Abstract
We consider the M→0 limit for tagged particle diffusion in a 1-dimensional Rayleigh-gas, studied originaly by Sinai and Soloveichik [Ya. G. Sinai, M. R. Soloveichik, Commun. Math. Phys. 104:423–443 (1986)], and by Szász and Tóth [D. Szász, B. Tóth, Commun. Math. Phys. 104:445–457 (1986)], respectively. In this limit we derive a new type of model for tagged particle diffusion, for which the two central particles, in addition to elastic collisions with the rest of the gas, interact with Calogero-Moser-Sutherland (i.e. inverse quadratic) potential. Computer simulations on this new model reproduce exactly the numerical value of the limiting variance obtained by Boldrighini, Frigio and Tognetti in [C. Boldrighini, S. Frigio, D. Tognetti, J. Stat. Phys. 108:703–712 (2002)].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Bálint, B. Tóth and P. Tóth, New upper bounds for the limiting variance of tagged particle diffusion in 1-d. In preparation.
P. Billingsley, Convergence of Probability Measures. (Wiley, New York, 1968).
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 (1990).
C. Boldrighini, S. Frigio and D. Tognetti, Numerical evidence for the low-mass behavior of the one-dimensional Rayleigh gas with local interaction. J. Stat. Phys. 108:703–712 (2002).
F. Calogero, Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potential. J. Math. Phys. 12:419–436 (1971).
D. Dürr, S. Goldstein and J. L. Lebowitz, A mechanical model of Brownian motion. Commun. Math. Phys. 78:507–530 (1981).
J. F. Fernandez and J. Marro, Diffusion in a one-dimensional gas of hard point particles. J. Stat. Phys. 71:225–233 (1993).
T. E. Harris, Diffusions with collisions between particles. J. Appl. Probability 2:323–338 (1965).
R. Holley, The motion of a heavy particle in an infinite one-dimensional gas of hard spheres. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 17:181–219 (1971).
M. L. Khazin, Investigation of the diffusion of a massive particle in a one-dimensional ideal gas. Theoret. Math. Phys. 71:299–303 (1987).
J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16:197–220 (1975).
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. 44:339–346 (1986).
A. I. Neishtadt and Ya. G. Sinai, Adiabatic piston as a dynamical system. J. Stat. Phys. 116:815–820 (2004).
D. Ruelle and Ya. G. Sinai, From dynamical systems to statistical mechanics and back. Physica A140:1–8 (1986).
Ya. G. Sinai, Dynamics of a massive particle surrounded by a finite number of light particles. Theor. Math. Phys. 121:1351–1357 (1999).
Ya. G. Sinai and M. R. Soloveichik, One-dimensional classical massive particle in the ideal gas. Commun. Math. Phys. 104:423–443 (1986).
M. Soloveitchik, Mechanical background of Brownian motion. In: Sinai's Moscow Seminar on Dynamical Systems, vol. 171, L. A. Bunimovich, B. M. Gurevich, and Ya. B. Pesin (eds.) (AMS Translations, Series 2, 1996).
M. Soloveitchik, Hydrodynamic scale for a driven tracer particle: rigorous results. Rendiconti di Matematica, Serie VII 22:1–146 (2002).
F. Spitzer, Uniform motion with elastic collisions of an infinite particle system. J. Math. Mech. 18:973–989 (1969).
B. Sutherland, Exact results for a quantum many body problem in one dimension, II. Phys. Rev. A5:1372–1376 (1972).
D. Szász and B. Tóth, Bounds on the limiting variance of the “heavy particle” in ℝ1. Commun. Math. Phys. 104:445–457 (1986).
D. Szász and B. Tóth, Towards a unified dynamical theory of the Brownian particle in an ideal gas. Commun. Math. Phys. 111:41–62 (1987).
D. Szász and B. Tóth, Dynamical theory of the Brownian particle in a Rayleigh-gas. J. Stat. Phys. 47:681–695 (1987).
P. Wright, A simple piston problem in one dimension. Nonlinearity 19:2365–2389 (2006).
Description of the numerical simulation and program code available online at http://www.renyi.hu/mogy/publications/szimulacio/brown1d
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Domokos Szász on his 65th birthday.
Rights and permissions
About this article
Cite this article
Bálint, P., Tóth, B. & Tóth, P. On the Zero Mass Limit of Tagged Particle Diffusion in the 1-d Rayleigh-Gas. J Stat Phys 127, 657–675 (2007). https://doi.org/10.1007/s10955-007-9304-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-007-9304-2