Rigorous results on spontaneous symmetry breaking in a one-dimensional driven particle system

TL;DR

This paper rigorously analyzes spontaneous symmetry breaking in a one-dimensional driven two-species cellular automaton, providing asymptotic estimates for symmetry breaking and stability times, supported by simulations.

Contribution

It offers the first rigorous asymptotic estimates for symmetry breaking times in a driven particle system with open boundaries.

Findings

Expected symmetry breaking time scales as L ln(L).

Symmetry breaking stability time grows exponentially with system size.

Flipping times follow an algebraic distribution at the phase transition line.

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> ~ L ln(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.