Abstract
We consider the trajectoryQ M(t) of a Brownian particle of massM in an ideal gas of identical particles of mass 1 and of density 1 in equilibrium at inverse temperature 1 (the dynamics is uniform motion plus elastic collisions with the Brownian particle). Our theory, in dimension one, describes a variety of limiting processes — containing the Wiener process and the Ornstein-Uhlenbeck process — forA −1/2 Q M(A)(At) depending on the asymptotic behaviour ofM(A). Part of the theory is hypothetical while another part relies upon known results. We also prove that, ifA 1/2+ε≪M(A)≪A, thenA −1/2 Q M(A) (At) converges to a Wiener process whose variance is known from papers of Sinai-Soloveichik and of the present authors.
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References
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Bürger, G.A.: Des Freiherrn von Münchhausen Reisen und Abenteuer zu Wasser und Lande, wie er dieselben bei einer Flasche im Zirkel seiner Freunde zu erzählen pflegte. Göttingen, × 1788
Dürr, D., Goldstein, S.: Remarks on the central limit theorem for weakly dependent random variables. In: Stochastic processes — Mathematics and physics. Albeverid, S., Blanchard, P., Streit, L. (eds.). Lecture Notes in Mathematics, Vol. 1158. Berlin, Heidelberg, New York: Springer 1986
Dürr, D., Goldstein, S., Lebowitz, J.L.: A mechanical model of Brownian motion. Commun. Math. Phys.78, 507–530 (1981)
Goldstein, S., Guetti, J.: On the diffusion of the fast molecule. J. Stat. Phys. (to appear)
Gordin, M.I., Lifshic, B.A.: Central limit theorem for stationary Markov chains. Dokl. Akad. Nauk. SSSR239, 766–767 (1978)
Helland, I.: Central limit theorems for martingales with discrete or continuous time. Scand. J. Stat.9, 79–94 (1982)
Harris, T.E.: Diffusions with collisions between particles. J. Appl. Probab.2, 323–338 (1965)
Holley, R.: The motion of a heavy particle in an infinite one dimensional gas of hard spheres. Z. Wahrscheinlichkeitstheor. Verw. Geb.17, 181–219 (1971)
Kipnis, C., Varadhan, S.: A central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Commun. Math. Phys.104, 1–19 (1986)
Nelson, E.: Dynamical theory of Brownian motion. Princeton, NJ: Princeton University Press 1967
Neveu, J.: Bases mathématiques du calcul des probabilités. Paris: Masson 1964
Omerti, E., Ronchetti, M., Dürr, D.: Numerical evidence for mass dependence in the diffusive behaviour of the “Heavy Particle” on the line. J. Stat. Phys.44, 339–346 (1986)
Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press 1978
Sinai, Ya.G.: Oral communication
Sinai, Ya. G., Soloveichik, M.R.: One-dimensional classical massive particle in the ideal gas. Commun. Math. Phys.104, 423–443 (1986)
Spitzer, F.: Uniform motion with elastic collisions of an infinite particle system. J. Math. Mech.18, 973–989 (1969)
Szász, D., Tóth, B.: Bounds on the limiting variance of the heavy particle. Commun. Math. Phys.104, 445–455 (1986)
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Communicated by J. L. Lebowitz
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Szász, D., Tóth, B. Towards a unified dynamical theory of the Brownian particle in an ideal gas. Commun.Math. Phys. 111, 41–62 (1987). https://doi.org/10.1007/BF01239014
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DOI: https://doi.org/10.1007/BF01239014