Kinetic energy and microcanonical nonanalyticities in finite and infinite systems

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.

14 pages, 2 figures; v2: minor corrections, final version

References in corpus (7)

• Phase transitions and configuration space topology
• Partial equivalence of statistical ensembles and kinetic energy
• Phase transitions induced by saddle points of vanishing curvature
• On the mean-field spherical model
• Nonanalyticities of entropy functions of finite and infinite systems
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Topological conditions for discrete symmetry breaking and phase transitions

Cited by in corpus (4)

• Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian
• Exploring the energy landscape of XY models
• On a microcanonical relation between continuous and discrete spin models
• Stationary point approach to the phase transition of the classical XY chain with power-law interactions