Poincar\'{e} cycle of a multibox Ehrenfest urn model with directed transport

TL;DR

This paper introduces a generalized directed Ehrenfest urn model with multiple urns arranged in a circle, analyzing its dynamics, oscillatory behavior, and calculating the Poincaré cycle based on microstate counting.

Contribution

It presents a new generalized model with directed transport, provides a solution method for the N-ball M-urn problem, and analyzes the system's oscillations and recurrence time.

Findings

Average urn occupancy oscillates before reaching stationarity

Poincaré cycle is derived from microstate enumeration

Directed transport induces unique oscillatory behavior

Abstract

We propose a generalized Ehrenfest urn model of many urns arranged periodically along a circle. The evolution of the urn model system is governed by a directed stochastic operation. Method for solving an $N$-ball, $M$-urn problem of this model is presented. The evolution of the system is studied in detail. We find that the average number of balls in a certain urn oscillates several times before it reaches a stationary value. This behavior seems to be a peculiar feature of this directed urn model. We also calculate the Poincar\'{e} cycle, i.e., the average time interval required for the system to return to its initial configuration. The result can be easily understood by counting the total number of all possible microstates of the system.