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

Takuma Akimoto, Eli Barkai, Günter Radons

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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 we devise a useful approximation for the momentum distribution in the HRW model. While the models are kinetically nonidentical, 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 nonintegrable observables, the models in general exhibit nonidentical statistical laws. This implies that focusing on nonintegrable observables, we discover nonuniversal features of the cooling process, which 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.

Original languageEnglish
Article number064126
JournalPhysical Review E
Issue number6
StatePublished - Jun 2022

Bibliographical note

Publisher Copyright:
© 2022 authors. Published by the American Physical Society.


T.A. was supported by JSPS Grant-in-Aid for Scientific Research (No. C JP18K03468). The support of Israel Science Foundation's Grant No. 1898/17 is acknowledged (E.B.).

FundersFunder number
Japan Society for the Promotion of ScienceJP18K03468
Israel Science Foundation1898/17


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