Description
Through an explicit construction, we assign to any infinite temperature autocorrelation function $C(t)$ a function $\alpha^R(t)$, called "arrow of time function". The construction of $\alpha^R(t)$ from $C(t)$ requires the first $2R$ temporal derivatives of $C(t)$ at times $0$ and $t$. For correlation functions of few body observables we numerically observe the following: There is a overall tendency of the $\alpha^R(t)$ to become more monotonously decreasing with increasing $R$. However, while this tendency may by weak or absent in nonchaotic models, it is rather pronounced in chaotic regimes. The $R$ at which the $\alpha^R(t)$ become essentially monotonous may exceed 100 in nonchaotic models, but we always find this monotonicity at lower two digit R in the chaotic regime. All $\alpha^R(t)$ put upper bounds to the respective autocorrelation functions, i.e. $\alpha^R(t) \geq C^2(t)$. Hence $\alpha^R(t)$ may be interpreted as a distance from equilibrium. Thus the above construction is called an arrow of time since it states that a meaningful distance from equilibrium can (almost) only decrease, a statement similar in spirit to that of the H-theorem. We furthermore argue that finding may to some extend be traced back to the operator growth hypothesis. This argument is laid out in the framework of the so called recursion method.
