论文标题
Boltzmann方程可逆吗？关于不可逆性悖论的大偏差观点
Is the Boltzmann equation reversible? A large deviation perspective on the irreversibility paradox
论文作者
论文摘要
我们考虑了Boltzmann稀释气体的动力学理论 - Grad限制。我们根据经验分布动态的概率进行了基于大偏差估计的新观点。假设Boltzmann分子混乱假说（Stosszahlansatz），我们得出了一个较大的偏差率函数或作用，描述了经验分布的随机过程。如预期的那样，此动作的准跨性是熵的负。虽然Boltzmann方程似乎是最可能的演化，与大型定律相对应，但该动作描述了经验分布的真正可逆随机过程，与微观的可逆性一致。结果，这种较大的偏差视角赋予了Boltzmann方程的预期含义，并将其不可逆性解释为将物理描述限制为最可能进化的自然结果。更有趣的是，它还量化了与Boltzmann方程解决方案的任何动态演化的概率。这张图片与不可逆性的启发式经典观点完全兼容，但以各种方式使其更加精确。我们还解释说，这种大型偏差动作为玻尔兹曼方程提供了自然的梯度结构。
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, 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.
