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Exact analysis of phase transitions in mean-field Potts models
Paolo Lorenzoni1, Antonio Moro2
1Dipartimento di Matematica e Applicazioni, Edificio U5, via Roberto Cozzi 55, 20126 Milano, Italy.
Physical Review. E
|October 3, 2019
Summary
This study presents the exact partition function for the Potts model with external fields, revealing nematic phase transitions interpreted as shock dynamics. The method generalizes for various states and provides critical exponents.
Area of Science:
- Statistical Mechanics
- Complex Systems Theory
- Mathematical Physics
Background:
- The Potts model is a fundamental model in statistical mechanics.
- Understanding phase transitions in systems with external fields is crucial.
- Previous methods often lacked exact solutions for complex couplings.
Purpose of the Study:
- To derive the exact partition function for the Potts model on a complete graph with linear and nematic external fields.
- To analyze phase transitions and their interpretation in thermodynamic variable space.
- To provide critical asymptotics for various physical quantities.
Main Methods:
- Constructing the exact partition function as a solution to a linear diffusion equation.
- Utilizing the semiclassical limit to determine the free energy.
- Analyzing singularities in equations of state to identify phase transitions.
Main Results:
- The exact partition function was derived for the q-state Potts model (demonstrated for q=3, generalized for arbitrary q).
- Nematic-type phase transitions were identified at non-zero external fields.
- Phase transitions were interpreted using shock dynamics in thermodynamic variables.
- Critical asymptotics for magnetization, susceptibility, specific heat, and exponents (β, γ, α) were provided.
Conclusions:
- The developed method provides an exact solution for the Potts model under specific external field conditions.
- The interpretation of phase transitions via shock dynamics offers new insights.
- The generalization to arbitrary q-states enhances the applicability of the findings.
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