Entropy Current from Partition Function: One Example

TL;DR

This paper tests a proposed algorithm that derives an entropy current with non-negative divergence from the equilibrium partition function, applied to a parity-even charged fluid at second order in derivatives. It constructs an explicit entropy current as a sum of a canonical piece, a partition-function-derived piece, and an extension term, then demonstrates the divergence can be organized into a sum of squares, yielding constraints on transport coefficients. The analysis reveals 7 partition-function parameters and 17 total free coefficients at this order, with 10 further ambiguities intrinsic to the construction; these clarify how non-dissipative transport is fixed while leaving legitimate freedom in the dissipative sector. The work highlights how equilibrium data encode non-dissipative structure and sets a framework for extending to other sectors and to holographic contexts.

Abstract

In hydrodynamics the existence of an entropy current with non-negative divergence is related to the existence of a time-independent solution in a static background. Recently there has been a proposal for how to construct an entropy current from the equilibrium partition function of the fluid system. In this note, we have applied this algorithm for the charged fluid at second order in derivative expansion. From the partition function we first constructed one example of entropy current with non-negative divergence upto the required order. Finally we extended it to its most general form, consistent with the principle of local entropy production. As a by-product we got the constraints on the second order transport coefficients for a parity even charged fluid, but in some non-standard fluid frame.