Abstract
A dynamical theory of the Brownian motion is worked out for the Rayleigh gas and open problems of this theory are surveyed.
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C. Boldrighini, A. Pellegrinotti, E. Presutti, Ya. G. Sinai, and M. R. Soloveychik, Ergodic properties of a semi-infinite one-dimensional system of statistical mechanics,Commun. Math. Phys. 101:363–382 (1985).
D. Brydges and T. Spencer, Self-avoiding walk in 5 or more dimensions,Commun. Math. Phys. 97:125–148 (1985).
R. L. Dobrushin, presented at IAMP congress (1986).
D. Dürr and S. Goldstein, Remarks on the central limit theorem for weakly dependent random variables, inStochastic Processes—Mathematics and Physics, S. Albeverio, P. Blanchard, and L. Streit, eds. (Springer, 1986).
D. Dürr, S. Goldstein, and J. L. Lebowitz, A mechanical model of Brownian motion,Commun. Math. Phys. 78:507–530 (1981).
M. I. Gordin and B. A. Lifschic, Central limit theorem for stationary Markov chains,Dokl. Akad. Nauk SSSR 239:766–767 (1978).
T. G. Harris, Diffusions with collisions between particles,J. Appl. Prob. 2:323–338 (1965).
R. Holley, The motion of a heavy particle in an infinite one dimensional gas of hard spheres,Z. Wahrsch. Verw. Geb. 17:181–219 (1971).
C. Kipnis and S. Varadhan, A central limit theorem for additive functional of reversible Markov processes and applications to simple exclusion,Commun. Math. Phys. 104:1–19 (1986).
J. L. Lebowitz and J. K. Percus, Kinetic equations and density expansions,Phys. Rev. 155:48 (1967).
E. Nelson,Dynamical Theory of Brownian Motion (Princeton University Press, 1967).
E. Omerti, M. Ronchetti, and D. Dürr, Numerical evidence for mass dependence in the diffusive behaviour of the “heavy particle” on the line,J. Stat. Phys. 44:339 (1986).
Ya. G. Sinai, Personal communication.
Ya. G. Sinai and M. R. Soloveychik, One-dimensional classical massive particle in the ideal gas,Commun. Math. Phys. 104:423–443 (1986).
F. Spitzer, Uniform motion with elastic collisions of an infinite particle system,J. Math. Mech. 18:973–989 (1969).
H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits,Rev. Mod. Phys. 53:569 (1980).
D. Szász and B. Tóth, Bounds on the limiting variance of the heavy particle,Commun. Math. Phys. 104:445–455 (1986).
D. Szász and B. Tóth, Towards a unified dynamical theory of the Brownian particle in an ideal gas,Commun. Math. Phys., to appear.
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Szász, D., Tóth, B. A dynamical theory of Brownian motion for the Rayleigh gas. J Stat Phys 47, 681–693 (1987). https://doi.org/10.1007/BF01206152
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DOI: https://doi.org/10.1007/BF01206152