[Paper Review] Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics

Paolo Glorioso, Hong Liu|arXiv (Cornell University)|May 23, 2018

Advanced Thermodynamics and Statistical Mechanics8 references15 citations

TL;DR

This paper presents a symmetry-based effective field theory (EFT) framework for non-equilibrium systems at finite temperature, focusing on fluctuating hydrodynamics and the second law of thermodynamics. It introduces a closed-time-path (CTP) formalism with dynamical KMS symmetry to systematically treat dissipation and noise, providing a first-principles derivation of nonlinear fluctuating hydrodynamics and a new proof of the second law via emergent entropy in non-dissipative EFTs.

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.

Motivation & Objective

• To develop a systematic effective field theory (EFT) framework for non-equilibrium systems at finite temperature, where traditional equilibrium methods fail.
• To address the challenge of dissipation in action-based formulations by introducing dynamical KMS symmetry as a fundamental organizing principle.
• To provide a first-principles derivation of fluctuating hydrodynamics, including nonlinear noise and correlation functions, beyond phenomenological stochastic equations.
• To establish a new proof of the second law of thermodynamics using emergent entropy in non-dissipative EFTs.
• To clarify the role of IR variables in non-equilibrium systems, which differ fundamentally from equilibrium or vacuum-based EFTs.

Proposed method

• Formulates a non-equilibrium EFT using the closed-time-path (CTP) formalism with two copies of fields (r and a) to describe real-time dynamics and correlation functions.
• Introduces dynamical KMS symmetry as a fundamental symmetry of the non-equilibrium effective action, ensuring consistency with thermal equilibrium at late times.
• Derives the effective action for hydrodynamics via a fluid spacetime formulation, with field redefinitions ensuring equivalence to a single-copy action in the classical limit.
• Applies the CTP path integral to compute generating functionals for correlation functions, including non-Markovian and nonlinear noise effects.
• Uses discrete symmetries (T, CPT) to classify tensor transformations and derive the structure of dynamical KMS transformations in the action.
• Constructs the EFT action for hydrodynamics by demanding invariance under spacetime diffeomorphisms, internal symmetries, and dynamical KMS symmetry, leading to a consistent nonlinear fluctuating hydrodynamics.

Experimental results

Research questions

• RQ1 How can effective field theories be consistently formulated for non-equilibrium systems at finite temperature, where no partition function captures all dynamics?
• RQ2 What is the role of dynamical KMS symmetry in organizing non-equilibrium EFTs and ensuring consistency with thermal equilibrium?
• RQ3 How can fluctuating hydrodynamics be derived from first principles using an action principle, including nonlinear noise and correlations?
• RQ4 Can the second law of thermodynamics be derived from an EFT framework using emergent entropy and non-dissipative dynamics?
• RQ5 How do IR variables in non-equilibrium systems differ from those in equilibrium or vacuum, and how are they encoded in the EFT action?

Key findings

• The paper derives a first-principles action for nonlinear fluctuating hydrodynamics using the CTP formalism and dynamical KMS symmetry, enabling systematic treatment of noise and correlations beyond linear response.
• It establishes a new proof of the second law of thermodynamics by showing that entropy production emerges from the structure of the non-dissipative EFT action and its symmetry constraints.
• The effective action for hydrodynamics is constructed via a fluid spacetime formulation, with field redefinitions showing equivalence to a single-copy action in the classical limit.
• The framework reveals that the coupling between the two CTP copies in the generating functional arises from boundary conditions at late time, and is essential for capturing non-equilibrium correlations.
• The method systematically handles dissipation in the action formalism by embedding it within a larger symmetry structure—dynamical KMS symmetry—avoiding the usual problems of non-conservative actions.
• The approach generalizes the MSRJ formalism by deriving the action from symmetries and an action principle, rather than from phenomenological stochastic equations.