Classical solutions for the Boltzmann transport equations

for soft potentials with initial data near local Maxwellians

Irene M. Gamba

The University of Texas at Austin

Department of Mathematics and ICES

FRG - kinetic models

Brown University -- May 2010

Collaborators

Ricardo Alonso, Rice University, (Applied Math)

Emanuel Carneiro, Institute for Advanced Studies (Math)

Alexander Bobylev, Karlstad University,

Carlo Cercignani, Politecnico di Milano,

Ravi Srinivasan, UT Austin

Part II: new applications in kinetic collisional theory to multi agent dynamics

‘v

‘v*

v

v*

C = number of particle in the box

a = diameter of the spheres

d = space dimension

η

elastic

inelastic

u · η = uη := impact velocity

η:= impact direction

(random in S+d-1)

u · η = (v-v*) . η = - e ('v-'v*) · η = -e 'u . η

u · η ┴ = (v-v*) · η┴

= ('v-'v*) · η ┴ = 'u · η ┴

for hard spheres in 3-d: ( L. Boltzmann 1880's), in strong form:

For f (t; x; v) = f and f (t; x; v*) = f* describes the evolution of a

probability distribution function (pdf) of finding a particle centered at x ϵd,

with velocity v ϵd, at time t ϵ+ , satisfying

e := restitution coefficient : 0 < e ≤ 1

e = 1 elastic interaction , 0 < e < 1 inelatic interaction, ( e=0 „sticky‟ particles)

u = v-v* := relative velocity

|u · η| dη := collision rate

The classical Elastic/Inelastic Boltzmann Transport Equation

γ

θ

'v* and 'v are called pre-collisional velocities, and

v* and v are the corresponding post-collisional velocities

Grad’s assumption allows to split the collision operator in a gain and a loss part, Q( f, g) = Q+( f, g) − Q−( f, g) = Gain - Loss

The loss operator Q−( f, g) = f R(g), with R(g), called the collision frequency, given by

NOTE: The loss bilinear form is local in f and a weighted convolution in g.

while the gain is a bilinear form with a weighted symmetric convolution structure

Consider the Cauchy Boltzmann problem (Maxwell, Boltzmann 1860s-80s); Grad 1950s;

Cercignani 60s; Kaniel Shimbrot 80’s, Di Perna-Lions late 80’s):

Find a function f (t, x, v) ≥ 0 that solves the equation

Assumption on the model: A more general collision kernel reflecting intra molecular potentials and

higher dimensional spaces takes the form B(u, û · σ) with

(i) B(u, û · σ) = |u|λ b(û · σ) with -n < λ ≤ 1 ; we call soft potentials: -n < λ < 0

(ii) Grad‟s assumption: b(û · σ) є L1(S n−1), that is

Exchange of velocities in center of mass-relative velocity frame

Energy dissipation parameter β (or restitution parameter e)

Recall: Q+(v) operator in weak (Maxwell) form, and then it can easily be extended to

dissipative (inelastic) collisions

with

λβ

1-β

γ

Grad Cut-off

condition

λ = 0 for Maxwell Type (or Maxwell Molecule) models γ

= 1 for hard spheres models;

0< λ <1 for variable hard potential models,

-d < λ < 0 for variable soft potential models.

Average angular estimates & weighted Young‟s inequalities &

Hardy Littlewood Sobolev inequalities & sharp constantsR. Alonso and E. Carneiro’08 (to appear in Adv. Math.), and R. Alonso and E. Carneiro, IG, 09 (Comm.Math.Phys,10)

