Foundations of statistical mechanics for unstable interactions

Area of Science:

• Statistical Mechanics
• Thermodynamics
• Condensed Matter Physics

Background:

• Traditional Boltzmann-Gibbs statistical mechanics is limited to systems with stable interactions.
• Unstable systems lack a conventional thermodynamic limit due to unbounded ground state energy.
• Existing literature often assumes interaction stability is necessary for thermodynamic behavior.

Purpose of the Study:

• To develop a generalized statistical mechanics framework for systems with unstable interactions.
• To demonstrate that unstable systems can exhibit normal thermodynamic properties and possess a thermodynamic limit.
• To re-examine the foundational principles of statistical physics for broader applicability.

Main Methods:

• Revisiting foundational postulates of statistical physics, including extensivity, divisibility, and statistical independence.
• Introducing generalized ensembles that accommodate unstable interactions.
• Classifying system stability using an index σ of regular variation for ground state energy.

Main Results:

• Generalized ensembles ensure the existence of the thermodynamic limit for unstable systems.
• Systems with unstable interactions are shown to be thermodynamically normal and extensive.
• The Curie-Weiss-Ising model with σ=2 exhibits a novel first-order phase transition and a 1/2 order transition at absolute zero.

Conclusions:

• Interaction stability is not a prerequisite for the thermodynamic limit.
• The developed formalism provides a robust framework for analyzing unstable systems.
• Novel phase transitions and universality classes are identified in strongly coupled models.