Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics

Abstract

We review recent progress in developing effective field theories (EFTs) for non-equilibrium processes at finite temperature, including a new formulation of fluctuating hydrodynamics, and a new proof of the second law of thermodynamics. There are a number of new elements in formulating EFTs for such systems. Firstly, the nature of IR variables is very different from those of a system in equilibrium or near the vacuum. Secondly, while all static properties of an equilibrium system can in principle be extracted from the partition function, there appears no such quantity which can capture all non-equilibrium properties. Thirdly, non-equilibrium processes often involve dissipation, which is notoriously difficult to deal with using an action principle. The purpose of the review is to explain how to address these issues in a pedagogic manner, with fluctuating hydrodynamics as a main example.

Hydrodynamic effective field theories with discrete rotational symmetry

“…We have extended the effective field theory formalism proposed in [4][5][6] to non-relativistic fluids with only discrete rotational symmetry (point group). These fluids most naturally couple to the vielbein, whose indices transform in representations of the point group.…”

“…Recent years have seen a flurry of activity to develop a dissipative effective field theory for hydrodynamics [1][2][3][4][5][6]. While the actual field theory itself was well-known for a very long time (e.g.…”

“…For our purpose, we only consider the closed time path[6] 2. The momentum mimics the time-reversal-odd "charges".…”

Enstrophy from symmetry

“…We have chosen (47) since by doubling the fields (and adding appropriate ghosts) the action can be extended to include non dissipative fluids. We refer the reader to [35] for an extensive discussion. We claim that the following transformation of the dynamical fields…”

Unified Extended Irreversible Thermodynamics and the stability of relativistic theories for dissipation

“…However, almost all these dispersion relations are gapped (the associated Thermo-modes describe the tendency of the fluid elements to relax towards local thermodynamic equilibrium). The only modes that survive in the slow limit are the few gapless ones, which describe the transport of conserved quantities [57] and are (almost [1]) universally described by the Navier-Stokes equations (another reason we call them Hydro-modes). This mechanism will be explored in detail in subsection 4.4, starting from a kinetic description.…”