Analytical and numerical issues Analytical and numerical issues for non-conservativefor non-conservative

non-linear Boltzmann transport equationnon-linear Boltzmann transport equation

Irene M. Gamba

Department of Mathematics and ICES

The University of Texas at Austin

In collaboration with: Alexandre Bobylev , Karlstad University, SwedenKarlstad University, Sweden, and

Carlo Cercignani, Politecnico di Milano, ItalyPolitecnico di Milano, Italy, on selfsimilar asymptotics and decay rates to generalized models for multiplicative stochastic interactions.

Sri Harsha Tharkabhushanam , ICES- UT Austin, ICES- UT Austin, on Deterministic-Spectral solvers for non-conservative, non-linear Boltzmann transport equation

MAMOS workshop – UT Austin – October 07

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

•Development of deterministic schemes: spectral-Lagrangian methodsDevelopment of deterministic schemes: spectral-Lagrangian methods

• Rarefied ideal gases-elastic: conservativeconservative Boltzmann Transport eq.Boltzmann Transport eq.

• Energy dissipative phenomena: 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.

•(Soft) condensed matter at nano scale: Bose-Einstein condensates models and charge transport in solids: current/voltage transport modeling semiconductor.

•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…).

A general form for Boltzmann equation for binary interactionsA general form for Boltzmann equation for binary interactionswith external ‘heating’ sourceswith external ‘heating’ sources

For a Maxwell type model: a linear equation for the kinetic energy

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

Time irreversibility is expressed in this inequality stability

In addition:

( )

asymptotics

An important application:An important application:

The homogeneous BTE in Fourier space

Boltzmann Spectrum

A benchmark case: A benchmark case:

Deterministic numerical method: Spectral Lagrangian solversDeterministic numerical method: Spectral Lagrangian solvers

Numerical simulationsNumerical simulations

Comparisons of energy conservation vs dissipationComparisons of energy conservation vs dissipation

For a same initial state, we test the energy Conservative scheme and the scheme for the energy dissipative Maxwell-Boltzmann Eq.

Numerical simulationsNumerical simulations

Moments calculations:Moments calculations:

Thank you Thank you

very much very much

for your attention !!for your attention !!