Free energy fluxes and the Kubo-Martin-Schwinger relation

TL;DR

This paper derives a fundamental constraint on free energy fluxes in large-scale fluid models, linking them to entropy currents and the Kubo-Martin-Schwinger relations, with implications for integrable systems and hydrodynamics.

Contribution

It establishes a new relation showing free energy fluxes are perpendicular to inverse temperature vectors, ensuring physical consistency and entropy conservation in hydrodynamic models.

Findings

Free energy fluxes are perpendicular to inverse temperature vectors.

Entropy currents can be expressed as averages of local observables.

In integrable models, the thermodynamic Bethe ansatz satisfies a unitarity condition.

Abstract

A general, multi-component Eulerian fluid theory is a set of nonlinear, hyperbolic partial differential equations. However, if the fluid is to be the large-scale description of a short-range many-body system, further constraints arise on the structure of these equations. Here we derive one such constraint, pertaining to the free energy fluxes. The free energy fluxes generate expectation values of currents, akin to the specific free energy generating conserved densities. They fix the equations of state and the Euler-scale hydrodynamics, and are simply related to the entropy currents. Using the Kubo-Martin-Schwinger relations associated to many conserved quantities, in quantum and classical systems, we show that the associated free energy fluxes are perpendicular to the vector of inverse temperatures characterising the state. This implies that all entropy currents can be expressed as averages of local observables. In few-component fluids, it implies that the averages of currents follow from the specific free energy alone, without the use of Galilean or relativistic invariance. In integrable models, in implies that the thermodynamic Bethe ansatz must satisfy a unitarity condition. The relation also guarantees physical consistency of the Euler hydrodynamics in spatially-inhomogeneous, macroscopic external fields, as it implies conservation of entropy, and the local-density approximated Gibbs form of stationarity states. The main result on free energy fluxes is based on general properties such as clustering, and we show that it is mathematically rigorous in quantum spin chains.