Summary
We generalize the results of Spitzer, Jepsen and others [1–4] on the motion of a tagged particle in a uniform one dimensional system of point particles undergoing elastic collisions to the case where there is also an external potential U(x). When U(x) is periodic or random (bounded and statistically translation invariant) then the scaled trajectory of a tagged particle \({{y_A (t) = y(At)} \mathord{\left/ {\vphantom {{y_A (t) = y(At)} {\sqrt A }}} \right. \kern-\nulldelimiterspace} {\sqrt A }}\) converges, as A → ∞, to a Brownian motion W D (t) with diffusion constant \({{D = \rho _{\min } \left\langle {|v|} \right\rangle } \mathord{\left/ {\vphantom {{D = \rho _{\min } \left\langle {|v|} \right\rangle } {\hat \rho ^2 }}} \right. \kern-\nulldelimiterspace} {\hat \rho ^2 }}\), where \(\hat \rho \) is the average density, \(\left\langle {|v|} \right\rangle = \sqrt {{2 \mathord{\left/ {\vphantom {2 {\pi \beta m}}} \right. \kern-\nulldelimiterspace} {\pi \beta m}}} \) is the mean absolute velocity and β−1 the temperature of the system. When U(x) is itself changing on a macroscopic scale, i.e. \(U_A (x) = U({x \mathord{\left/ {\vphantom {x {\sqrt A }}} \right. \kern-\nulldelimiterspace} {\sqrt A }})\), then the limiting process is a spatially dependent diffusion. The stochastic differential equation describing this process is now non-linear, and is particularly simple in Stratonovich form. This lends weight to the belief that heuristics are best done in that form.
References
Spitzer, F.: Uniform motion with elastic collisions of an infinite particle system. J. Math. Mech. 18, 973–989 (1968)
Jepsen, D.W.: J. Math. Phys. 6, 405 (1965)
Lebowitz, J.L., Percus, J.K.: Kinetic equations and density expansions: Exactly solvable onedimensional model system. Phys. Rev. 155, 122 (1967)
Dürr, D., Goldstein, S., Lebowitz, J.L.: Asymptotics of particle trajectories in infinite one dimensional systems with collisions. Comm. Pure Appl. Math. 38, 573 (1985)
Einstein, A.: Investigations on the theory of the brownian movement. Dover 1965
Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1 (1934)
Uhlenbeck, G.E., Ornstein, L.S.: On the theory of the brownian motion. Phys. Rev. 36, 823 (1930)
cf.: Kubo, R., Toda, M., Hashitsume, N.: Statistical physics II. Springer Series in Solid-State Science 31 (1985)
van Kampen, N.: Stochastic processes in physics and chemistry. Amsterdam: North Holland 1981
cf: van Kampen, N.G.: Ito versus Stratonovich. J. Stat. Phys. 24, 175 (1981)
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Ferrari, P.A., Goldstein, S., Lebowitz, J.L.: Diffusion, mobility and the Einstein relation. In: Fritz, Jaffe, Szasz (eds.). Statistical physics and dynamical systems. Basel: Birkhauser 1985
Golden, K., Goldstein, S., Lebowitz, J.L.: Classical transport in modulated structures. Phys. Rev. Letters 55, 2629 (1985)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North Holland 1982
Author information
Authors and Affiliations
Additional information
Dedicated to Frank Spitzer on the occasion of his 60th birthday
Work supported in part by NSF Grants No. PHY 8201708 and No. DMR 81-14726
Heisenberg-fellow
Also Department of Physics
Rights and permissions
About this article
Cite this article
Dürr, D., Goldstein, S. & Lebowitz, J.L. Self-diffusion in a non-uniform one dimensional system of point particles with collisions. Probab. Th. Rel. Fields 75, 279–290 (1987). https://doi.org/10.1007/BF00354038
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00354038
Keywords
- Differential Equation
- Stochastic Process
- Brownian Motion
- Probability Theory
- Statistical Theory