Extensions of the Kac N-particle Extensions of the Kac N-particle model to model to

multi-particle interactionsmulti-particle interactions

Irene M. GambaDepartment of Mathematics and ICES

The University of Texas at Austin

IPAM KTW4, May 2009

Consider a spatially homogeneous d-dimensional ( d ≥ 2) rarefied gas of particles having a unit mass. Let f(v, t), where v R∈ d and t R∈ +, be a one-point pdf with the usual normalization

Assumption: I - collision frequency is independent of velocities of interacting particles (Maxwell-type) II - the total scattering cross section is finite.

Hence, one can choose such units of time such that the corresponding classical Boltzmann eqs. reads

with

Q+(f) is the gain term of the collision integral and Q+ transforms f to another probability density

Motivation: Connection between the kinetic Boltzmann eq.s and Kac probabilistic Motivation: Connection between the kinetic Boltzmann eq.s and Kac probabilistic interpretation of statistical mechanics -- Properties and Examplesinterpretation of statistical mechanics -- Properties and Examples

The structure of this equation follows from thr well-known probabilistic interpretation by M. Kac: Consider stochastic dynamics of N particles with phase coordinates (velocities) VN=vi(t) R∈ d, i = 1..N

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 equationKac 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

The corresponding “weak formulation” for f(v,t) for any test function φ(v) where the RHS has a bilinear structure from evaluating f(vi’,t) f(vl’, t) yields

the Boltzmann equation of Maxwell type in weak form

2

A general form statistical transport : The Boltzmann Transport Equation (BTE) with external heating sources: important examples from mathematical physics and social sciences:

The termmodels external heating sources:

Space homogeneous examples:•background thermostat (linear collisions), •thermal bath (diffusion)•shear flow (friction), •dynamically scaled long time limits (self-similar solutions).

Inelastic Collisionu’= (1-β) u + β |u| σ , with σ the direction of elastic post-collisional relative velocity

γ=0 Maxwell molecules γ=1 hard spheres

The same stochastic model admits other possible generalizationsThe 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

What properties of Q(j)+ are needed to make them consistent with the Maxwell-type interactions?

1. Temporal evolution of the system is invariant under scaling transformations of the 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.

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

is also invariant under scaling transformations k → λ k, k R∈ 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 equation,

2. in R1 = R should be replaced by reflections. An interesting one-dimensional system is based on the above discussed multi-particle stochastic model with non-negative phase

variables v = R+, for which the Laplace transform

If solutions are isotropic

then

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

Recall self-similarity:Recall self-similarity:

Back to molecular models of Maxwell type (as originally studied)Back to 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’sCarlen, Carvalho, Gabetta, Toscani, 80-90’s For inelastic interactions: Bobylev,Carrillo, I.M.G. 00Bobylev, Cercignani,Toscani, 03, Bobylev, Cercignani, I.M.G’06 and 08, 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/M of M particles associated to system of vectors randomly interchanging velocities pairwise while preserving momentum and local energy, independently of their relative velocities.independently of their relative velocities.

We work in the space of characteristic functions associated to ProbabilitiesWe work in the space of characteristic functions associated to Probabilities:

Bobylev operatorΓ

σ

1

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

λ1 := ∫(0,1) aβ(s) + bβ(s) ds = 1 kinetic energy is conserved

N

< 1 kinetic energy is dissipated

> 1 kinetic energy is generated

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

φt + φ = Γ(φ , φ ) ; φ(t,0)=1, φ(0,k)=F(f0)(k), θ(t)= - φ’(0)

Using the linearization of Γ(φ , φ ) about the stationary state φ=1, we can inferred the energy rate of change by looking at λ1

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’08)

ExamplesExamples

An example for multiplicatively interacting stochastic process (with Bobylev’08):

Phase variable: goods (monies or wealth) particles: M- indistinguishable players

• 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 (n-particle collision)

is a quadratic n x n matrix with non-negative random elements, and must satisfy a condition that ensures the model does not depend on numeration of identical particles.

Simplest example: a 2-parameter family

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

The corresponding transformation is

• Jumps are caused by interactions of 1 ≤ n ≤ N ≤ M particles (the case N =1 is understood as a interaction with background) • Relative probabilities of interactions which involve 1; 2; : : : ;N particles are given respectively by non-negative real

numbers β1; β2 ; …. βN such that β1 + β2 + …+ βN = 1 , so it is possible to reduce the hierarchy of the system to

Assume VM(t), n≥ M undergoes random jumps caused by interactions. Intervals between two successive jumps have the Poisson distribution with the average ΔtM = θ /M, θ const.

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

Model of M players participating in a N-linear ‘game’ according to the Kac rules (Bobylev, Cercignani,I.M.G.):

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

• Taking the Laplace transform of the probability f:

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

In the limit M ∞

Example: For the choice of rules of random interaction

With a jump process for θ a random variable with a pdf

So we obtain a model of the class being under discussion where self-similar asymptotics is possible

,N

Where μ(p) is a curve with a unique minima at p0>1 and approaches + ∞ as p 0

Also μ’(1) < 0 for and it is possible to find a second root conjugate to μ(1) for γ<γ*<1So a self-similar attracting state with a power law exists

whose spectral function is

N

So

is a multi-linear algebraic equation whose spectral properties can be well studied

In general we can see that

1. For more general systems multiplicatively interactive stochastic processes the lack of entropy functional does not impairsdoes not impairs the understanding and realization of global existence (in the sense of positive Borel measures), long

time behavior from spectral analysis and self-similar asymptotics.

