Abstract
Numerical investigations, supported by partial rigorous results suggest that the motion of a tagged particle of mass M on the line ℝ1 colliding with a free gas of particles of mass m in equilibrium is diffusive. It was conjectured that the diffusion constant D(M), for small mass M→0, should approach D(m). (In dimension 1 this is a kind of “continuity hypothesis.”) Previous results of computer simulations are inconclusive. We report on some new computer results, which show clearly that there is no continuity, and the limit of D(M) as M→0 is smaller than D(m). We compare with the corresponding results for a similar two-dimensional model, to which the “continuity argument” cannot be applied.
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REFERENCES
Ya. G. Sinai and M. R. Soloveichick, One-dimensional classical massive particle in the ideal gas, Commun. Math. Phys. 104:423-443 (1986).
D. Szász and B. Tóth, Bounds for the limiting variance of the “heavy particle” in ℝ1, 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, Comm. Math. Phys. 111:41-62 (1987).
P. Calderoni and D. Dürr, The Smoluchowski limit for a simple mechanical model, J. Statist. Phys. 55:695-738 (1989).
F. Spitzer, J. Math. Mech. 18:973 (1969).
T. E. Harris, Diffusion with “collisions” between particles, J. Appl. Prob. 2:323 (1965).
C. Boldrighini and M. Soloveichick, Drift and diffusion for a mechanical system, Prob. Theory Relat. Fields 103:349-379 (1995).
C. Boldrighini and M. Soloveitchik, On the Einstein relation for a mechanical system, Prob. Theory Relat. Fields 107:493-515 (1997).
M. L. Khazin, Investigation of the diffusion of a massive particle in a one-dimensional ideal gas, Teoret. Mat. Fiz. 71(2):299-303 (1987) [in Russian].
E. Omerti, M. Ronchetti, and D. Dürr, Numerical evidence for mass dependence in the diffusive behavior of the “heavy particle” on the line, J. Stat. Phys. 44:339-346 (1986).
C. Boldrighini, G. C. Cosimi, and S. Frigio, Diffusion and Einstein relation for a massive particle in a one-dimensional free gas: Numerical evidence, J. Stat. Phys. 59:1241-1250 (1989).
P. Erdös and M. Kac, On certain limit theorems of the theory of probability, Bull. Am. Math. Soc. 52:292-302 (1946).
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Boldrighini, C., Frigio, S. & Tognetti, D. Numerical Evidence for the Low-Mass Behavior of the One-Dimensional Rayleigh Gas with Local Interaction. Journal of Statistical Physics 108, 703–712 (2002). https://doi.org/10.1023/A:1015738209984
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DOI: https://doi.org/10.1023/A:1015738209984