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Self-diffusion in a non-uniform one dimensional system of point particles with collisions
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  • Published: 01 June 1987

Self-diffusion in a non-uniform one dimensional system of point particles with collisions

  • Detlef Dürr1,
  • Sheldon Goldstein2 &
  • Joel L. Lebowitz2 

Probability Theory and Related Fields volume 75, pages 279–290 (1987)Cite this article

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.

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References

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Author information

Authors and Affiliations

  1. BIBOS, Universität Bielefeld, D-4800, Bielefeld, Germany

    Detlef Dürr

  2. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    Sheldon Goldstein & Joel L. Lebowitz

Authors
  1. Detlef Dürr
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  2. Sheldon Goldstein
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  3. Joel L. Lebowitz
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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

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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

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  • Received: 28 April 1986

  • Revised: 29 December 1986

  • Published: 01 June 1987

  • Issue Date: June 1987

  • DOI: https://doi.org/10.1007/BF00354038

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Keywords

  • Differential Equation
  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Statistical Theory
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