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Dynamics of states of infinite quantum systems as a cornerstone of the second law of thermodynamics

Walter F. Wreszinski

TL;DR

This work addresses how a second-law-like behavior can be formulated for quantum spin systems with infinite degrees of freedom. By introducing mean entropy $s(\omega)$ and analyzing adiabatic transformations—including sudden perturbations—the paper proves a modified Clausius law for two distinct one-dimensional universality classes at $T=0$: an exactly solvable exponential model and a Dyson model exhibiting quantum chaos. The results reveal that, depending on the universality class and perturbation type (extended vs local), the system evolves toward a maximally mixed, tracial equilibrium state, with a time-arrow emerging from the preparation step of the transformation. The analysis links microscopic dynamics to macroscopic irreversibility, discusses implications for non-equilibrium steady states, and outlines open challenges for extending these ideas to algebraic quantum field theory (AQFT). Overall, the paper provides a dynamical, microscopic foundation for the second law in infinite quantum systems and clarifies how universality classes shape the route to equilibrium and entropy increase.

Abstract

We improve on our version of the second law of thermodynamics as a deterministic theorem for quantum spin systems in two basic aspects. The first concerns the general statement of the second law: spontaneous changes in an adiabatically closed system will always be in the direction of increasing mean entropy, which rises to a maximal value. Two specific examples concern the transition from pure to mixed states in two different universality classes of dynamics in one dimension, one being the exponential model, the other the Dyson model, the dynamics of the latter exhibiting strong graphical evidence of quantum chaos, as a consequence of the results of Albert and Kiessling on the Cloitre function.

Dynamics of states of infinite quantum systems as a cornerstone of the second law of thermodynamics

TL;DR

This work addresses how a second-law-like behavior can be formulated for quantum spin systems with infinite degrees of freedom. By introducing mean entropy and analyzing adiabatic transformations—including sudden perturbations—the paper proves a modified Clausius law for two distinct one-dimensional universality classes at : an exactly solvable exponential model and a Dyson model exhibiting quantum chaos. The results reveal that, depending on the universality class and perturbation type (extended vs local), the system evolves toward a maximally mixed, tracial equilibrium state, with a time-arrow emerging from the preparation step of the transformation. The analysis links microscopic dynamics to macroscopic irreversibility, discusses implications for non-equilibrium steady states, and outlines open challenges for extending these ideas to algebraic quantum field theory (AQFT). Overall, the paper provides a dynamical, microscopic foundation for the second law in infinite quantum systems and clarifies how universality classes shape the route to equilibrium and entropy increase.

Abstract

We improve on our version of the second law of thermodynamics as a deterministic theorem for quantum spin systems in two basic aspects. The first concerns the general statement of the second law: spontaneous changes in an adiabatically closed system will always be in the direction of increasing mean entropy, which rises to a maximal value. Two specific examples concern the transition from pure to mixed states in two different universality classes of dynamics in one dimension, one being the exponential model, the other the Dyson model, the dynamics of the latter exhibiting strong graphical evidence of quantum chaos, as a consequence of the results of Albert and Kiessling on the Cloitre function.
Paper Structure (14 sections, 5 theorems, 93 equations)

This paper contains 14 sections, 5 theorems, 93 equations.

Table of Contents

  1. Introduction: states and observables for classical systems, chaotic states, and the smoothing property of densities
  2. Dynamics of states and the Schrödinger paradox. The case of states of infinite systems
  3. Models of qss and their dynamical properties
  4. The mean entropy, the Schrödinger paradox and the time-arrow: the necessity of states of infinite systems
  5. Two different ways to introduce a time arrow: 1) The Boltzmann approach; 2) Adiabatic transformations and the Lieb-Yngvason approach
  6. Dynamical properties of states of infinite systems related to the modified second law
  7. Equilibrium properties, basic structural transitions underlying the modified second law and Boltzmann's dictum
  8. Adiabatic transformations as generalizations of the barrier model. Sudden interactions
  9. Quantum chaotic states. The Krylov mechanism of approach to equilibrium: the role of dynamics and interactions
  10. The second law as a theorem for two classes of models of interacting spins at $T=0$: the necessity of extended perturbations
  11. Epilogue: Why does the mean entropy increase? The Universe, the time-arrow and the role of quantum fields, and some problems posed by AQFT
  12. Why does the mean entropy increase? The role of dynamics and universality classes in selecting the physical mechanism of approach to a NESS
  13. The Universe and the time-arrow, and the role of quantum fields
  14. Some problems posed by AQFT

Key Result

Theorem 2.1

For the gIm with interaction either of the exponential model $E_{\xi}$ with $\xi$ a transcendental number, or the Dyson model $D_{\alpha}$, and any initial state (at $t=0$) $\omega$ satisfying (2.19), where the limit on the l.h.s. of (2.14) is taken in the weak* topology.

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.1
  • Definition 2.4
  • Remark 2.2
  • Remark 2.3
  • ...and 6 more