by means of radial symmertrization (rearrangement) techniques

and the angular mixing operator

Bobylev‟s (‟75) The angular operator on Maxwell type interactions λ=0

is the well know identity for the Fourier transform of the Q+

translations and reflections

is invariant under rotations

The weak formulation of the gain operator is a symmetric weighted convolution

Where the weight is an invariant under rotation operator involving

Young‟s inequality for variable hard potentials : 0 ≤ λ ≤ 1

Hardy-Littlewood-Sobolev type inequality for soft potentials : -n < λ < 0

for

0 ≤ λ ≤ 1

-n < λ < 0

Sketch of proof: important facts

1- Radial symmetrization

2- Preservation of the Lp - norm

and for set

α corresponds to moments weights

3- angular mixing operator on radial functions

Angular averaging lemma4-

5- Young‟s inequality for hard potentials for general 1 ≤ p , q, r ≤ ∞

The main idea is to establish a connection between the Q+ and P operators, and then use the knowledge

from the previous estimates. For α = 0 = λ (Maxwell type interactions) no weighted norms

The exponents p, q, r in Theorem 1 satisfy

Regroup and use Holder’s inequality and the angular averaging estimates on Lr’/q’ to obtain

These estimates resemble a Beckner and Brascamp-Lieb type inequality argument (for a nonlinear

weight) with best/exact constants approach to obtain Young‟s inequality

Remark: 1- Previous LP estimates by Gustafsson 88, Villani-Mouhot ‘04 for pointwise bounded b(u . σ),

I.M.G-Panferov-Villani ’03 for (p,1,p) with σ -integrable b(u . σ) in Sn-1.

2-The dependence on the weight α may have room to improvement. One may expect estimates with

polynomial (?) decay in α , like in L1α as shown Bobylev,I.M.G, Panferov and recently with Villani (97, 04,08)

(also previous work of Wennberg ’94, Desvilletes, 96, without decay rates.)

Maxwell type interactions λ=0 with β constant: the constants are sharp in (1,2, 2) and in (2,1,2)

Corollary:

The constant is achieved by the sequences:

and

So approximate

a Dirac

in x

(see Alonso and Carneiro, to appear in Adv Math 2009)

Hardy-Littlewood-Sobolev inequality for soft potentials -n < λ < 0 :

Applying Holder’s inequality and then the angular averaging lemma to the inner integral with (p, q, r) =

(a,1, a), a to be determined, one obtains

The choice of integrability exponents allowed to get

rid of the integrand singularity at s = −1, producing a

uniform control with respect to the inelasticity β.

Indeed, combining with the complete integral above, using triple Holder’s inq. yields

Is it possible to make such choice of a ?

This is a type of Brascamp-Lieb inequality argument (for a non-linear coordinate transformation)

Then: for

Using the classical Hardy-Littlewood-Sobolev inequality to obtain (Lieb ’83) with explicit constants

where the exponents satisfy

In fact, it is possible to find

1/a in the non-empty interval

so with C= D1 = C1 C21/a C3

1/a’

and

provided r < q !!

and r < q

The case r < p follows using the symmetric convolution structure Q+(f,g) (crucial for relating

to the classical HLS inequality for convolutions with singular kernels

Finally, the case r r max{ p,q } follows using the Riez-Thorin interpolation theorem

so with C=D2= C4 C51/a C6

1/a’ and r < p

so with C=D1t D2

1-t

However the loss operator Q- (f,g) lacks the symmetric convolution structure !!

• The locality in f does not support a commutative convolution structure in its weak form.

--

for r < q

Counterexample for the HLS loss operator Q-(f,g) estimates:

One should not expect the HLS type inequality for the loss term Q-(f,g) to hold outside the

range 1 b p b n/(n-l) .

For instance, set p = q = r = 2 and l = n/2, ( so n/(n- l) = 2)

Then taking the potential F(x) = |x|-n/2 and the function g(x)

Then Q-(f,g) (v) = f(v) has a pointwise blow up.

Then, compute

Consider the Cauchy Boltzmann problem:

B(u, û · σ) = |u|−λ b(û · σ) with 0 ≤ λ < n-1 with the Grad‟s assumption:

Q−( f, g) = f

Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials

with integrable angular cross section (Ricardo Alonso & I.M.G., Jour. Stat Phys. 09 )

with

Notation and spaces: For

Set with the norm

(1)

and

Set , so problem one reduces to

Theorem: Let B(u, û · σ) = |u|−λ b(û · σ) with -n < λ ≤ 0 with the Grad‟s assumption

In addition, assume that f0 is ε–close to the local Maxwellian distribution

M(x, v) = C Mα,β(x − v, v ) (0 < α, 0 < β).

