Generalised Ornstein-Uhlenbeck processes

TL;DR

This paper extends the classical Ornstein-Uhlenbeck process to include position-dependent forces, revealing anomalous diffusion and non-Maxwellian distributions, using spectral and propagator methods.

Contribution

It introduces a generalized Ornstein-Uhlenbeck model with position-dependent forces and analyzes its properties with novel spectral and propagator techniques.

Findings

Exhibits anomalous diffusion at short times

Shows non-Maxwellian velocity distributions in equilibrium

Provides closed-form propagator for specific initial conditions

Abstract

We solve a physically significant extension of a classic problem in the theory of diffusion, namely the Ornstein-Uhlenbeck process [G. E. Ornstein and L. S. Uhlenbeck, Phys. Rev. 36, 823, (1930)]. Our generalised Ornstein-Uhlenbeck systems include a force which depends upon the position of the particle, as well as upon time. They exhibit anomalous diffusion at short times, and non-Maxwellian velocity distributions in equilibrium. Two approaches are used. Some statistics are obtained from a closed-form expression for the propagator of the Fokker-Planck equation for the case where the particle is initially at rest. In the general case we use spectral decomposition of a Fokker-Planck equation, employing nonlinear creation and annihilation operators to generate the spectrum which consists of two staggered ladders.