Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics

TL;DR

This work presents a first-principles framework for non-equilibrium effective field theories at finite temperature, anchored by a two-copy (closed time path) action and a dynamical KMS symmetry that enforces fluctuation-dissipation relations and the second law. It systematically builds the general EFT structure, shows how unitarity and DKMS constrain the theory, and demonstrates how an emergent entropy current arises, guaranteeing nonnegative entropy production. The authors develop concrete EFTs for distinct regimes: a critical O(n) model (Model A), a diffusion theory for conserved currents, and a comprehensive action-based formulation of fluctuating hydrodynamics (including a fluid spacetime formulation). They also discuss non-dissipative limits, ghost considerations, and field redefinitions, outlining broad avenues for future applications and generalizations to other media and low-temperature settings.

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.

Figures (4)

• Figure 1: Relaxation of different types of excitations. The horizontal direction is along some spatial direction. The straight dashed lines denote the global equilibrium values and the solid lines denote values of some perturbed quantities. (a) Perturbations in non-conserved quantities can relax back to equilibrium values locally--deviations separated at length scales larger than the relaxation length $\ell$ relax independently--in a time of order of the relaxation time $\tau$. (b) Conserved quantities can only relax through transports, i.e. excesses have to be transported to regions with deficits to achieve equilibrium. (c) In a spacetime region with $\ell \ll {{\delta}} x \ll \lambda, \tau \ll {{\delta}} t \ll t_\lambda$ a system can be considered as in local equilibrium specified by the local values of conserved quantities.
• Figure 2: (a) Path integral segments for evolution of a general initial density matrix $\rho_0$. Paths of integration are indicated by arrows. (b) Equation \ref{['oo1']} can be obtained by inserting $V$ at time $t$ on either segment, and joining the future ends at some time $t_f >t$. (c) An example of the path integral contour for a general correlation function \ref{['oo2']}. Depending on the relative magnitudes of $t_1, t_2, t_3, \cdots$ in \ref{['oo2']}, the path integrals can have different number of segments. Shown in figure is an example which require four segments, $\mathop{\rm Tr}(\rho_0W(t_4) V(t_2)W(t_3)V(t_1))$ with $t_1 < t_2 < t_4 < t_3$. To measure such an observable requires that we evolve experimental apparatus both forward and "backward" in time. (d) An example of correlation function on CTP, corresponding to Eq. (\ref{['mp']}).
• Figure 3: (a) Integration contour corresponding to $W$. (b) Integration contour corresponding to $W_T$ as defined in \ref{['newfdt']}.
• Figure 4: Relations between the fluid spacetime and two copies of physical spacetimes. The red straight line in the fluid spacetime with constant $\sigma^i$ is mapped by $X^\mu_{1,2} (\sigma^0, \sigma^i)$ to physical spacetime trajectories (also in red) of the corresponding fluid element.