Abstract
We review the derivation of the Kac master equation model for random collisions of particles, its relationship to the Poisson process, and existing algorithms for simulating values from the marginal distribution of velocity for a single particle at any given time. We describe and implement a new algorithm that efficiently and more fully leverages properties of the Poisson process, show that it performs at least as well as existing methods, and give empirical evidence that it may perform better at capturing the tails of the single particle velocity distribution. Finally, we derive and implement a novel “ε-perfect sampling” algorithm for the limiting marginal distribution as time goes to infinity. In this case the importance is a proof of concept that has the potential to be expanded to more interesting (DSMC) direct simulation Monte Carlo applications.
References
[1] Babovsky H. and Illner R., A convergence proof for Nanbu’s simulation method for the full Boltzmann equation, SIAM J. Numer. Anal. 26 (1989), 45–65. 10.1137/0726004Search in Google Scholar
[2] Bird G. A., Molecular Gas Dynamics, Clarendon Press, Oxford, 1994. Search in Google Scholar
[3] Carlen E., Carvalho M. and Loss M., Kinetic theory and the Kac master equation, Entropy & the Quantum II, Contemp. Math. 552, American Mathematical Society, Providence (2011), 1–20. 10.1090/conm/552/10908Search in Google Scholar
[4] Carlen E. A., Carvalho M. C., Le Roux J., Loss M. and Villani C., Entropy and chaos in the Kac model, Kinet. Relat. Models 3 (2010), no. 1, 85–122. 10.3934/krm.2010.3.85Search in Google Scholar
[5] Casella G., Lavine M. and Robert C. P., Explaining the perfect sampler, Amer. Statist. 55 (2001), no. 4, 299–305. 10.1198/000313001753272240Search in Google Scholar
[6] Corcoran J. N. and Tweedie R. L., Perfect sampling of ergodic Harris chains, Ann. Appl. Probab., 11 (2001), no. 2, 438–451. 10.1214/aoap/1015345299Search in Google Scholar
[7] Kac M., Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Volume 3: Contributions to Astronomy and Physics, University of California Press, Berkeley (1956), 171–197. 10.1525/9780520350694-012Search in Google Scholar
[8] Krook M. and Wu T. T., Formation of Maxwellian tails, Phys. Rev. Lett. 36 (1976), no. 19, 1107–1109. 10.1103/PhysRevLett.36.1107Search in Google Scholar
[9] Nanbu K., Direct simulation scheme derived from the Boltzmann equation. I: Monocomponent gases, J. Phys. Soc. Japan 49 (1980), no. 5, 2042–2049. 10.1143/JPSJ.49.2042Search in Google Scholar
[10] Wagner W., A convergence proof for Bird’s direct simulation Monte Carlo method for the Boltzmann equation, J. Statist. Phys. 66 (1992), 1011–1044. 10.1007/BF01055714Search in Google Scholar
© 2016 by De Gruyter
