1. Analytical and numerical issues for non-conservative non-linear Boltzmann transport equation non-linear Boltzmann transport equation Irene M. Gamba Department of Mathematics and ICES The University of Texas at Austin In collaboration with: Karlstad University, Sweden Alexandre Bobylev, Karlstad University, Sweden, and Politecnico di Milano, Italy Carlo Cercignani, Politecnico di Milano, Italy, on selfsimilar asymptotics and decay rates to generalized models for multiplicative stochastic interactions. ICES- UT Austin, Sri Harsha Tharkabhushanam, ICES- UT Austin, on Deterministic-Spectral solvers for non-conservative, non-linear Boltzmann transport equation MAMOS workshop – UT Austin – October 07

3. Goals: 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 conservativeBoltzmann Transport eq. Rarefied ideal gases-elastic: conservative Boltzmann Transport eq. Energy dissipative phenomena: Gas of elastic or inelastic interacting systems in the presence of a thermostat with a fixed background temperature ө b 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…).

4. A general form for Boltzmann equation for binary interactions with external ‘heating’ sources

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

7. The Boltzmann Theorem: The Boltzmann Theorem: there are only N+2 collision invariants Time irreversibility is expressed in this inequalitystability In addition:

11. ()

12. asymptotics

13. An important application: The homogeneous BTE in Fourier space

16. Boltzmann Spectrum

19. A benchmark case:

22. Deterministic numerical method: Spectral Lagrangian solvers

31. Numerical simulations

35. Comparisons 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.

38. Numerical simulations

41. Moments calculations: Thank you very much very much for your attention !!