The approach to thermal equilibrium in quantized chaotic systems

TL;DR

This work shows that chaotic many-body quantum systems with banded random-matrix matrix elements for observables naturally equilibrate: the infinite-time average of an observable matches the thermal value and fluctuations are suppressed as $O(e^{-S})$, with quantum fluctuations mapping onto thermal fluctuations. By treating time as a random variable, the authors derive a universal, effectively deterministic evolution for the time-dependent expectation value, $A_t \approx C(t)A_0$, where $C(t)$ is a state-independent correlation function determined by the observable’s spectral bandwidth $W$. In the linear-response regime, $C(t)$ coincides with the Kubo correlation function $C_{\rm Kubo}(t)$, connecting Onsager’s regression to microscopic chaos via the bandwidth $|f(E,\omega)|^2$; Markovian behavior arises when a pole in this spectrum is present with a small decay rate $\Gamma \ll T$. Overall, the paper provides a robust, state-independent mechanism for Boltzmann-like relaxation in chaotic quantum systems and clarifies the role of spectral properties and linear response in determining the approach to equilibrium.

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.