Staggered Ladder Spectra

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. The ladders are staggered: the odd-even separation differs from even-odd. The Fokker-Planck equation corresponds, in the limit of weak damping, to a generalized Ornstein-Uhlenbeck process where the random force depends upon position as well as time. The process describes damped stochastic acceleration, and exhibits anomalous diffusion at short times and a stationary non-Maxwellian momentum distribution.

Generalized Ornstein-Uhlenbeck processes

“…for positive integers l. This result is consistent with the result obtained in [2] (eq. (8) in that paper).…”

“…The corresponding eigenfunctions are generated by a raising operator. A concise account of our work on these staggered ladder spectra appeared earlier [2]. In the following we show how the results summarised in [2] were obtained.…”

“…A concise account of our work on these staggered ladder spectra appeared earlier [2]. In the following we show how the results summarised in [2] were obtained. We also derive new results, not included in our earlier report: a closed-form solution for example, and the generalisation of our previous results to a continuous family of diffusion processes.…”

“…We remark that a brief summary of many of the results of this paper was already published [2]. The closed-form solution of section 3, the WKB analysis and most of the results for general values of ζ were not discussed in this earlier work.…”

Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

“…This scaling is responsible for the interesting effects discussed in the following. The addition of a linear force to the logarithmic potential breaks this scaling and leads to a very different behavior [30][31][32][33]. ) and their time average (thick red) for the Brownian particle moving in an asymptotically logarithmic potential U (x) = (U0/2) ln(1 + x 2 ) (top panel, inset).…”