Evidence for simple "arrow of time functions" in closed chaotic quantum systems

TL;DR

This paper constructs and analyzes 'arrows of time functions' (AOTFs) from autocorrelation functions in chaotic quantum systems, showing they generally decrease monotonically and indicate a directed approach to equilibrium, akin to an H-Theorem.

Contribution

It introduces a method to derive AOTFs from autocorrelation functions and demonstrates their monotonic behavior in chaotic quantum systems, linking to operator growth and equilibration.

Findings

AOTFs can be constructed from autocorrelation functions using derivatives.

Most systems exhibit AOTFs unless near nonchaotic regimes.

AOTFs provide bounds and indicate a directed approach to equilibrium.

Abstract

Through an explicit construction, we assign to any infinite temperature autocorrelation function $C(t)$ a set of functions $\alpha^n(t)$. The construction of $\alpha^n(t)$ from $C(t)$ requires the first $2n$ temporal derivatives of $C(t)$ at times $0$ and $t$. Our focus is on $\alpha^n(t)$ that (almost) monotonously decrease, we call these ``arrows of time functions" (AOTFs). For autocorrelation functions of few body observables we numerically observe the following: An AOTF featuring a low $n$ may always be found unless the the system is in or close to a nonchaotic regime with respect to a variation of some system parameter. All $\alpha^n(t)$ put upper bounds to the respective autocorrelation functions, i.e. $\alpha^n(t) \geq C^2(t)$. Thus the implication of the existence of an AOTF is comparable to that of the H-Theorem, as it indicates a directed approach to equilibrium. We furthermore argue that our numerical finding may to some extent be traced back to the operator growth hypothesis. This argument is laid out in the framework of the so-called recursion method.