Non-Conservative Boltzmann equations

Maxwell type models

Irene Martinez GambaDepartment of Mathematics and ICES

The University of Texas at Austin

Buenos Aires, Diciembre 06

In collaboration with:

A. Bobylev, Karlstad Univesity, Sweden

C. Cercignani, Politecnico di Milano, Italy.

The Boltzmann Transport Equation (BTE) is a model from an statistical

description of a flow of ``particles'' moving and colliding or interacting in a

describable way ‘by a law’; and the average free flight time between stochastic

interactions (mean free path Є) inversely proportional to the collision frequency.

Example: Think of a `gas': particles are flowing moving around “billiard-like”

interacting into each other in such setting that

• The particles are so tightly pack that only a few average quantities will

described the flow so, Є << 1 (macroscopic or continuous mechanical system)

• There are such a few particles or few interactions that you need a complete description

of each particle trajectory so Є >> 1, (microscopic or dynamical systems),

• There are enough particles in the flow domain to have “good” statistical

assumptions such that Є =O(1) (Boltzmann-Grad limit):

mesoscopic or statistical models and systems.

Goals: • Understanding of analytical properties: large energy tailsUnderstanding of analytical properties: large energy tails

•long time asymptotics and characterization of asymptotics stateslong time asymptotics and characterization of asymptotics states

•A unified approach for Maxwell type interactions.A unified approach for Maxwell type interactions.

Examples are Non-Equilibrium Statistical States (NESS) in N dimensions, Variable Hard Potentials (VHP , 0<λ ≤ 1) and Maxwell type potentials interactions (λ =0). • Rarefied ideal gases-elastic (conservative) classical theoryclassical theory,

• Gas of elastic or inelastic interacting systems in the presence of a thermostat with a fixed background temperature өb or Rapid granular flow dynamics: (inelastic hard sphere interactions): homogeneous cooling states, randomly heated states, shear flows, shockwaves past wedges, etc.

•Bose-Einstein condensates models and charge transport in solids: current/voltage transport modeling semiconductor nano-devices.

•Emerging applications from Stochastic dynamics for multi-linear Maxwell type interactions : Multiplicatively Interactive Stochastic Processes:Multiplicatively Interactive Stochastic Processes: Pareto tails for wealth distribution, non-conservative dynamics: opinion dynamic models, particle swarms in population dynamics, etc (Fujihara, Ohtsuki, Yamamoto’ 06,Toscani, Pareschi, Caceres 05-06…).

The Boltzmann Transport EquationThe Boltzmann Transport Equation

u’= |u| ω and u . ω = cos(ө)

|u|

Notice: ω direction of specular reflection = σ

α < 1 loss of translational component for u

but conservation of the rotational component of u

In addition: Classical n-D-Boltzmann equation formulation for binary elastic or inelastic collisions for VHP or Maxwell interactions, in the (possible) presence of ‘heating sources’ or dynamical rescaling

For a Maxwell type model:

The Boltzmann Theorem: there are only N+2 collision invariants

But…

What kind of solutions do we get?

Molecular models of Maxwell typeMolecular models of Maxwell type

Bobylev, ’75-80, for the elastic, energy conservative case-Bobylev, Cercignani, I.M.G’06, for general non-conservative problem

Where, for the Fourier Transform of f(t,v) in v:

The transformed collisional operator satisfies, by symmetrization in v and v*

with

Since

then

For

Typical Spectral function μ(p) for Maxwell type models

• 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

Kintchine type CLT

)

ExampleExample

References• Cercignani, C.; Springer-Verlag, 1988

• Cercignani, C.;Illner R.; Pulvirenti, M. ; Springer-Verlag;1992

• Villani;C.; Notes on collisional transport theory; Handbook of Fluid dynamics, 2004.

For recent preprints and reprints see:www.ma.utexas.edu/users/gamba/research and references therein

For references see www.ma.utexas.edu/users/gamba/research and references therein

Thank you !