Scaling of Ergodicity in Binary Systems

TL;DR

This paper investigates how the number of sub-sequences in a binary sequence scales with their length to understand the emergence of ergodic behavior, using the Mean Ergodic Time as a key metric.

Contribution

It introduces a method to estimate the scaling of sub-sequences needed for ergodicity in binary systems based on the Mean Ergodic Time.

Findings

Derived a relationship between sub-sequence count and length for ergodicity.

Provided a quantitative metric (MET) to assess ergodic behavior.

Analyzed the conditions under which binary sequences exhibit ergodicity.

Abstract

Given pseudo-random binary sequence of length $L$, assuming it consists of $k$ sub-sequences of length $N$. We estimate how $k$ scales with growing $N$ to obtain a {\it limiting} ergodic behaviour, to fulfill the basic definition of ergodicity (due to Boltzmann). The average of the consecutive sub-sequences plays the role of time (temporal) average. This average then compared to ensemble average to estimate quantitative value of a simple metric called Mean Ergodic Time (MET), when system is ergodic.