Kinetic energy and microcanonical nonanalyticities in finite and infinite systems

TL;DR

This paper investigates nonanalyticities in microcanonical thermodynamic functions for finite and infinite systems, revealing how kinetic energy influences these nonanalyticities and explaining peculiar behaviors observed in specific models.

Contribution

It establishes the relationship between nonanalyticities of configurational and total microcanonical entropy and analyzes the effects of kinetic energy and system size.

Findings

Nonanalyticities in configurational entropy induce similar nonanalyticities in total entropy for finite systems.

Kinetic energy weakens nonanalyticities, increasing differentiability of entropy.

In the thermodynamic limit, nonanalyticities shift to higher energies, explaining observed model behaviors.

Abstract

In contrast to the canonical case, microcanonical thermodynamic functions can show nonanalyticities also for finite systems. In this paper we contribute to the understanding of these nonanalyticities by working out the relation between nonanalyticities of the microcanonical entropy and its configurational counterpart. If the configurational microcanonical entropy $\omega_N^c(v)$ has a nonanalyticity at $v=v_c$, then the microcanonical entropy $\omega_N(\epsilon)$ has a nonanalyticity at the same value $\epsilon=v_c$ of its argument for any finite value of the number of degrees of freedom $N$. The presence of the kinetic energy weakens the nonanalyticities such that, if the configurational entropy is $p$ times differentiable, the entropy is $p+\lfloor N/2 \rfloor$-times differentiable. In the thermodynamic limit, however, the behaviour is very different: The nonanalyticities do not longer occur at the same values of the arguments, but the nonanalyticity of the microcanonical entropy is shifted to a larger energy. These results give a general explanation of the peculiar behaviour previously observed for the mean-field spherical model. With the hypercubic model we provide a further example illustrating our results.