Then, for sufficiently small ε the Boltzmann equation has a unique solution satisfying

C1(t) Mα1,β1(x − (t + 1)v, v ) ≤ f ( t, x-v t , v ) ≤ C2(t) Mα2,β2

(x − (t + 1)v, v )

for some positive functions 0 < C1(t) ≤ C ≤ C2(t) < ∞, and parameters 0 < α2 ≤ α ≤ α1 and

0 < β2 ≤ β ≤ β1.

Moreover, the case β = 0 (infinite mass) is permitted as long as β1 = β2 = 0.

(this last part extends the result of Mishler & Perthame ’97 to soft potentials)

Distributional solutions near local Maxwellians : Ricardo Alonso, IMG’08

Previous work by Toscani ’88, Goudon’97, Mischler –Perthame ‘97

As a consequence, one concludes that the distributional solution f is controlled by a traveling Maxwellian,

and that

It behaves like the heat equation, as

mass spreads as t grows

Classical solutions (Different approach from Guo’03, our methods follow some of the those by

Boudin & Desvilletes ‘00, plus new ones )

Definition. A classical solution in [0, T] of problem our is a function such that

,

Theorem (Application of HLS inequality to Q+ for soft potentials) : Let the collision kernel satisfying

assumptions λ < n and the Grad cut-off,

Loss

Gain

Then, for 1 < p, q, r < ∞, with 1/p + 1/q + /n = 1 + 1/r,

However, for 1 < p, q, r < ∞, with 1/p + 1/q + /n = 1 + 1/r and r < p

Remark: the singularity at s = 1 that appears in the constant formula is treated by

symmetrazing b(s) when f = g, so that

• The corresponding HLS estimate for general kernels B(u, û · σ) = F(u) b(û · σ),

with F e Ls radially symmetric and non-increasing.

Then, for 1 ≤ p, q, r ≤ ∞ , with 1/p + 1/q + 1/s = 1 + 1/r and r ≤ p

Therefore, in order to estimate the gradients of the constructed solutions in L∞-Maxwellian

weighted space we need to impose more integrability on the angular part of cross section

when we use to study the regularity of the constructed solution

Remark: the estimate for Q + is the analog one with just 1/p + 1/q + 1/s = 1 + 1/r

Gradient estimates

Theorem (R. Alonso, I.M.G): Assume b e La (Sn-1) for some a >1 , and that the initial

state f0 is small or is near a local Maxwellian. Assume also that f0e (L1 ∩ Lp0) for p0

>1.

Then, there is a unique classical solution f to the Cauchy problem in the interval [0, T]

satisfying the L∞-Maxwellian weighted estimates. Furthermore, there exists

1< b(a)<min(p0, n/(n-l) ) such that for any p e [1, b(a)] the following

estimates hold

Remarks: b e La secures C+ is finite. There is no constrain on the size of f0

Proof: set

: ∫

Case p>1:

1. Estimate the Q+ part of the operator by HLS

for

now taking 1< b(a) < min(p0, n/(n-l) ) such that for any p e (1, b(a)) implies

2. The corresponding estimate for Q+( tf, Df) is direct since b(·) was chosen to have

support in [0, 1].

3. The estimate for Q−(tf,Df) follows using the HLS theorem for the loss operator by

choosing b(a) < min{p0, n/ (n-l)}.

with

By Gronwall inequality with a = n/(n−λ)

4.

p e (1, b(a))

Case p = 1:

1. Separate the potential in two radially symmetric potentials

2. Then

3. Where, using the upper Mawxellian control on f

with n/s’ > 1 for s e ( n/(n−1) , n/(n – l) ).

