VOLUME 55, NUMBER 7 PHYSICAL REVIEW LETTERS 12 AUGUST 1985 Role of Irreversibility in Stabilizing Complex and Nonergodic Behavior in Locally Interacting Discrete Systems Charles H. Bennett ' and G. Grinstein IBM T. J. 8 atson Research Laboratory, Yorktown Heights, New York 10598 (Received 18 March 1985) Irreversibility stabilizes certain locally interacting discrete systems against the nucleation and growth of a most-stable phase, thereby enabling them to behave in a computationally complex and nonergodic manner over a set of positive measure in the parameter space of their local transition probabilities, unlike analogous reversible systems. PACS numbers: 05.90. +m The dynamics of a statistical-mechanical system in contact with a larger environment is often modeled as a random walk on the system's state space (e.g. , kinet- ic Ising model). If the environment is at equilibrium, this random walk will be "microscopically reversible" (its matrix of transition probabilities being of the form DSD ', where D is diagonal and S symmetric), and the system's stationary distribution of states will be an equilibrium one (e.g. , canonical ensemble) defined by a Hamiltonian simply related to the transition probabil- ities. On the other hand, if the environment is not at equilibrium, the system's transition matrix in general will be irreversible, and the resulting nonequilibrium stationary distribution may be very hard to character- ize. Though an irreversible system's distribution of states is not simply related to the transition probabili- ties, its distribution of histories is. More specifically, the stationary distribution of histories for any stochas- tic model, whether reversible or not, may be viewed as a canonical distribution under an effective Hamil- tonian on the space of histories, in which each confi- guration interacts with its predecessor in time with an "interaction energy, " equal simply to the logarithm of the corresponding transition probability. In particular, we consider the case in which the underlying stochastic model is a probabilistic cellular automaton (CA), in other words, a d-dimensional lat- tice with finitely many states per site, in which each site, at each discrete time step, undergoes a transition depending probabilistically on the states of its neigh- bors. In this case'2 the stationary distribution of his- tories of the CA is equivalent to the equilibrium statis- tics of a corresponding generalized Ising model (GIM) in 2+1 dimensions. This appears paradoxical, be- cause CA's are known to be capable of complex, nonergodic behavior even when all local transition probabilities are positive, whereas the behavior of GIM is generally simple and ergodic (a stochastic pro- cess is "ergodic" if its stationary distribution is unique). For example, a standard kinetic Ising model, at a generic point in its temperature-magnetic field parameter space, undergoes nucleation and growth of a unique most-stable phase, thereby relaxing to a stationary distribution independent of the initial conditions. Here we note the resolution of the paradox, and il- lustrate an essential difference between reversible and irreversible systems by characterizing the phase dia- gram, equation of state, domain growth kinetics, and equivalent (d+1)-dimensional GIM of one of the simplest nonergodic irreversible CA, viz. , Toom's north-east-center (NEC) voting model, The resolution of the paradox lies in the fact that when a d- dimensional CA is represented as a (d + 1)- dimensional GIM, the parameters (coupling constants) of the latter system are not all independent, but are constrained in such a way as to cause the free energy of the (2+1)-dimensional system to be identically zero, no matter how the parameters (transition proba- bilities) of the underlying CA are varied. It is there- fore possible for irreversible systems such as the NEC model to be nonergodic, and in particular to have two or more stable phases, over a finite region in their phase diagrams, whereas reversible systems can exhib- it this behavior only over a subset of zero measure consisting of points in the phase diagram where two or more phases, by symmetry or accident, have exactly equal free energy. North east center m-odel, -and reasons for its nonergo dicity. The NEC — model is one of a class of voting rules for CA shown by Toom3 to be nonergodic in the presence of small but arbitrary probabilistic perturba- tions. The model consists of a square lattice of spins, each of which may be up or down. The spins are up- dated synchronously, with a spin's future state decided by majority vote of the spins in an unsymmetric neigh- borhood, consisting of the spin itself and its northern and eastern neighbors. The rule just described is deterministic; we consider two-parameter noisy pertur- bations of the rule, in which a spin whose present neighborhood majority is up, instead of going up with certainty at the next time step, goes up with probability 1 — p and down with probability p; and a spin whose present neighborhood majority is down goes down with probability 1 — q and up with probability q. Alter- natively, the noise may be characterized by its "ampli- tude" p+ q, analogous to temperature, and its "bias" (p — q)/(p+ q), analogous to magnetic field. Because probabilistic CA typified by the NEC model 657