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Entropy current and equilibrium partition function in fluid dynamics

Sayantani Bhattacharyya

Abstract

Existence of an entropy current with non-negative divergence puts a lot of constraints on the transport coefficients of a fluid, so does the existence of equilibrium. In all the cases we have studied so far we have seen an overlap between these two sets of constraints. In this note we shall try to explore the reason for such an overlap. We shall also see how a part of the entropy current could be determined from the equilibrium partition function.

Entropy current and equilibrium partition function in fluid dynamics

Abstract

Existence of an entropy current with non-negative divergence puts a lot of constraints on the transport coefficients of a fluid, so does the existence of equilibrium. In all the cases we have studied so far we have seen an overlap between these two sets of constraints. In this note we shall try to explore the reason for such an overlap. We shall also see how a part of the entropy current could be determined from the equilibrium partition function.

Paper Structure

This paper contains 23 sections, 116 equations.

Table of Contents

  1. Introduction
  2. Rough sketch of the method
  3. The set up
  4. Entropy current as a function of background
  5. Total entropy in appropriate form
  6. Rate of entropy change at leading order in $(\omega)$
  7. Entropy current in terms of fluid variables
  8. The prescription to go from $\hat{J}^\mu$ to $J^\mu$
  9. The divergence of $J^\mu$ and the 'entropy condition'
  10. Construction of the higher order pieces in $J^\mu$
  11. Ambiguities
  12. Terms with zero divergence
  13. Ambiguity in covariantizing $\hat{J}^\mu$
  14. One 'toy' example
  15. Conclusion
  16. ...and 8 more sections