A fundamental postulate of statistical mechanics is that all microstates in an isolated system are equally probable. This postulate, which goes back to Boltzmann, has often been criticized for not having a clear physical foundation. In this note, we provide a derivation of the canonical (Boltzmann) distribution that avoids this postulate. In its place, we impose two axioms with physical interpretations. The first axiom (thermal equilibrium) ensures that, as our system of interest comes into contact with different heat baths, the ranking of states of the system by probability is unchanged. Physically, this axiom is a statement that in thermal equilibrium, population inversions do not arise. The second axiom (energy exchange) requires that, for any heat bath and any probability distribution on states, there is a universe consisting of a system and heat bath that can achieve this distribution. Physically, this axiom is a statement that energy flows between system and heat bath are unrestricted. We show that our two axioms identify the Boltzmann distribution.

中文翻译：

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玻尔兹曼分布公理

统计力学的一个基本假设是，孤立系统中的所有微观状态都是等概率的。这个可以追溯到玻尔兹曼的假设经常因为没有明确的物理基础而受到批评。在本说明中，我们提供了避免此假设的规范 (Boltzmann) 分布的推导。取而代之的是，我们强加了两个具有物理解释的公理。第一个公理（热平衡）确保，当我们感兴趣的系统与不同的热浴接触时，系统状态的概率排序不变。从物理上讲，这个公理是一个陈述，即在热平衡中，不会出现种群倒置。第二个公理（能量交换）要求，对于任何热浴和状态的任何概率分布，有一个由系统和热浴组成的宇宙可以实现这种分布。从物理上讲，这个公理表明系统和热浴之间的能量流动是不受限制的。我们证明了我们的两个公理确定了 Boltzmann 分布。