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

Abstract

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 WD(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.