4. Thenby Gronwall

Velocity regularity

Proof : Take for a fix h > 0 and ˆv ∈ S n−1 and the

corresp. translation operator and transforming

v∗ → v∗ + hˆv in the collision operator.

: ∫

Theorem Let f be a classical solution in [0, T] with f0 satisfying the condition of smallness

assumption or is near to a local Maxwellian and ∇x f0 ∈Lp(R2n) for some

1 < p < b(a) < min{p0 , n/(n-l) ).

In addition assume that ∇v f0 ∈ Lp(R2n). Then, f satisfies

the estimate

with

then

Which is solved by

Just set

Then, by the regularity estimate

with 0 < λ < n-1

Bernoulli ODE

for 1 ≤ p < b(a) < min{ p0 , n/(n-l) }

Lp and Mα,β stability

Set

Now, since f and g are controlled by traveling Maxwellians one has

Theorem Let f and g distributional solutions of problem associated to the initial datum

f0 and g0 respectively. Assume that these datum satisfies the condition of theorems for small data or

near Maxwellians solutions (0 < λ < n-1) . Then, there exist C > 0 independent of time such that

Moreover, for f0 and g0 sufficiently small in Mα,β

Our result is for integrable b(û · σ)

with 0 < λ < n-1

for 1 b p < b(a) < min{ p0 , n/(n-

l) }

Consider a spatially homogeneous d-dimensional ( d ≥ 2) “rarefied gas of particles” having a

unit mass.

Let f(v, t), where v ∈d and t ∈+, be a one-point pdf with the usual normalization

Assumptions:

I – interaction (collision) frequency is independent of the phase-space variable (Maxwell-type)

II - the total “scattering cross section” (interaction frequency w.r.t. directions) is finite.

Choose such units of time such that the corresponding classical Boltzmann eqs. reads as

a birth-death rate equation for pdfs

with

Q+(f) is the gain term of the collision integral which Q+ transforms f into another

probability density

Motivation: Connection between the kinetic Boltzmann equations and Kac probabilistic

interpretation of statistical mechanics (Bobylev, Cercignani and IMG, arXiv.org’06, 09, CMP’09)

Part II : Extensions of the Kac N-particle model to multi-linear interactions connection to

dynamics of information

The same stochastic model admits other possible generalizations.

For example we can also include multiple interactions and interactions with a background (thermostat).

This type of model will formally correspond to a version of the kinetic equation for some Q+(f).

where Q(j)+ , j = 1, . . . ,M, are j-linear positive operators describing interactions of j ≥ 1 particles,

and αj ≥ 0 are relative probabilities of such interactions, where

Assumption: Temporal evolution of the system is invariant under scaling transformations

in phase space: if St is the evolution operator for the given N-particle system such that

St{v1(0), . . . , vM(0)} = {v1(t), . . . , vM(t)} , t ≥ 0 ,

then St{λv1(0), . . . , λ vM(0)} = {λv1(t), . . . , λvM(t)} for any constant λ > 0

which leads to the property

Q+(j) (Aλ f) = Aλ Q+

(j) (f), Aλ f(v) = λd f(λ v) , λ > 0, (j = 1, 2, .,M)

Note that the transformation Aλ is consistent with the normalization of f with respect to v.

Note: this property on Q(j)+ is needed to make the consistent with the classical BTE for Maxwell-type interactions

Property: Temporal evolution of the system is invariant under scaling transformations of the phase

space: Makes the use of the Fourier Transform a natural tool

so the evolution eq. is transformed into an

evolution eq. for characteristic functions

which is also invariant under scaling transformations k → λ k, k ∈d

All these considerations remain valid for d = 1, the only two differences are:

1. The evolving Boltzmann Eq should be considered as the one-dimensional Kac master

equation, and one uses the Laplace transform

2. We discussed a one dimensional multi-particle stochastic model with non-negative phase

variables v in R+,

If solutions are isotropic

then

where Qj(a1, . . . , aj) can be an generalized functions of j-non-negative variables.

