Unified Gauge-Geometry Symmetry for Equilibrium Statistical Mechanics

Abstract

We present a symmetry-based framework for equilibrium statistical mechanics that formulates a single Lie group combining conventional spacetime symmetries with a recently identified phase-space gauge-shifting invariance [Muller et al., Phys. Rev. Lett. 133, 217101 (2024)]. Using Noether's theorem, we obtain a set of general Ward identities together with previously unexplored cross-relations arising from the noncommutation of different symmetry generators. The approach extends standard many-body symmetries, such as translations, rotations, Galilean boosts, dilations, and particle exchange, by incorporating an internal gauge-shift symmetry within a unified group structure. The resulting Lie algebra suggests a hierarchy of exact identities that encompass established sum rules and indicate possible cross-coupling relations between distinct response and correlation functions. We also identify a Wigner-Eckart-Ward reduction that simplifies tensor-hyperforce correlators to two scalar radial spectra in isotropic fluids, and we outline an equivariant gauge-constrained DFT formulation whose Euler-Lagrange equations are constructed to satisfy the corresponding Ward and cross-Ward constraints. This framework provides a consistent organizational basis for phenomena in liquids, mixtures, and interfaces, and may offer a symmetry-based perspective connecting structure, mechanics, and dynamics in many-body systems.

Figures (3)

• Figure 1: Radial validation of the gauge–$\mathrm U(1)$ identity $\nabla\!\cdot \mathbf C_{\delta\rho F_{\mathrm{int}}}=k_{\mathrm B} T\,\rho^2\,\nabla^2 h$ in a bulk Lennard–Jones fluid ($T^\ast=1.5$, $\rho^\ast=0.85$). Solid line: shell–averaged divergence $\langle \nabla\!\cdot \mathbf C_{\delta\rho F_{\mathrm{int}}}\rangle_r$ computed from the density–internal–force cross–correlator measured in MD. Dashed line: $k_{\mathrm B} T\,\rho^2\,\nabla^2 h(r)$ obtained from the total correlation $h(r)=g(r)-1$ of the same run.
• Figure 2: Shell‑averaged magnitudes of the longitudinal (circles) and transverse (squares) projections of the density–internal‑force cross correlator $\mathbf C_{\delta\rho F_{\mathrm{int}}}(\mathbf{k}) = V^{-1}\langle \delta\hat{\rho}_{-\mathbf{k}}\,\hat{\mathbf F}_{\mathrm{int}}(\mathbf{k})\rangle$ for a bulk Lennard–Jones fluid at $T^\ast=1.5$ and $\rho^\ast=0.85$. The transverse signal is strongly suppressed relative to the longitudinal one, consistent with the gauge$\times$rotation cross identity $\mathbf P_T(\mathbf{k})\,\mathbf C_{\delta\rho F_{\mathrm{int}}}(\mathbf{k})=\mathbf 0.$ [Eq. \ref{['eq:C2']}], while the small‑$k$ growth of the longitudinal branch follows the gauge–$\mathrm U(1)$ prediction $|P_L\mathbf C_{\delta\rho F_{\mathrm{int}}}(\mathbf{k})| \propto k\,|S(k)-1|$ from Eq. \ref{['eq:C1']}.
• Figure 3: Test of the momentum--fiber/gauge identity, Eq. \ref{['eq:PiG-current-vanish']}, using the signed longitudinal current cross correlator $\mathrm{Re}\!\left[\hat{\mathbf{k}}\!\cdot\!\langle B\,\hat{\mathbf j}_{\mathbf{k}}\rangle_{\rm shell}\right]/V$ (circles: $B=\hat{\rho}_{-\mathbf{k}}$; squares: $B=\delta U$). The exact prediction is zero (horizontal line); error bars are block-averaged standard errors. Units are Lennard--Jones reduced, with $\tau\equiv\sigma\sqrt{m/\varepsilon}$.