统计物理中的量子和经典方法：一些基本不等式
Quantum and classical approaches in statistical physics: some basic inequalities
摘要：
给出了自由能、熵和平均能的经典值与量子值之间的一些基本不等式。我们研究了从确定性情形（经典力学）到概率性情形（量子力学）的过渡。在本文的第一部分中，我们假设约化普朗克常数$\hbar$、绝对温度$T$、振荡器频率$ω$和系统自由度$N$是固定的。这种比较量子力学和经典力学的方法是新的（见[35]-[37]）。在本文的第二部分中，我们同时导出了四种情形的半经典极限，即$\hbar{\to}0$，$T{\to}\infty$，$ω{\to}0$和$N{\to}\infty$ .我们注意到，在量子力学中通常只考虑$\hbar{\to}0$的情况（见[21]）。量子力学中的事例$T{\to}\infty$和$ω{\to}0$是M. Planck和A.爱因斯坦，分别。
英文摘要：
We present some basic inequalities between the classical and quantum values of free energy, entropy and mean energy. We investigate the transition from the deterministic case (classical mechanics) to the probabilistic case (quantum mechanics). In the first part of the paper, we assume that the reduced Planck constant $\hbar$, the absolute temperature $T$, the frequency of an oscillator $ω$, and the degree of freedom of a system $N$ are fixed. This approach to the problem of comparing quantum and classical mechanics is new (see [35]--[37]). In the second part of the paper, we simultaneously derive the semiclassical limits for four cases, that is, for $\hbar{\to}0$, $T{\to}\infty$, $ω{\to}0$, and $N{\to}\infty$. We note that only the case $\hbar{\to}0$ is usually considered in quantum mechanics (see [21]). The cases $T{\to}\infty$ and $ω{\to}0$ in quantum mechanics were initially studied by M. Planck and by A. Einstein, respectively.
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