Transcript Maxwellian models

Non-Conservative
Boltzmann equations
Maxwell type models
Irene Martinez Gamba
Department 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
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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.
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 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:
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…).
Goals:
• Understanding of analytical properties: large energy tails
•long time asymptotics and characterization of asymptotics states
•A unified approach for Maxwell type interactions.
The 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 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
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Self similar asymptotics for:
For p0 >1 and 0<p< (p +Є) < p0
For any initial state φ(x) = 1 – xp + x(p+Є) , p ≤ 1.
μ(p)
Decay rates in Fourier space: (p+Є)[ μ(p) - μ(p +Є) ]
For finite (p=1) or infinite (p<1) initial energy.
Power tails
For μ(1) = μ(s*) , s* >p0 >1
1
p0
s*
Kintchine type CLT
μ(s*) =μ(1)
μ(po)
•For p0< 1 and p=1
No self-similar asymptotics with finite energy
)
Example
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
Thank you !
For references see www.ma.utexas.edu/users/gamba/research and references therein