Akimoto Takuma, Barkai Eli, Radons Günter
Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan.
Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel.
Phys Rev E. 2022 Jun;105(6-1):064126. doi: 10.1103/PhysRevE.105.064126.
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.
我们比较了亚反冲激光冷却气体的三种随机模型中动能的遍历性质。一种模型基于异质随机游走（HRW），另一种是具有长程跳跃的HRW（指数模型），还有一种是指数模型的类平均场近似（确定性模型）。所有模型在长时间极限下都显示出动量在零处的积累，并且形式上的稳态无法归一化，即存在无限不变密度。我们得到了指数模型和确定性模型的无限不变密度的精确形式以及标度函数，并为HRW模型中的动量分布设计了一种有用的近似。虽然这些模型在动力学上不相同，但鉴于它们都由无限不变密度描述，人们自然会想它们的遍历性质是否有共同特征。我们表明，这个问题的答案取决于所研究的可观测量的类型。如果可观测量是可积的，那么遍历性质，比如时间平均的统计行为，是普遍的，因为它们由达林 - 卡茨定理描述。相反，对于不可积的可观测量，这些模型通常表现出不同的统计规律。这意味着关注不可积可观测量时，我们发现了冷却过程的非普遍特征，这有望使我们更好地理解最适合对该过程进行统计描述的特定模型。预计这个结果在激光冷却之外的许多其他系统中也成立。
