A Stochastic process behind Boltzmann's kinetic equation and issues of coarse graining

Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena

Volumn , Issue , 2006, Pages 207-224

  Öttinger, Hans Christian ^a  

^a ETH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 58749111983     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/3-540-35888-9_9     Document Type: Chapter

References (46)

1. C. Cercignani: The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, vol. 67 (Springer, Berlin Heidelberg New York 1988).
2. H. Struchtrup: Macroscopic Transport Equations for Rarefied Gas Flows (Springer, Berlin Heidelberg New York 2005).
3. H.C. Ottinger: Beyond Equilibrium Thermodynamics (Wiley, Hoboken 2005).
4. H.C. Ottinger: Derivation of two-generator framework of nonequilibrium thermodynamics for quantum systems. Phys. Rev. E 62, 4720-4724 (2000).
5. A.N. Gorban, I.V. Karlin, H.C. Ottinger, L.L. Tatarinova: Ehrenfest's argument extended to a formalism of nonequilibrium thermodynamics. Phys. Rev. E 63, 1-6 (2001).
6. A.N. Gorban: Basic types of coarse-graining. Preprint cond-mat/0602024 (2006).
7. H. Brenner: Is the tracer velocity of a fluid continuum equal to its mass velocity? Phys. Rev. E 70, 1-10 (2004).
8. H. Brenner: Kinematics of volume transport. Physica A 349, 11-59 (2005).
9. H. Brenner: Navier-Stokes revisited. Physica A 349, 60-132 (2005).
10. H.C. Ottinger: Stochastic Processes in Polymeric Fluids: Tools and Examples for Developing Simulation Algorithms (Springer, Berlin Heidelberg New York 1996).
11. H. Bauer: Probability Theory and Elements of Measure Theory, second edition (Academic Press, London 1981).
12. R.J. Di Perna, P.L. Lions: On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. 130, 321-366 (1989).
13. L. Desvillettes, C. Villani: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation. Invent. Math. 159, 245-316 (2005).
14. M. Kac: Foundations of kinetic theory. In J. Neyman, editor, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. III, 171-197 (Univ. of California Press. Berkeley 1956).
15. H.P. McKean: Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arch. Rational Mech. Anal. 21, 343-367 (1966).
16. F.A. Grunbaum: Propagation of chaos for the Boltzmann equation. Arch. Rational Mech. Anal. 42, 323-345 (1971).
17. H. Tanaka: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrscheinlichkeitstheorie verw. Gebiete 46, 67-105 (1978).
18. A.S. Sznitman: Equations de type de Boltzmann, spatialment homogenes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 66, 559-592 (1984).
19. C. Graham, S. Meleard: Probabilistic tools and Monte-Carlo approximations for some Boltzmann equations. In: Presentation du CEMRACS 1999, ed. by F. Coquel and S. Cordier, ESAIM Proceedings, vol. 10, 77-126 (EDP Sciences, Les Ulis, 2001) www.edpsciences.org/articlesproc/Vol.10/contents.htm.
20. O.E. Lanford III: Time evolution of large classical systems. In: Dynamical Systems, Theory and Applications, Batelle Seattle 1974 Rencontres, ed. by J. Moser, Lecture Notes in Physics, vol. 38, 1-111 (Springer, Berlin 1975).
21. L. Arnold: Stochastic Differential Equations: Theory and Applications (Wiley, New York 1974).
22. W. Feller: An Introduction to Probability Theory and Its Applications, vol. 2, second edition (Wiley, New York 1971).
23. C.W. Gardiner: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer Series in Synergetics, vol. 13, second edition (Springer, Berlin Heidelberg New York 1990).
24. G. Kallianpur: Stochastic Filtering Theory. Applications of Mathematics, vol. 13 (Springer, Berlin Heidelberg New York 1980).
25. T.D. Frank: Nonlinear Fokker-Planck Equations. Springer Series in Synergetics. (Springer, Berlin Heidelberg New York 2005).
26. K. Oelschlager: A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12, 458-479 (1984).
27. J. Gartner: On the McKean-Vlassov limit for interacting diffusions. Math. Nachr. 137, 197-248 (1988).
28. A.S. Sznitman: A fluctuation result for nonlinear diffusions. In: Infinite Dimensional Analysis and Stochastic Processes, ed. by S. Albeverio, vol. 124, 145-160 (Pitman, Boston 1985).
29. H.P. McKean: Propagation of chaos for a class of non-linear parabolic equations. In: Stochastic Differential Equations, ed. by A. K. Aziz, Lecture Series in Differential Equations, vol. 2, 177-194 (Van Nostrand, Amsterdam 1969).
30. A.S. Sznitman: Topics in propagation of chaos. In: Ecole d'Ete de Probabilites de Saint-Flour, ed by P. L. Hennequin, Lecture Notes in Computer, vol. 1464, 165-251 (Springer, Berlin Heidelberg New York 1991).
31. N. Fournier, S. Meleard: A Markov process associated with a Boltzmann equation without cutoff and for non-Maxwell molecules. J. Stat. Phys. 104, 359-385 (2001).
32. K. Nanbu: Interrelations between various direct simulation methods for solving the Boltzmann equation. J. Phys. Soc. Jpn. 52, 3382-3388 (1983).
33. G.A. Bird: Molecular Gas Dynamics, Oxford Engineering Science Series (Clarendon, Oxford 1976).
34. A.M. Kogan: Derivation of Grad's type equations and study of their relaxation properties by the method of maximization of entropy. J. Appl. Math. Mech. 29, 130-142 (1965).
35. R.M. Lewis: A unifying principle in statistical mechanics. J. Math. Phys. 8, 1448-1459 (1967).
36. A.N. Gorban, I.V. Karlin: General approach to constructing models of the Boltzmann equation. Physica A 206, 401-420 (1994).
37. J.G. Kirkwood: The statistical mechanical theory of transport processes. I. General theory. J. Chem. Phys. 14, 180-201 (1946).
38. J.L. Lebowitz, H.L. Frisch, E. Helfand: Nonequilibrium distribution functions in a fluid. Phys. Fluids 3, 325-338 (1960).
39. Y.L. Klimontovich: On the need for and the possibility of a unified description of kinetic and hydrodynamic processes. Theor. Math. Phys. 92, 909-921 (1992).
40. S. Hess: Boltzmann equation approach to polymer statistics. Physica A 112, 287-302 (1982).
41. M. Rubinstein, R.H. Colby: Polymer Physics. (Oxford University Press 2003).
42. H. Matsumoto: Variable sphere molecular model for inverse power law and Lennard-Jones potentials in Monte Carlo simulations. Phys. Fluids 14, 4256- 4265 (2002).
43. H.C. Ottinger: GENERIC formulation of Boltzmann's kinetic equation. J. Non- Equilib. Thermodyn. 22, 386-391 (1997).
44. M. Hutter, H.C. Ottinger: Modification of linear response theory for mean field approximations. Phys. Rev. E 54, 2526-2530 (1996).
45. M. Hutter, I.V. Karlin, H.C. Ottinger: Dynamic mean-field models from a nonequilibrium thermodynamics perspective. Phys. Rev. E 68, 016115 (2003).
46. H.A. Kramers: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284-304 (1940).