Generalised Ornstein-Uhlenbeck processes

Details Generalised Ornstein-Uhlenbeck processes Generalised Ornstein-Uhlenbeck processes

2006-02-23

arxiv.org/abs/cond-mat/0602548v1

J. Math. Phys. 47, 073301 (2006)

Physics

Condensed Matter

Statistical Mechanics

24 pages, 2 figures

Scientific paper

10.1063/1.2206878

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.

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