15
speaker again thank you so much for
66:17
joining us to deliver this beautiful
66:19
talk
66:21
and i pass over to jan if you want to
66:24
say a few words
66:26
yeah thanks a lot as for my set for
66:28
speaker today pm manuel
66:30
thank you very much for being here and
66:32
also thanks everybody else for joining
66:33
thanks eduardo for
66:35
being the host this morning uh yes and
66:37
please
66:38
uh as i just posted in the chats feel
66:40
free to join us again next week by for
66:42
talk by
66:43
jose carrillo
66:46
same time same place and enjoy your week
66:57
everybody