-∞-∞

The structure of this equation follows from the well-known probabilistic interpretation by

M. Kac: Consider stochastic dynamics of N particles with phase coordinates (velocities)

VN=vi(t) ∈ Ωd, i = 1..N , with Ω= or +

A simplified Kac rules of binary dynamics is: on each time-step t = 2/N , choose randomly a pair of

integers 1 ≤ i < l ≤ N and perform a transformation (vi, vl) →(v′i , v′l) which corresponds to an

interaction of two particles with ‘pre-collisional‟ velocities vi and vl.

Then introduce N-particle distribution function F(VN, t) and consider a weak form of the

Kac Master equation (we have assumed that V‟ N j= V‟N j ( VN j , UN j · σ) for pairs j=i,l with σ

in a compact set)

The assumed rules lead (formally, under additional assumptions)

to molecular chaos, that is

Introducing a one-particle distribution function (by setting v1 = v) and the hierarchy reduction

The corresponding “weak formulation” for f(v,t) for any test function υ(v) where the RHS has a bilinear

structure from evaluating f(vi‟,t) f(vl‟, t) M. Kac showed yields the the Boltzmann equation of

Maxwell type in weak form (or Kac’s walk on the sphere)

2Ωd

N

B BBfor B= -∞ or B=0

dσ

Molecular models of Maxwell type (as originally studied)

Bobylev, ’75-80, for the elastic, energy conservative case.

Drawing from Kac’s models and Mc Kean work in the 60’s

Carlen, Carvalho, Gabetta, Toscani, 80-90’s

For inelastic interactions: Bobylev,Carrillo, I.M.G. JSP’00

Bobylev, Cercignani,Toscani, 03, Bobylev, Cercignani, I.M.G’06 and 09, for general non-conservative problem

characterized by

so is also a probability distribution function in v.

The Fourier transformed problem:

One may think of this model as the generalization original Kac (‟59) probabilistic interpretation of rules of dynamics on

each time step Δt=2/N of N particles associated to system of vectors randomly interchanging velocities pairwise while

preserving momentum and local energy, independently of their relative velocities.

Then: work in the space of “characteristic functions” associated to Probabilities: “positive probability

measures in v-space are continuous bounded functions in Fourier transformed k-space”

Fourier transformed operatorΓ

accounts for the integrability of the function b(1-2s)(s-s2)(N-3)/2

λ1 := ∫1

0 (aβ(s) + bβ(s)) G(s) ds = 1 kinetic energy is conserved

< 1 kinetic energy is dissipated

> 1 kinetic energy is generated

For isotropic solutions the equation becomes (after rescaling in time the dimensional constant)

υt + υ = Γ(υ , υ ) ; υ(t,0)=1, υ(0,k)=F (f0)(k), Θ(t)= - υ‟(0)

In this case, using the linearization of Γ(υ , υ ) about the stationary state υ=1, we can inferred the

energy rate of change by looking at λ1 defined by

For isotropic (x = |k|2/2 ) or self similar solutions (x = |k|2/2 eμt ), μ is the energy dissipation

rate, that is: Θt = - μ Θ , and

Recall from Fourier transform: nthmoments of f(., v) are nth derivatives of υ(.,k)|k=0

Θ

the Fourier transformed collisional gain operator is written

, with

Kd

The existence theorems for the classical elastic case ( β=e = 1) of Maxwell type of interactions were

proved by Morgenstern, ,Wild 1950s, Bobylev 70s and for inelastic ( β<1) by Bobylev,Carrillo, I.M.G.JSP’00

using the Fourier transform

Note that if the initial coefficient |υ0|≤1, then |Фn|≤1 for any n≥ 0.

the series converges uniformly for τ ϵ [0; 1).

Classical Existence approach : Wild's sum in the Fourier representation.

