Parallel updating cellular automaton models of driven diffusive Frenkel-Kontorova-type systems

TL;DR

This paper introduces parallel updating cellular automaton models for driven diffusive systems related to Frenkel-Kontorova models, capturing key features like phase transitions and hysteresis, and provides analytical solutions validated by simulations.

Contribution

It presents new parallel updating CA models for FK systems, enabling analytical treatment and capturing essential physical phenomena.

Findings

Models reproduce phase transitions and jamming.

Analytical solutions agree with numerical data.

Parallel updating simplifies analysis.

Abstract

Three cellular automaton (CA) models of increasing complexity are introduced to model driven diffusive systems related to the generalized Frenkel-Kontorova (FK) models recently proposed by Braun [Phys.Rev.E58, 1311 (1998)]. The models are defined in terms of parallel updating rules. Simulation results are presented for these models. The features are qualitatively similar to those models defined previously in terms of sequentially updating rules. Essential features of the FK model such as phase transitions, jamming due to atoms in the immobile state, and hysteresis in the relationship between the fraction of atoms in the running state and the bias field are captured. Formulating in terms of parallel updating rules has the advantage that the models can be treated analytically by following the time evolution of the occupation on every site of the lattice. Results of this analytical approach are given for the two simpler models. The steady state properties are found by studying the stable fixed points of a closed set of dynamical equations obtained within the approximation of retaining spatial correlations only upto two nearest neighboring sites. Results are found to be in good agreement with numerical data.