Generalizations to Boltzmann-Maxwell Generalizations to Boltzmann-Maxwell Interaction DynamicsInteraction Dynamics

Irene M. GambaDepartment of Mathematics and ICES

The University of Texas at Austin

IAS, February 2009

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

Molecular models of Maxwell type as originally studied: 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, 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.

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

Bobylev operatorΓ

1

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

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.

Existence, (Bobylev, Cercignani, I.M.G.;. 06-08)

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

θ

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

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 tails

CLT 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.

)

An example for multiplicatively interacting stochastic process (Bobylev, Cercignani, I.M.G’08):

Phase variable: goods (monies or wealth) particles: n- 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 hat 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 θ a random variable with a pdf

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

,N N

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)So a self-similar attracting state with a power law exists

Another benchmark case: Another benchmark case: Self-similar asymptotics for a for a slowdown process given by elastic BTE with a thermostat

Soft condensed matter Soft condensed matter phenomenaphenomena

Remark: The numerical algorithm is based on the evolution of the continuous spectrum of the solution in the same spirit of Greengard-Lin’00 spectral calculation of the free space heat kernel, i.e. self-similar of the heat equation in all space.

Explicit formula in Fourier space

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

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,  ’08 submitted.

A.V. Bobylev and I. M. Gamba, On special solutions for linear Maxwell-Boltzmann models for slow down processes. In preparation.

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 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. Jour. Computational Physics, (2008).

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