Γ

Γ

Γ

• rescale time t → τ

and solve the initial value problem

by a power series expansion in time where the phase-space dependence is in the coefficients

Wild's sum in the Fourier representation.(It can be also obtained by a Picard interation

of the integral ODE above)

β/2β/21-β/2

Existence, asymptotic behavior - self-similar solutions and power like tails:

From a unified point of energy dissipative Maxwell type models: λ1 energy

dissipation rate (Bobylev, I.M.G.JSP‟06, Bobylev,Cercignani,I.G. arXiv.org‟06- CMP‟09)

Examples

Applications to agent interactions

Two examples:

• A couple of information percolation models (Dauffie, Malamud and Manso, 08-09)

•M-game multi agent model (Bobylev Cercignani, Gamba, CMP‟09)

Example 1 : Information aggregation model I (Duffie, Malamud & Giroux 09)

1. The higher the type f of the set of signals, the higher is the likelihood ratio between states H

and L and the higher the posterior probability that X is high.

2. Any particular agent is matched to other agents at each of a sequence of Poisson arrival times

with a mean arrival rate (intensity) l , which is the same across agents.

3. At each meeting time, m−1 other agents are randomly selected from the population of agents

Definition of “phase space” Basic probability by Bayes’ rule: the logarithm of the

likelihood ratio between states H and L conditional on signals

{s1, . . . , sn}

“type” q of the set of signals

• A random variable X of potential concern to all agents has 2 possible outcomes,

H (“high”) with probability n , and L (“low”) 1 − n.

• Each agent is initially endowed with a sequence of signals {s1, . . . , sn} that may be informative

about X.

• The signals {s1, . . . , sn} (primitively observed by a particular agent are, conditional on X,

independent with outcomes 0 and 1 (Bernoulli trials).

• W.l.g assume P(si = 1|H) r P(si = 1|L). A signal i is informative if P(si = 1|H) > P(si = 1|L).

• Binary: for almost every pair of agents, the matching times and counterparties of one agent are

independent of those of the other: whenever an agent of type qmeets an agent with type f

and they communicate to each other their posterior distributions of X,

they both attain the posterior type q+f .

• m-ary : whenever m agents of respective types 1, . . . , m share

their beliefs, they attain the common posterior type 1 + · · · + m.

Interaction law : The meeting group size m is a parameter of the information model that varies

Aggregation model

We let μt denote the cross-sectional distribution of posterior types in the population at time t.

• The initial distribution μ0 of types induced by an initial allocation of signals to agents.

• Assume that there is a positive mass of agents that has at least one informative signal.

• That is, the first moment m1(μ0(q) ) > 0 if X = H, and m1(μ0(q) ) < 0 if X = L.

• Assume that the initial law μ0 has a moment generating function, finite on a neighborhood of

z = 0 , where z = ⌠ ezq d(μ0(q)) (Laplace transform)

aggregation equation in integral formBinary

or

“m-ary”

Multi-agent

Existence by „Wild sums‟ methods

Self-similarity, Pareto tails formation and dynamically scaled solutions (with Ravi Srinivasan)

Statistical equation: (Smolukowski type)

with V’m = G Vm ; Vm = (v1; …; vm) ; V’m = (v’1, … ; v’m) ; where

G is a square m x m matrix with entries G = {gik = 1 , for all i, k = 1, . . . , m} ,

We notice the similarity with the the Kac model: let the type signals Vm and its posterior V‟m

Then the m-particle distribution function F(VN, t) and the weak form of the

Kac Master equation

The assumed rules lead (formally, under additional assumptions)

to molecular chaos, that is

Introducing a one-particle distribution function (by setting v1 = v) and the hierarchy reduction

2for N=m

Then the aggregation models hold for f(vm , t ) for either binary or multi-agent forms

The approach extends to more general information percolation models where the signal type

do not necessarily aggregate but “distributes ” itself between the posterior types

(in collaboration with Ravi Srinivasan)

For any search-effort policy function C(n), the cross-sectional distribution ft of precisions and

posterior means of the i-agents is almost surely given by

ft(n; x; w) = μC(n,t) pn(x |Y (w))

Another Example: Information aggregation model with equilibrium search dynamics

(Duffie, Malamud & Manso 08)

where μt(n) is the fraction of agents with information precision n at time t, which

is the unique solution of the differential equation below (of generalized Maxwell type)

and pn( x| Y(w) ) is the Y-conditional Gaussian density of E(Y |X1; …. ;Xn), for any n signals

X1; … ;Xn.

