玻尔兹曼等先验概率假设的一个简单数学证明
A simple mathematical proof of Boltzmann's equal a priori probability hypothesis
摘要：
利用耗散定理和涨落定理的一个推论，即第二定律不等式，给出了微正则系综在相空间中先验概率相等的玻尔兹曼假设的第一性原理推导.我们证明，如果初始分布不同于能量超曲面上的均匀分布，那么在非常广泛和普遍满足的条件下，初始分布将松弛到均匀分布。这个结果有点类似于玻尔兹曼H定理，但与该定理不同的是，它适用于稠密流体以及稀气体，并且还允许非单调松弛到平衡。我们还证明了在遍历系统中的均匀（微正则）分布是唯一的固定的，耗散的恒定能量合奏分布。
英文摘要：
Using the Dissipation Theorem and a corollary of the Fluctuation Theorem, namely the Second Law Inequality, we give a first-principles derivation of Boltzmann's postulate of equal a priori probability in phase space for the microcanonical ensemble. We show that if the initial distribution differs from the uniform distribution over the energy hypersurface, then under very wide and commonly satisfied conditions, the initial distribution will relax to that uniform distribution. This result is somewhat analogous to the Boltzmann H-theorem but unlike that theorem, applies to dense fluids as well as dilute gases and also permits a nonmonotonic relaxation to equilibrium. We also prove that in ergodic systems the uniform (microcanonical) distribution is the only stationary, dissipationless distribution for the constant energy ensemble.
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