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The second law of thermodynamics from symmetry and unitarity

Paolo Glorioso, Hong Liu

TL;DR

The paper derives a first-principles, non-equilibrium effective field theory for slow degrees of freedom in quantum many-body systems in local equilibrium, constrained by unitarity and a $\mathcal{Z}_2$ dynamical-KMS symmetry. This structure yields a local entropy current with a nonnegative divergence, establishing a global second law and a perturbative local second law, and in some cases a nonperturbative positivity proof using special-number identities. It also shows that the equilibrium entropy arises as a zeroth-order Noether current and that the thermodynamic arrow of time is tied to boundary conditions of the dynamical-KMS symmetry, with direct implications for hydrodynamics and fluctuating hydrodynamics. The results provide a constructive, first-principles route to the entropy current in continuum theories and connect microscopic unitarity to macroscopic irreversibility and causality.

Abstract

The second law of thermodynamics states that for a thermally isolated system entropy never decreases. Most physical processes we observe in nature involve variations of macroscopic quantities over spatial and temporal scales much larger than microscopic molecular collision scales and thus can be considered as in local equilibrium. For a many-body system in local equilibrium a stronger version of the second law applies which says that the entropy production at each spacetime point should be non-negative. In this paper we provide a proof of the second law for such systems and a first derivation of the local second law. For this purpose we develop a general non-equilibrium effective field theory of slow degrees of freedom from integrating out fast degrees of freedom in a quantum many-body system and consider its classical limit. The key elements of the proof are the presence of a $Z_2$ symmetry, which can be considered as a proxy for local equilibrium and micro-time-reversibility, and a classical remnant of quantum unitarity. The $Z_2$ symmetry leads to a local current from a procedure analogous to that used in the Noether theorem. Unitarity leads to a definite sign of the divergence of the current. We also discuss the origin of an arrow of time, as well as the coincidence of causal and thermodynamical arrows of time. Applied to hydrodynamics, the proof gives a first-principle derivation of the phenomenological entropy current condition and provides a constructive procedure for obtaining the entropy current.

The second law of thermodynamics from symmetry and unitarity

TL;DR

The paper derives a first-principles, non-equilibrium effective field theory for slow degrees of freedom in quantum many-body systems in local equilibrium, constrained by unitarity and a dynamical-KMS symmetry. This structure yields a local entropy current with a nonnegative divergence, establishing a global second law and a perturbative local second law, and in some cases a nonperturbative positivity proof using special-number identities. It also shows that the equilibrium entropy arises as a zeroth-order Noether current and that the thermodynamic arrow of time is tied to boundary conditions of the dynamical-KMS symmetry, with direct implications for hydrodynamics and fluctuating hydrodynamics. The results provide a constructive, first-principles route to the entropy current in continuum theories and connect microscopic unitarity to macroscopic irreversibility and causality.

Abstract

The second law of thermodynamics states that for a thermally isolated system entropy never decreases. Most physical processes we observe in nature involve variations of macroscopic quantities over spatial and temporal scales much larger than microscopic molecular collision scales and thus can be considered as in local equilibrium. For a many-body system in local equilibrium a stronger version of the second law applies which says that the entropy production at each spacetime point should be non-negative. In this paper we provide a proof of the second law for such systems and a first derivation of the local second law. For this purpose we develop a general non-equilibrium effective field theory of slow degrees of freedom from integrating out fast degrees of freedom in a quantum many-body system and consider its classical limit. The key elements of the proof are the presence of a symmetry, which can be considered as a proxy for local equilibrium and micro-time-reversibility, and a classical remnant of quantum unitarity. The symmetry leads to a local current from a procedure analogous to that used in the Noether theorem. Unitarity leads to a definite sign of the divergence of the current. We also discuss the origin of an arrow of time, as well as the coincidence of causal and thermodynamical arrows of time. Applied to hydrodynamics, the proof gives a first-principle derivation of the phenomenological entropy current condition and provides a constructive procedure for obtaining the entropy current.

Paper Structure

This paper contains 16 sections, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Non-equilibrium effective theories
  3. A theorem of local second law of thermodynamics
  4. Proof of the second law
  5. Proof of the local second law
  6. Discussions
  7. Constraints from unitarity
  8. Motivation for dynamical KMS symmetry
  9. Imposing the dynamical KMS condition
  10. Examples
  11. Model A
  12. Fluctuating hydrodynamics for relativistic charged fluids
  13. Details of proof
  14. Bernoulli and Genocchi numbers
  15. Two identities
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Closed time path contour in a state given by density matrix $\rho_0$. The two lines represent path integrals along two copies of spacetime with arrows indicating the direction of integrations. (a) and (b) correspond to having $\rho_0$ as the initial and final state respectively.