Quantum and classical approaches in statistical physics: some basic inequalities

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 $\mathrm{\hslash }$, the absolute temperature $T$, the frequency of an oscillator $\omega$, 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 $\mathrm{\hslash }\to 0$, $T\to \mathrm{\infty }$, $\omega \to 0$, and $N\to \mathrm{\infty }$. We note that only the case $\mathrm{\hslash }\to 0$ is usually considered in quantum mechanics (see [21]). The cases $T\to \mathrm{\infty }$ and $\omega \to 0$ in quantum mechanics were initially studied by M. Planck and by A. Einstein, respectively.