Cellular automaton models of driven diffusive Frenkel-Kontorova-type systems

Three cellular automaton models of increasing complexity are introduced to model driven diffusive systems related to the generalized Frenkel-Kontorova (FK) models recently proposed by Braun el. nl. [Phys. Rev. E 58, 1311 (199 8)]. The models are defined in terms of parallel updating rules. Simulation results are presented for these models. The features are qualitatively sim ilar to those models defined previously in terms of sequentially updating r ules. Essential features of the FK model such as phase transitions, jamming due to atoms in the immobile state, and hysteresis in the relationship bet ween the fraction of atoms in the running state and the bias field are capt ured. Formulating in terms of parallel updating rules has the advantage tha t the models can be treated analytically by following the time evolution of the occupation on every site of the lattice. Results of this analytical ap proach are given for the two simpler models. The steady state properties ar e found by studying the stable fixed points of a closed set of dynamical eq uations obtained within the approximation of retaining spatial correlations only up to two nearest-neighboring sites. Results are found to be in good agreement with numerical data. [S1063-651- X(99)03807-6].