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Is the Boltzmann Equation Reversible? A Large Deviation Perspective on the Irreversibility Paradox
Journal of Statistical Physics ( IF 1.6 ) Pub Date : 2020-07-03 , DOI: 10.1007/s10955-020-02588-y
Freddy Bouchet

We consider the kinetic theory of dilute gases in the Boltzmann–Grad limit. We propose a new perspective based on a large deviation estimate for the probability of the empirical distribution dynamics. Assuming Boltzmann molecular chaos hypothesis (Stosszahlansatz), we derive a large deviation rate function, or action, that describes the stochastic process for the empirical distribution. The quasipotential for this action is the negative of the entropy when the conservation laws are verified, as should be expected. While the Boltzmann equation appears as the most probable evolution, corresponding to a law of large numbers, the action describes a genuine reversible stochastic process for the empirical distribution, in agreement with the microscopic reversibility. As a consequence, this large deviation perspective gives the expected meaning to the Boltzmann equation and explains its irreversibility as the natural consequence of limiting the physical description to the most probable evolution. More interestingly, it also quantifies the probability of any dynamical evolution departing from solutions of the Boltzmann equation. This picture is fully compatible with the heuristic classical view of irreversibility, but makes it much more precise in various ways. We also explain that this large deviation action provides a natural gradient structure for the Boltzmann equation.

中文翻译:

玻尔兹曼方程可逆吗?不可逆悖论的大偏差视角

我们在 Boltzmann-Grad 极限中考虑稀释气体的动力学理论。我们提出了一个基于经验分布动态概率的大偏差估计的新视角。假设 Boltzmann 分子混沌假设 (Stosszahlansatz),我们推导出一个大的偏差率函数或动作,它描述了经验分布的随机过程。正如预期的那样,当守恒定律得到验证时,该动作的准势是熵的负值。虽然玻尔兹曼方程看起来是最可能的进化,对应于大数定律,但该动作描述了经验分布的真正可逆随机过程,与微观可逆性一致。作为结果,这种大偏差的观点赋予了玻尔兹曼方程预期的意义,并将其不可逆性解释为将物理描述限制为最可能的进化的自然结果。更有趣的是,它还量化了偏离 Boltzmann 方程解的任何动力学演化的概率。这张图与启发式经典的不可逆性观点完全兼容,但在各种方面使其更加精确。我们还解释了这种大偏差动作为玻尔兹曼方程提供了自然的梯度结构。它还量化了偏离 Boltzmann 方程解的任何动力学演化的概率。这张图与启发式经典的不可逆性观点完全兼容,但在各种方面使其更加精确。我们还解释了这种大偏差动作为玻尔兹曼方程提供了自然的梯度结构。它还量化了偏离 Boltzmann 方程解的任何动力学演化的概率。这张图与启发式经典的不可逆性观点完全兼容,但在各种方面使其更加精确。我们还解释了这种大偏差动作为玻尔兹曼方程提供了自然的梯度结构。
更新日期:2020-07-03
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