This density has conditional mean and conditional variance

mt(n) satisfies the dynamic equation

with π(n) a given distribution independent of

any pair of agents

Where μtC (n) = C(n) μ(n,t) is the effort-weighted measure such that: C(n) is the search-effort policy function

For μt(n) for the fraction of agents with precision n (related to the cross-sectional distribution μt of

information precision at time t in a given set) its the evolution equation is given by

Where μtC (n) = C(n) μ(n,t) is the effort-weighted measure such that: C(n) is the search-effort policy function

Linear term: represents the replacement of agents with newly entering agents.

Gain Operator: The convolution of the two measures μtC

* μtC represents the gross rate at which

new agents of a given precision are created through matching and information sharing.

Example from information search (percolation) model not of Maxwell type!!

Loss operator: The last term of captures the rate μtC μt

C(N) of replacement of agents with

prior precision n with those of some new posterior precision that is obtained through

matching and information sharing, where

is the cross-sectional average search effort

Remark: The stationary model can be viewed as a form of algebraic Riccati equation.

An example for multiplicatively interacting stochastic process:

M-game multi linear models (Bobylev, Cercignani, I.M.G.; CMP’09):

particles: j ≥ 1 indistinguishable players

phase state: individual capitals (goods) is characterized by a vector Vj = (v1;…; vj) ϵ j+

• A realistic assumption is that a scaling transformation of the phase variable (such as a change of

goods interchange) does not influence a behavior of player.

• The game of these n partners is understood as a random linear transformation (j-particle collision)

The parameters (a,b) can be fixed or randomly distributed in +2 with some probability density

Bn(a,b).

The corresponding transformation is

Set V’j = G Vj ; Vj = (v1; …; vj) ; V’j = (v’1, … ; v’j ) ; where

G is a square j x j matrix with non-negative random elements such that the model does not depend on

numeration of identical particles.

Simplest class: a 2-parameter family

G = {gik , i, k = 1, . . . , j} , such that gik = a, if i = k and gik= b, otherwise,

• Jumps are caused by interactions of 1 ≤ j ≤ M ≤ N particles (the case M =1 is understood as a

interaction with background)

• Relative probabilities of interactions which involve 1; 2; …;M particles are given by non-negative

real numbers β1; β2 ; …. βM such that β1 + β2 + …+ βM = 1 ,

Assume

•VN(t), N >> M undergoes random jumps caused by interactions.

•Intervals between two successive jumps have the Poisson distribution with the average

ΔtM = l = Θ /N, interaction frequency with

Then we introduce M-particle distribution function F(VN; t) and consider a weak form as in the Kac

Master eq:

Model of M players participating in a M-linear „game‟ according to the Kac rules(Bobylev, Cercignani,I.M.G.) CMP’09

So, it is possible to reduce the hierarchy of the system to

In the limit N ∞

• Taking the test function on the RHS of the equation for f:

• Taking the Laplace transform of the probability f:

• And assuming the “molecular chaos” assumption (factorization)

The evolution of the corresponding characteristic function is given by

with initial condition u|t=0 = e−x (the Laplace transformed condition from f|t=0 =δ(v − 1) )

Existence, stability,uniqueness,

(Bobylev, Cercignani, I.M.G.;.arXig.org „06 - CPAM 09)

with 0 < p < 1 infinity energy,

or p ≥ 1 finite energy

Θ

Rigorous results for Maxwell type interacting models

Relates to the work of Toscani, Gabetta,Wennberg, Villani,Carlen, Carvallo,…..

(for initial data with finite energy)

Boltzmann Spectrum

- I

Aggregation Spectrum Wealth distribution

Spectrum

References and preprints http://rene.ma.utexas.edu/users/gamba/publications-web.htm

Thank you very much for your attention