2. “power tail formation for high energy tails” of self similar states is due to lack of total energy conservation, independent independent of the process being micro-reversible (elastic) or micro-irreversible (inelastic).

It is also possible to see Self-similar solutions may be singular at zeroSelf-similar solutions may be singular at zero.

3. The long time asymptotic dynamics and decay rates are fully described by the continuum spectrum associated to the linearization about continuum spectrum associated to the linearization about singular measuressingular measures.

Explicit solutions an elastic model in the presence of a thermostat for d ≥ 2 Explicit solutions an elastic model in the presence of a thermostat for d ≥ 2 Mixtures of colored particles (same mass β=1 ): (Bobylev & I.M.G., JSP’06)

=

Set β=1=

and set

1. Laplace transform of ψ: Transforms The eq. into

2- set and y(z) =z-2 u(zq) + B , B constant

Transforms The eq. into

and

3- Hence, choosing α=β=0 = B(B-1)

Painleve eq.

= 0 with θ=μ -1 -5μq and 6μq2 = ± 1

, with

Theorem: the equation for the slowdown process in Fourier space, has exact self-similar solutions satisfying the condition

for the following values of the parameters θ(p) and μ(p):

Case 1: Case 2:

where the solutions are given by equalities

with

Case 1:and Case 2:

Infinity energySS solutions

Finite energySS solutions

For p = 1/3 and p=1/2 then θ=0 the Fourier transf. Boltzmann eq. for one-component gas These exact solutions were already obtained by Bobylev and Cercignani, JSP’03

after transforming Fourier back in phase space

Computations: spectral Lagrangian methods in collaboration with Harsha Tharkabhushaman JCP 2009

Also, rescaling back w.r.t. to M^(k) and Fourier transform back f0ss(|v|) = MT(v) and

the similarity asymptotics holds as well.

Qualitative results for Qualitative results for Case 2Case 2 with finite energy: with finite energy:

, both for infite and finite energy cases

Maxwell Molecules modelRescaling of spectral modes exponentially by the continuous spectrum with λ(1)=-2/3

Testing: BTE with Thermostatexplicit solution problem of colored particles

Moments calculations:Moments calculations:Testing: BTE with Thermostat

Existence,

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

with 0 < p < 1 infinity energy, or p ≥ 1 finite energy

θ

Rigorous resultsRigorous results

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

(for initial data with finite energy)

Boltzmann SpectrumBoltzmann Spectrum

- I

Stability estimate for a weighted pointwise distance

for finite or infinite initial energy

These representations explain the connection of self-similar solutions with stable distributions

Similarly, by means of Laplace transform inversion, for v ≥0 and 0 < p ≤ 1

with

In addition, the corresponding Fourier Transform of the self-similar pdf admits an integral representation by distributions Mp(|v|) with kernels Rp(τ) , for p = μ−1(μ ). They are given by:∗

Theorem: appearance of stable law (Kintchine type of CLT)

For p0 >1 and 0<p< (p +Є) < p0

p01

μ(p)

μ(s*) = μ(1)

μ(po)

Self similar asymptotics for:

For any initial state φ(x) = 1 – xp + x(p+Є) , p ≤ 1. Decay rates in Fourier space: (p+Є)[ μ(p) - μ(p +Є) ]

For finite (p=1) or infinite (p<1) initial energy.

For p0< 1 and p=1

No self-similar asymptotics with finite energy

s*

For μ(1) = μ(s*) , s* >p0 >1

Power tailsCLT to a stable law

Finite (p=1) or infinite (p<1) initial energy

Study of the spectral function μ(p) associated to the linearized collision operator

p

ms> 0 for all s>1.

)

In the limit M ∞

So we obtain a model of the class being under discussion where self-similar asymptotics is possible:

,N N

Where μ(p) is a curve with a unique minima at p0>1 and approaches + ∞ as p 0 and μ’(1) < 0 for

And it is possible to find a second root conjugate to μ(1) for γ<γ*<1So a self-similar attracting state with a power law exists

whose spectral function is

Non-Equilibrium Stationary Statistical States -- γ - homogeneity of kernels vs. high energy tails for stationary states

Elastic caseElastic case

Inelastic Inelastic casecase

Thank you very much for your attention

A.V. Bobylev, C. Cercignani and I. M. Gamba, On the self-similar asymptotics for generalized  non-linear kinetic Maxwell models, to appear CMP’09

A.V. Bobylev, C. Cercignani and I. M. Gamba, Generalized kinetic Maxwell models of granular gases; Mathematical models of granular matter Series: Lecture Notes in Mathematics Vol.1937, Springer, (2008) .

A.V. Bobylev, C. Cercignani and I. M. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, arXiv:math-ph/0608035 A.V. Bobylev and I. M. Gamba, Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails. J. Stat. Phys. 124, no. 2-4, 497--516. (2006).

A.V. Bobylev, I.M. Gamba and V. Panferov, Moment inequalities and high-energy tails for Boltzmann equations wiht inelastic interactions, J. Statist. Phys. 116, no. 5-6, 1651-1682.(2004).

A.V. Bobylev, J.A. Carrillo and I.M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, Journal Stat. Phys., vol. 98, no. 3?4, 743--773, (2000).

I.M. Gamba and Sri Harsha Tharkabhushaman, Spectral - Lagrangian based methods applied to computation of Non - Equilibrium Statistical States. Journal of Computational Physics 228 (2009) 2012–2036

I.M. Gamba and Sri Harsha Tharkabhushaman, Shock Structure Analysis Using Space Inhomogeneous Boltzmann Transport Equation, To appear in Jour. Comp Math. 09

And references therein