Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling

TL;DR

This paper investigates the ergodic properties of three stochastic models related to subrecoil laser cooling, revealing universal behavior for integrable observables and model-specific laws for non-integrable ones, with implications for understanding cooling processes.

Contribution

It provides exact forms of infinite invariant densities for two models and an approximation for the third, analyzing how ergodic properties depend on observable types across different models.

Findings

Universal ergodic behavior for integrable observables via Darling-Kac theorem.

Non-universal, model-dependent laws for non-integrable observables.

Exact invariant densities and a new approximation method for the models.

Abstract

We compare ergodic properties of the kinetic energy for three stochastic models of subrecoil-laser-cooled gases. One model is based on a heterogeneous random walk (HRW), another is an HRW with long-range jumps (the exponential model), and the other is a mean-field-like approximation of the exponential model (the deterministic model). All the models show an accumulation of the momentum at zero in the long-time limit, and a formal steady state cannot be normalized, i.e., there exists an infinite invariant density. We obtain the exact form of the infinite invariant density and the scaling function for the exponential and deterministic models and devise a useful approximation for the momentum distribution in the HRW model. While the models are kinetically non-identical, it is natural to wonder whether their ergodic properties share common traits, given that they are all described by an infinite invariant density. We show that the answer to this question depends on the type of observable under study. If the observable is integrable, the ergodic properties such as the statistical behavior of the time averages are universal as they are described by the Darling-Kac theorem. In contrast, for non-integrable observables, the models in general exhibit non-identical statistical laws. This implies that focusing on non-integrable observables, we discover non-universal features of the cooling process, that hopefully can lead to a better understanding of the particular model most suitable for a statistical description of the process. This result is expected to hold true for many other systems, beyond laser cooling.