The fact that the entropy of an isolated system can never decrease during any transformation has a very clear interpretation from the statistical point of view. Boltzmann has proved that the entropy of a given state of a thermo-dynamical system is connected by a simple relationship to the probability of the state.
We have already emphasized the difference between the dynamical and thermodynamical concepts of the state of a system. To define the dynamical state, it is necessary to have the detailed knowledge of the position and motion of all the molecules that compose the system. The thermo- dynamical state, on the other hand, is defined by giving only a small number of parameters, such as the temperature, pressure, and so forth. It follows, therefore, that to the same thermodynamical state there corresponds a large number of dynamical states. In statistical mechanics, criteria are given for assigning to a given thermodynamical state the number $\pi$ of corresponding dynamical states. (See also section 30.) This number $\pi$ is usually called the probability of the given thermodynamical state, although, strictly speaking, it is only proportional to the probability in the usual sense. The latter can be obtained by dividing $\pi$ by the total number of possible dynamical states.
从统计学的角度来看，孤立系统的熵在任何变换过程中都不会减少这一事实有非常清晰的解释。 玻尔兹曼已经证明，热动力系统给定状态的熵与该状态的概率之间存在着简单的联系。
我们已经强调了系统状态的动力学和热力学概念之间的差异。 要定义动力学状态，需要详细了解组成系统的所有分子的位置和运动。 另一方面，热力学状态是通过仅给出少量参数（例如温度、压力等）来定义的。 因此，对于相同的热力学状态，存在大量的动力学状态。 在统计力学中，给出了将给定热力学状态分配给相应动力学状态的数量 $\pi$ 的标准。 （另见第30节。）这个数字 $\pi$ 通常称为给定热力学状态的概率，尽管严格来说，它只是概率的比例。 后者可以通过将 $\pi$ 除以可能的动力学状态的总数来获得。
We shall now assume, in accordance with statistical considerations, that in an isolated system only those spontaneous transformations occur which take the system to states of higher probability, so that the most stable state of such a system will be the state of highest probability consistent with the given total energy of the system.
We see that this assumption establishes a parallelism between the properties of the probability $\pi$ and the entropy $S$ of our system, and thus suggests the existence of a functional relationship between them. Such a relationship was actually established by Boltzmann, who proved that
$$ \begin{aligned} S = k \log \pi,\tag{75} \end{aligned} $$
where $k$ is a constant called the Boltzmann Constant and is equal to the ratio,
$$ \begin{aligned} \frac{R}{A}\tag{76} \end{aligned} $$
of the gas constant $R$ to Avogadro’s number $A$.
Without giving a proof of (75), we can prove, assuming the existence of a functional relationship between $S$ and $\pi$,
$$ \begin{aligned} S = f(\pi),\tag{77} \end{aligned} $$
that the entropy is proportional to the logarithm of the probability.
我们现在将根据统计考虑，假设在孤立系统中只发生那些将系统转变为更高概率状态的自发变换，因此这样一个系统的最稳定状态将是与系统的给定总能量一致的最高概率状态。
我们看到，这个假设在我们的系统的概率 $\pi$ 和熵 $S$ 的性质之间建立了一个平行关系，从而暗示了它们之间存在一个功能关系。 事实上，玻尔兹曼确实建立了这样一个关系，证明了
$$ \begin{aligned} S = k \log \pi,\tag{75} \end{aligned} $$
