Thermodynamic Invariants of Coupled Channels: A Many-Channel Tolman-Ehrenfest Effect

TL;DR

This paper extends thermodynamic invariants to coupled channels, revealing a universal relation and geometric insights into granular materials, with testable predictions near jamming.

Contribution

It introduces a unique invariant for multi-channel thermodynamics and links geometric curvature to granular stress and dilatancy.

Findings

Derived the invariant $ abla_i imes T_i = C$ for coupled channels.

Connected off-diagonal curvature to dilatancy ratio and energy restrictions.

Predicted a testable relation $ abla_V imes ext{chi} = ext{const}$ across shear bands.

Abstract

When multiple thermodynamic channels are coupled, single-channel equilibrium conditions fail. Extending the Tolman--Ehrenfest effect to the entropy manifold, we derive the unique $n$-channel invariant $\zeta_i T_i = C$, where $\zeta_i$ is the holonomy of the Ruppeiner connection. For the granular volume--stress ensemble, Rowe's dilatancy ratio and energy restriction emerge as geometric consequences of the off-diagonal curvature $g_{V\sigma}$, and the 60-year puzzle of state-dependent $K_\mu$ is resolved: the correction $\zeta_V$ reaches $O(1)$ near jamming. The prediction $\zeta_V\chi=\mathrm{const}$ across a shear band is experimentally testable.