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Unified gauge-geometry symmetry for equilibrium statistical mechanics
1Hanoi National University of Education, Hanoi National University of Education, Department of Physics, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam and Institute of Natural Sciences, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam.
Physical Review. E
|June 19, 2026
Summary
This study introduces a unified symmetry framework for statistical mechanics, combining spacetime and phase-space symmetries. This approach generates new identities and simplifies complex calculations in many-body systems.
Area of Science:
- Theoretical Physics
- Statistical Mechanics
- Condensed Matter Physics
Background:
- Conventional statistical mechanics relies on spacetime symmetries.
- A recent phase-space gauge-shifting invariance was identified.
- Existing frameworks lack a unified approach to these symmetries.
Purpose of the Study:
- To develop a unified symmetry framework for equilibrium statistical mechanics.
- To incorporate both spacetime and phase-space gauge symmetries.
- To derive new theoretical identities and simplify analyses of many-body systems.
Main Methods:
- Formulation of a single Lie group combining spacetime and phase-space symmetries.
- Application of Noether's theorem to derive Ward identities and cross-relations.
- Development of an equivariant gauge-constrained-density functional theory (DFT).
Main Results:
- A unified Lie group structure encompassing diverse symmetries.
- A hierarchy of exact identities, including Ward and cross-Ward relations.
- A Wigner-Eckart-Ward reduction for tensor-hyperforce correlators in isotropic fluids.
Conclusions:
- The framework offers a consistent organizational basis for phenomena in liquids, mixtures, and interfaces.
- It provides a symmetry-based perspective connecting structure, mechanics, and dynamics.
- The developed methods simplify complex correlators and satisfy theoretical constraints.
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