Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model

TL;DR

This paper analyzes the persistence probability of a particle in power-law and logarithmic potentials, revealing how noise relevance varies with potential strength and applying findings to vortex dynamics in the 2D XY model.

Contribution

It provides a detailed theoretical analysis of persistence in logarithmic and power-law potentials, connecting these results to vortex annihilation in the 2D XY model.

Findings

Persistence probability decays as a power-law for the logarithmic potential case.

The mean vortex-antivortex annihilation time scales as r^2 ln(r/a).

The Langevin equation can be transformed to a standard form despite multiplicative noise.

Abstract

The Langevin equation for a particle (`random walker') moving in d-dimensional space under an attractive central force, and driven by a Gaussian white noise, is considered for the case of a power-law force, F(r) = - Ar^{-sigma}. The `persistence probability', P_0(t), that the particle has not visited the origin up to time t, is calculated. For sigma > 1, the force is asymptotically irrelevant (with respect to the noise), and the asymptotics of P_0(t) are those of a free random walker. For sigma < 1, the noise is (dangerously) irrelevant and the asymptotics of P_0(t) can be extracted from a weak noise limit within a path-integral formalism. For the case sigma=1, corresponding to a logarithmic potential, the noise is exactly marginal. In this case, P_0(t) decays as a power-law, P_0(t) \sim t^{-theta}, with an exponent theta that depends continuously on the ratio of the strength of the potential to the strength of the noise. This case, with d=2, is relevant to the annihilation dynamics of a vortex-antivortex pair in the two-dimensional XY model. Although the noise is multiplicative in the latter case, the relevant Langevin equation can be transformed to the standard form discussed in the first part of the paper. The mean annihilation time for a pair initially separated by r is given by t(r) \sim r^2 ln(r/a) where a is a microscopic cut-off (the vortex core size). Implications for the nonequilibrium critical dynamics of the system are discussed and compared to numerical simulation results.