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.

arXiv comment4 pages, 1 figure

References in corpus (8)

• Classification of phase transitions and ensemble inequivalence, in systems with long range interactions
• An Introduction to the Thermodynamic and Macrostate Levels of Nonequivalent Ensembles
• Topological Signature of First Order Phase Transitions
• Phase transitions and topology changes in configuration space
• Non-analytic microscopic phase transitions and temperature oscillations in the microcanonical ensemble: An exactly solvable 1d-model for evaporation
• On the mean-field spherical model
• Topology, phase transitions and the spherical model
• Topological approach to phase transitions and inequivalence of statistical ensembles

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• Kinetic energy and microcanonical nonanalyticities in finite and infinite systems
• Thermostatistics of small systems: Exact results in the microcanonical formalism
• The Geometric Theory of Phase Transitions
• A simple topological model with continuous phase transition
• Models with symmetry-breaking phase transitions triggered by dumbbell-shaped equipotential surfaces
• Non-additive properties of finite 1D Ising chains with long-range interactions
• Density of states of the $XY$ model: an energy landscape approach
• Simplified energy landscape of the ${\varphi }^4$ model and the phase transition