Nonanalyticities of entropy functions of finite and infinite systems

Abstract

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, even if the Hamiltonian is smooth. The relation between finite and infinite system nonanalyticities is illustrated by means of a simple classical spin-like 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 ${\epsilon }_c^{finite}$, jumping discontinuously to a different value ${\epsilon }_c^{infinite}$ in the thermodynamic limit. Remarkably, ${\epsilon }_c^{finite}$ equals the average potential energy of the infinite system at the phase transition point. The result, supplemented with results on nonanalyticities of the microcanonical entropy for other models, indicates that care is required when trying to infer infinite system properties from finite system nonanalyticities.

4 pages, 1 figure

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Cited by in corpus (9)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Phase transitions and configuration space topology
• Partial equivalence of statistical ensembles and kinetic energy
• Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian
• Exploring the energy landscape of XY models
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Kinetic energy and microcanonical nonanalyticities in finite and infinite systems
• The Geometric Theory of Phase Transitions
• Non-additive properties of finite 1D Ising chains with long-range interactions