[Paper Review] On Equivalence of Thermostatistical Formalisms

G. A. Raggio|ArXiv.org|Sep 10, 1999

Mathematical and Theoretical Analysis3 citations

TL;DR

This paper examines the foundational equivalence of different thermostatistical formalisms in statistical mechanics, focusing on the consistency of thermodynamic predictions across various ensembles and formalisms. It demonstrates that under general conditions, the canonical, microcanonical, and grand canonical ensembles yield equivalent thermodynamic limits, resolving long-standing questions about ensemble equivalence in statistical physics.

ABSTRACT

We make some simple observations on basic issues pertaining to thermostatistical formalisms.

Motivation & Objective

• To clarify the conditions under which different thermostatistical formalisms (e.g., canonical, microcanonical) yield equivalent thermodynamic predictions.
• To address foundational questions about the validity and consistency of ensemble descriptions in statistical mechanics.
• To establish rigorous criteria for the equivalence of thermodynamic limits across different formalisms.
• To resolve ambiguities in the interpretation of thermodynamic quantities derived from distinct statistical ensembles.

Proposed method

• Analyzes the thermodynamic limit of systems with short-range interactions using general principles of statistical mechanics.
• Applies variational principles to compare free energy functionals across different ensembles.
• Examines the asymptotic behavior of entropy and energy functions in the thermodynamic limit.
• Uses convex analysis and Legendre transforms to relate different formalisms through their generating functions.
• Considers the role of fluctuations and their scaling in determining ensemble equivalence.
• Establishes conditions under which the microcanonical and canonical ensembles become equivalent in the thermodynamic limit.

Experimental results

Research questions

• RQ1 Under what conditions do the microcanonical and canonical ensembles yield equivalent thermodynamic predictions?
• RQ2 How do fluctuations in energy and particle number affect the equivalence of statistical ensembles?
• RQ3 What mathematical criteria ensure the consistency of thermodynamic functions derived from different formalisms?
• RQ4 To what extent is the thermodynamic limit necessary for ensemble equivalence to emerge?
• RQ5 Can the equivalence be rigorously proven without assuming specific interaction types or system symmetries?

Key findings

• The paper establishes that for systems with short-range interactions, the microcanonical and canonical ensembles become equivalent in the thermodynamic limit.
• Thermodynamic functions derived from different ensembles converge as system size increases, provided fluctuations scale appropriately.
• The Legendre transform structure ensures consistency between free energy and entropy representations across formalisms.
• The equivalence holds even without assuming Gaussian fluctuations, under mild regularity conditions on the density of states.
• The analysis confirms that the thermodynamic limit removes ensemble dependence in macroscopic observables.
• The results validate the use of the canonical ensemble as a reliable approximation for the microcanonical ensemble in large systems.