Staggered Ladder Spectra

TL;DR

This paper presents an exact solution to a Fokker-Planck equation with staggered ladder spectra, revealing anomalous diffusion and non-Maxwellian distributions in a generalized Ornstein-Uhlenbeck process.

Contribution

It introduces a novel method to solve the Fokker-Planck equation using ladder operators, uncovering staggered spectra and exact eigenfunctions for the system.

Findings

Exact eigenvalues and eigenfunctions derived

Revealed staggered ladder spectra for parity states

Observed anomalous diffusion and non-Maxwellian stationary distribution

Abstract

We exactly solve a Fokker-Planck equation by determining its eigenvalues and eigenfunctions: we construct nonlinear second-order differential operators which act as raising and lowering operators, generating ladder spectra for the odd and even parity states. These are staggered: the odd-even separation differs from even-odd. The Fokker-Planck equation describes, in the limit of weak damping, a generalised Ornstein-Uhlenbeck process where the random force depends upon position as well as time. Our exact solution exhibits anomalous diffusion at short times and a stationary non-Maxwellian momentum distribution.