Fast Condensation in a tunable Backgammon model

TL;DR

This paper investigates a tunable Monte Carlo model of the Backgammon system, revealing how varying a parameter accelerates condensation dynamics and differs from similar urn models.

Contribution

Introduces a parameterized Backgammon model with tunable dynamics, demonstrating fast condensation phenomena and contrasting its behavior with the Zeta Urn model.

Findings

For </2, the system exhibits rapid condensation.

Condensation time depends on  and system size N.

The probability distribution of particles per box evolves differently than in Zeta Urn models.

Abstract

We present a Monte Carlo study of the Backgammon model, at zero temperature, in which a departure box is chosen at random with a probability proportional to $(2\omega - 1)k + (1 - \omega)N$, where $k$ is the number of particles in the departure box and $N$ is the total number of particles (equivalently, boxes) in the system. The parameter $\omega \in [0,1]$ tunes the dynamics from being slow ($\omega = 1$) to being fast ($\omega = 0$). This parametrization tacitly assumes a two-box representation for the system at any instant of time and $\omega$ is formally related to the 'memory' parameter of a correlated binary sequence. For $\omega < 1/2$, the system undergoes a fast condensation beyond a certain time that depends on $\omega$ and the system size $N$. This condensation provides an interesting contrast to that studied with Zeta Urn model in that the probability that a box contains $k$ particles evolves differently in the model discussed here.