Predicting the future from the past: An old problem from a modern perspective

The Qohelet’s Book 1:9, NIV.

We sketch the derivation in the simple case of a discrete-state system as obtained by partitioning the phase-space of the given system into a discrete number of regions ${ρj}$⁠, each one defining a coarse-grained state j, such that the probability $pj$ of a state j is given by $pj=μ(ρj)$⁠. We can focus on a state s. Assume that during the time interval $T$⁠, a system trajectory has visited $Ns$ times the sth state (representing here the set $σ$⁠). Denoting with $τs(i)$ the time taken by the ith return to the state s, the mean recurrence time is $∑i=1Nsτs(i)/Ns=T/Ns$⁠, that is, the inverse of the fraction of time spent by the system in the sth state, $fs=Ns/T$⁠. Invoking ergodicity, in the limit $T→∞$⁠, we have $T/Ns→〈τs〉$ and $fs→ps$⁠, where $ps$ is the probability of the sth state (i.e., the probability $μ(σ)$⁠) so that $〈τs〉=1/ps$⁠, which represents Eq. (14). Replacing $T$ and $Ns$ with M – 1 and $M(ϵ)$⁠, respectively, we obtain Eqs. (10) and (11), with $Δt=1$⁠, so that for large M, Eq. (11) gives again Kac’s result.