Constraints on fluid dynamics from equilibrium partition functions

Abstract

We study the thermal partition function of quantum field theories on arbitrary stationary background spacetime, and with arbitrary stationary background gauge fields, in the long wavelength expansion. We demonstrate that the equations of relativistic hydrodynamics are significantly constrained by the requirement of consistency with any partition function. In examples at low orders in the derivative expansion we demonstrate that these constraints coincide precisely with the equalities between hydrodynamical transport coefficients that follow from the local form of the second law of thermodynamics. In particular we recover the results of Son and Surowka on the chiral magnetic and chiral vorticity flows, starting from a local partition function that manifestly reproduces the field theory anomaly, without making any reference to an entropy current. We conjecture that the relations between transport coefficients that follow from the second law of thermodynamics agree to all orders in the derivative expansion with the constraints described in this paper.

Effective actions for anomalous hydrodynamics

“…Since this has been already established in [13] we will be relatively brief. We also then demonstrate that our action when restricted to stationary fluid flows reproduces the anomalous part of the equilibrium partition described in [22].…”

“…Secondly, the transgression form is very easy to vary with respect to the gauge field, so the charge current is easily obtained from it. 22 Furthermore, since the transgression formalism constructs covariant actions, we immediately are able to generalize the discussion to non-abelian symmetries. Finally, the fact that S anom is given by a transgression form, shines some light on the physical significance of the transverse gauge field as originating from a horizon field that treats the chemical shift symmetry as another gauge symmetry.…”

“…The logic exploited in [22,23] is that equilibrium configurations of a fluid in the presence of arbitrary time-independent spatially varying sources (which thereby allow non-trivial stationary fluid flows) can be equivalently described by a partition function of the sources. By writing down the most general partition function compatible with the symmetries in a systematic low energy gradient expansion, together with introduction of parity-odd terms to account for anomalies, it was shown in [22] that all known effects of global anomalies in hydrodynamics could be captured explicitly. Furthermore, this was shown to be explicitly in agreement with the general entropy analysis of [13] in [24] ( [25,26] use this formalism to explain anomalous transport in two dimensions with the latter focusing on the Lorentz anomaly; cf., also [27,28]).…”

“…There is no low energy Lagrangian, and the stress-energy tensor and conserved currents are not constructed from more fundamental degrees of freedom of a true effective field theory (the equilibrium partition functions of [22,23] are functionals of JHEP03 (2014)034 background sources). It is therefore interesting to examine the extent to which an effective field theory of hydrodynamics can be formulated in terms of appropriate degrees of freedom.…”

“…5 2. It reduces quite simply to the anomalous equilibrium partition function derived in [22,30] upon restricting to stationary flows.…”