Rigorous Results on Spontaneous Symmetry Breaking in a One-Dimensional Driven Particle System

Abstract

We study spontaneous symmetry breaking in a one-dimensional driven two-species stochastic cellular automaton with parallel sublattice update and open boundaries. The dynamics are symmetric with respect to interchange of particles. Starting from an empty initial lattice, the system enters a symmetry broken state after some time T 1 through an amplification loop of initial fluctuations. It remains in the symmetry broken state for a time T 2 through a traffic jam effect. Applying a simple martingale argument, we obtain rigorous asymptotic estimates for the expected times 〈 T 1〉 ∝ Lln L and ln 〈 T 2〉 ∝ L, where L is the system size. The actual value of T 1 depends strongly on the initial fluctuation in the amplification loop. Numerical simulations suggest that T 2 is exponentially distributed with a mean that grows exponentially in system size. For the phase transition line we argue and confirm by simulations that the flipping time between sign changes of the difference of particle numbers approaches an algebraic distribution as the system size tends to infinity.