其中 $k$ 是一个称为玻尔兹曼常数的常数，等于，
$$ \begin{aligned} \frac{R}{A}\tag{76} \end{aligned} $$
气体常数 $R$ 与阿伏伽德罗数 $A$ 之比。
在不给出 (75) 的证明的情况下，我们可以证明，假设熵 $S$ 和概率 $\pi$ 之间存在一个函数关系，
$$ \begin{aligned} S = f(\pi),\tag{77} \end{aligned} $$
其中熵与概率的对数成正比。
Consider a system composed of two parts, and let $S_{1}$ and $S_{2}$ be the entropies and $\pi_{1}$ and $\pi_{2}$ the probabilities of the states of these parts. We have from (77):
$$ S_{1} = f(\pi_{1}),\quad S_{2} = f(\pi_{2}). $$
But the entropy of the total system is the sum of the two entropies:
$$ S = S_{1} + S_{2} $$
and the probability of the total system is the product of the two probabilities,
$$ \pi = \pi_{1} \pi_{2}. $$
From these equations and from (77) we obtain the following:
$$ f(\pi_{1}\pi_{2}) = f(\pi_{1}) + f(\pi_{2}). $$
The function f must accordingly obey the functional equation:
$$ f(xy) = f(x) +f(у). $$
This property of $f$ enables us to determine its form. Since (78) is true for all values of $x$ and $y$, we may take $y = 1 + \epsilon$, where $\epsilon$ is an infinitesimal of the first order. Then,
$$ f(x + x\epsilon) = f(x) + f(1 + \epsilon). $$
Expanding both sides by Taylor’s theorem and neglecting all terms of an order higher than the first, we have:
$$ f(x) + x\epsilon f^{\prime}(x) = f(x) + f(1) + \epsilon f^{\prime}(1). $$
For $\epsilon = 0$, we find $f(1) = 0$. Hence,
$$ xf^{\prime}(x) = f^{\prime}(1) = k. $$
where $k$ represents a constant, or:
$$ f^{\prime}(x) = \frac{k}{x}. $$
Integrating, we obtain:
$$ f(x) = k \log{x} + \text{const} $$
Remembering (77), we finally have:
$$ S = k \log{\pi} + \text{const}. $$
We can place the constant of integration equal to zero. This is permissible because the entropy is indeterminate to the extent of an additive constant. We thus finally obtain (75).
Of course, it should be clearly understood that this constitutes no proof of the Boltzmann equation (75), since we have not demonstrated that a functional relationship between $S$ and $\pi$ exists, but have merely made it appear plausible.
考虑一个由两部分组成的系统，设 $S_{1}$ 和 $S_{2}$ 为这些部分的熵，$\pi_{1}$ 和 $\pi_{2}$ 为这些部分的状态的概率。 我们从 (77) 得到：
$$ S_{1} = f(\pi_{1}),\quad S_{2} = f(\pi_{2}). $$
但是整个系统的熵是两个熵的和：
$$ S = S_{1} + S_{2} $$
整个系统的概率是两个概率的乘积，
$$ \pi = \pi_{1} \pi_{2}. $$
从这些方程和 (77) 我们得到以下结果：
$$ f(\pi_{1}\pi_{2}) = f(\pi_{1}) + f(\pi_{2}). $$
因此，函数 $f$ 必须遵守函数方程：
$$ f(xy) = f(x) +f(у). $$
$f$ 的这个性质使我们能够确定它的形式。 由于 (78) 对所有 $x$ 和 $y$ 的值都成立，我们可以取 $y = 1 + \epsilon$，其中 $\epsilon$ 是一阶无穷小。 然后，
$$ f(x + x\epsilon) = f(x) + f(1 + \epsilon). $$
通过泰勒定理展开两边并忽略所有高于一阶的项，我们有：
$$ f(x) + x\epsilon f^{\prime}(x) = f(x) + f(1) + \epsilon f^{\prime}(1). $$
对于 $\epsilon = 0$，我们发现 $f(1) = 0$。 因此，
$$ xf^{\prime}(x) = f^{\prime}(1) = k. $$
其中 $k$ 代表一个常数，或：
$$ f^{\prime}(x) = \frac{k}{x}. $$
积分，我们得到：
$$ f(x) = k \log{x} + \text{const} $$
记住 (77)，我们最终得到：
$$ S = k \log{\pi} + \text{const}. $$
我们可以将积分常数等于零。 这是允许的，因为熵在一个可加常数的程度上是不确定的。 因此，我们最终得到 (75)。
当然，应该清楚地理解，这并不构成玻尔兹曼方程 (75) 的证明，因为我们没有证明 $S$ 和 $\pi$ 之间存在函数关系，而只是使其看起来合理。
