Meeting time distributions in Bernoulli systems

TL;DR

This paper demonstrates that in Bernoulli systems, the distribution of meeting times between orbits follows an exponential law, with a specific rate depending on the system's probabilities, especially as the neighborhood size shrinks to zero.

Contribution

It establishes the exponential distribution of meeting times in Bernoulli systems and derives an explicit formula for the rate parameter based on system probabilities.

Findings

Meeting time distribution is exponential in Bernoulli systems.

The rate parameter α equals the sum of p_l(1 - p_l) over all states.

Distribution converges to exp(-ατ) as ε approaches zero.

Abstract

Meeting time is defined as the time for which two orbits approach each other within distance $\epsilon$ in phase space. We show that the distribution of the meeting time is exponential in $(p_1,...,p_k)$-Bernoulli systems. In the limit of $\epsilon\to0$, the distribution converges to exp(-\alpha\tau), where $\tau$ is the meeting time normalized by the average. The exponent is shown to be $\alpha=\sum_{l=1}^{k}p_l(1-p_l)$ for the Bernoulli systems.