理想气体的精确熵公式及其信息论解释
The Exact Entropy Formula of the Ideal Gas and its Information-Theoretic Interpretation
AI总结：
本文提出多项式分布的熵公式即为非相互作用不可区分粒子系统的物理熵，揭示了信息论与统计力学的联系，并给出理想气体Sackur-Tetrode公式的精确量子修正。
摘要：
本文分析了由非相互作用且不可区分的粒子组成的系统的熵，这些粒子的量子态编号被建模为独立同分布的经典随机变量。关键观察是，在此假设下，无论构成系统的粒子数量是多少，系统量子（微观）态的占据数都服从多项式分布。这一观察导出了物理系统的熵公式，它正是多项式分布的熵公式，我们主张其具有新颖性，因为这里首次提出多项式分布的熵即为物理系统的熵。多项式分布的熵公式揭示了信息论与统计力学之间尚未探索的联系，其中包括：给定随机占据数时随机微观态的条件熵与Boltzmann-Planck熵$\log(W)$之间的联系，以及这两者与Gibbs修正项$\log(N!)$的联系，热化与热态的通信论制备，热态的可访问信息与热化系统的物理熵。从我们的方法中得出的一个显著具体结果是，对容器中处于热平衡的理想气体的熵，给出了教科书Sackur-Tetrode公式的精确量子修正。
英文摘要：
The paper analyzes the entropy of a system composed by non-interacting and indistinguishable particles whose quantum state numbers are modelled as independent and identically distributed classical random variables. The crucial observation is that, under this assumption, whichever is the number of particles that constitute the system, the occupancy numbers of system's quantum (micro)states are multinomially distributed. This observation leads to an entropy formula for the physical system, which is nothing else than the entropy formula of the multinomial distribution, for which we claim novelty, in the sense that it is proposed here for the first time that the entropy of the multinomial distribution is the entropy of the physical system. The entropy formula of the multinomial distribution unveils yet unexplored connections between information theory and statistical mechanics, among which we mention the connection between conditional entropy of the random microstate given the random occupancy numbers and the Boltzmann-Planck entropy $\log(W)$ and between these two and the Gibbs correction term $\log(N!)$, thermalization and communication-theoretic preparation of a thermal state, accessible information of the thermal state and physical entropy of the thermalized system. A noticeable specific result that descends from our approach is the exact quantum correction to the textbook Sackur-Tetrode formula for the entropy of an ideal gas at the thermal equilibrium in a container.
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