The approach to thermal equilibrium in quantized chaotic systems

TL;DR

This paper investigates how quantum chaotic many-body systems naturally evolve towards thermal equilibrium, demonstrating that expectation values tend to relax over time following the Kubo correlation function, regardless of initial states.

Contribution

It provides a probabilistic framework showing that expectation values in quantum chaotic systems tend to equilibrium, aligning with Onsager's postulate, independent of initial conditions.

Findings

Expectation values move towards equilibrium over time.

The relaxation follows the Kubo correlation function.

Results are independent of the initial quantum state.

Abstract

We consider many-body quantum systems that exhibit quantum chaos, in the sense that the observables of interest act on energy eigenstates like banded random matrices. We study the time-dependent expectation values of these observables, assuming that the system is in a definite (but arbitrary) pure quantum state. We induce a probability distribution for the expectation values by treating the zero of time as a uniformly distributed random variable. We show explicitly that if an observable has a nonequilibrium expectation value at some particular moment, then it is overwhelmingly likely to move towards equilibrium, both forwards and backwards in time. For deviations from equilibrium that are not much larger than a typical quantum or thermal fluctuation, we find that the time dependence of the move towards equilibrium is given by the Kubo correlation function, in agreement with Onsager's postulate. These results are independent of the details of the system's quantum state.