Nonanalyticities of Entropy Functions of Finite and Infinite Systems

In contrast to the canonical ensemble where thermodynamic functions are smooth for all finite system sizes, the microcanonical entropy can show nonanalytic points also for finite systems. The relation between finite and infinite system nonanalyticities is illustrated by means of a simple classical spinlike model which is exactly solvable for both finite and infinite system sizes, showing a phase transition in the latter case. The microcanonical entropy is found to have exactly one nonanalytic point in the interior of its domain. For all finite system sizes, this point is located at the same fixed energy value ${\epsilon }_c^{\mathrm{finite}}$, jumping discontinuously to a different value ${\epsilon }_c^{\mathrm{infinite}}$ in the thermodynamic limit. Remarkably, ${\epsilon }_c^{\mathrm{finite}}$ equals the average potential energy of the infinite system at the phase transition point. The result indicates that care is required when trying to infer infinite system properties from finite system nonanalyticities.

Figure 1

The graph of the microcanonical entropy $s_N$ as a function of the energy $\epsilon$, plotted for system sizes $N=4$, 8, 16, 32, 64, 128, and for the infinite system.