Continuity of the entropy of macroscopic quantum systems

The proper definition of entropy is fundamental to the relationship between statistical mechanics and thermodynamics. It also plays a major role in the recent debate about the validity of the concept of negative temperature. In this paper, I analyze and calculate the thermodynamic entropy for large but finite quantum mechanical systems. A special feature of this analysis is that the thermodynamic energy of a quantum system is shown to be a continuous variable, rather than being associated with discrete energy eigenvalues. Calculations of the entropy as a function of energy can be carried out with a Legendre transform of thermodynamic potentials obtained from a canonical ensemble. The resultant expressions for the entropy are able to describe equilibrium between quantum systems having incommensurate energy-level spacings. This definition of entropy preserves all required thermodynamic properties, including satisfaction of all postulates and laws of thermodynamics. It demonstrates the consistency of the concept of negative temperature with the principles of thermodynamics.

Figure 1

The solid line is a plot of the entropy per object, $S_{\text{2-level}}/N$, of a collection of $N$ two-level objects, as a function of the dimensionless thermodynamic energy $U/N\epsilon$, as given in Eq. (58). Since $S_{\text{2-level}}$ is exactly extensive, this plot is valid for all values of $N$. The circles show the values of $S_B/N$, as defined in Eqs. (66) and (68), and the squares show $S_G/N$, as defined in Eqs. (67), both for $N=10$. Both $S_B$ and $S_G$ are defined only on the discrete set of energy eigenvalues.

Figure 2

Plot of the predictions of the volume entropy are compared with those of the entropy given by $S_{\text{2-level}}$ in Eq. (58) for the distribution of energy between two subsystems of two-level objects in thermal equilibrium. The exact condition of equipartition of energy is indicated by the straight line, which is also the prediction of $S_{\text{2-level}}$. The three curves giving the volume entropy predictions are for subsystem sizes $[N_1,N_2]$ of $[10,1],[100,10]$, and $[1000,100]$. In each case, the value of $N_2$ is given in the legend. Larger deviations of the curves for the volume entropy from the exact results correspond to larger subsystems. Increasing the ratio of the subsystem sizes also increases the violation of equipartition of energy.