Single-vacancy induced motion of a tracer particle in a two-dimensional lattice gas

Abstract

We study the random motion of a tracer particle in a two-dimensional dense lattice gas. Repeated encounters of asingle vacancy displace the tracer particle from its initial position by a vector y of which we calculate the time-dependent distributionPt(y). On an infinite lattice and for large times $$P_t (y) \simeq \frac{{2(\pi - 1)}}{{\ln t}}K_0 \left( {\left( {\frac{{4\pi (\pi - 1)}}{{\ln t}}} \right)^{1/2} y} \right)$$ whereK0 is a modified Bessel function. The same problem is studied on a finiteL×L lattice with periodic boundary conditions; therePt(y) is shown to be a Gaussian on a time scaleL2 InL. On an ∞×L strip and for large times,Pt(y) is an explicitly given (but nonelementary) function of the scaling variable ξy1/t1/4, identical to the function occurring in the problem of a random walker on a random one-dimensional path.