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Quantum chaos and the arrow of time

Nilakash Sorokhaibam

TL;DR

The paper addresses whether quantum chaotic many-body systems exhibit a thermodynamic arrow of time. It extends Berry's conjecture to the energy change under perturbations and employs ensemble equivalence to link microscopic matrix elements with macroscopic thermodynamics, yielding ETH-monotonicity constraints on off-diagonal terms. The leading-order analysis shows that the energy change ΔE has the same sign as the temperature T (Kelvin form), with the f-function in ETH becoming effectively constant in the thermodynamic limit, and numerical XXZ-chain results supporting these predictions. This work provides a microscopic mechanism for the arrow of time in chaotic quantum systems, clarifies the role of off-diagonal ETH terms, and introduces testable constraints for quantum thermalization beyond pure entropic arguments.

Abstract

We show that quantum chaotic many-body systems possess the thermodynamic arrow of time in the thermodynamic limit. Berry's conjecture in quantum chaotic systems and equivalence of ensembles imply the Kelvin statement of the second law of thermodynamics at leading order in perturbation theory. We verify this result using numerical calculations. We also show that this gives rise to new constraints on the off-diagonal terms in eigenstate thermalization hypothesis (ETH) statement. We call the new constraints collectively as ETH-monotonicity. These constraints arise because pure entropic consideration is not enough for the emergence of the thermodynamic arrow of time.

Quantum chaos and the arrow of time

TL;DR

The paper addresses whether quantum chaotic many-body systems exhibit a thermodynamic arrow of time. It extends Berry's conjecture to the energy change under perturbations and employs ensemble equivalence to link microscopic matrix elements with macroscopic thermodynamics, yielding ETH-monotonicity constraints on off-diagonal terms. The leading-order analysis shows that the energy change ΔE has the same sign as the temperature T (Kelvin form), with the f-function in ETH becoming effectively constant in the thermodynamic limit, and numerical XXZ-chain results supporting these predictions. This work provides a microscopic mechanism for the arrow of time in chaotic quantum systems, clarifies the role of off-diagonal ETH terms, and introduces testable constraints for quantum thermalization beyond pure entropic arguments.

Abstract

We show that quantum chaotic many-body systems possess the thermodynamic arrow of time in the thermodynamic limit. Berry's conjecture in quantum chaotic systems and equivalence of ensembles imply the Kelvin statement of the second law of thermodynamics at leading order in perturbation theory. We verify this result using numerical calculations. We also show that this gives rise to new constraints on the off-diagonal terms in eigenstate thermalization hypothesis (ETH) statement. We call the new constraints collectively as ETH-monotonicity. These constraints arise because pure entropic consideration is not enough for the emergence of the thermodynamic arrow of time.
Paper Structure (14 sections, 46 equations, 17 figures)

This paper contains 14 sections, 46 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Thermalization does not imply the arrow of time
  3. Set-up and results
  4. First pass: the arrow of time
  5. Second pass: new constraints on ETH statement
  6. Higher order terms of the change in energy
  7. Large perturbations
  8. Conclusions
  9. Change in energy starting from an energy eigenstate
  10. Change in energy starting from a thermal state
  11. Derivation using linear response theory
  12. Comparison of canonical and microcanonical ensembles
  13. The entropic factor in finite dimensional state space
  14. Numerical methods and further numerical results

Figures (17)

  • Figure 1: The density of states ($e^S$) versus energy ($E$) plot for the chaotic XXZ spin chain. $T$ is effective temperature of the energy eigenstates. Perturbation of the system leads to a change in energy $\Delta E$ which agrees with the Kelvin statement of the second law of thermodynamics. Using the relation $\Delta E= T\Delta S$, it also implies the Planck statement $\Delta S\geq 0$.
  • Figure 2: The leading term of the change in energy $\Delta E_n$ for the chaotic spin chains (left panels) and non-chaotic spin chains (right panels) after a unit delta function perturbation $\lambda(t)=\delta(t)$ with spin current operator $\mathcal{O_{SC}}$ (panels a, b, c and d), and with the kinetic energy term $\mathcal{O}_{KE}$ (panels e and f). The blue dots are for the different energy eigenstates of energy $E_n$ as the initial state, the yellow plots are for canonical ensembles and the green dots are for microcanonical ensembles. We have considered spin chain of system size $L=14, N=7$ (panel a and b), and $L=16, N=8$ (panel c, d, e and f). The dimension of the state space are $\binom{14}{7}=3432$ and $\binom{16}{8}=12870$. So, there are 12870 blue dots in each of the panels c, d, e and f.
  • Figure 3: $f(\bar{E})$ plots with fixed $\omega$'s for the spin current operator with different system sizes $L=14, N=7$, $L=16, N=8$, and $L=18, N=9$.
  • Figure 4: $1/f\partial f/\partial \bar{E}$ plots with fixed $\omega$'s for the spin current operator with different system sizes $L=14, N=7$, $L=16, N=8$, and $L=18, N=9$.
  • Figure 5: Panel (a) is plots of slope of linear fit of $1/f\partial f/\partial \bar{E}$ for $|\bar{E}|\leq 1$ for the spin current operator with different system sizes $L=14, N=7$, $L=16, N=8$, and $L=18, N=9$. Panels (b), (c) and (d) are plots of the slopes after scaling with different powers of the system size. Panel (c) shows that the slope is decreasing as $L^{-1}$ where $L$ is the system size.
  • ...and 12 more figures