The case of reversible coalescence, when the back reaction A → A + A is allowed, is special in that the steady state is a true equilibrium state. The process can then be analyzed by standard thermodynamics techniques and one expects simple, classical behavior. It is therefore surprising to find that the approach to equilibrium is characterized by a dynamical phase transition in the typical time of relaxation. This phase transition can be exactly analyzed in finite lattices, providing us with a unique opportunity to study finite-size effects in a dynamical phase transition.

The equilibrium steady state

We now turn to the case of reversible coalescence, when the back reaction A → A + A occurs at rate υ > 0 (and with no input, R = 0). The process is illustrated in Fig. 17.1. After a short transient the system arrives at a steady state with a finite concentration of particles. Because coalescence is now reversible, this steady state is in fact an equilibrium state, which satisfies detailed balance. The statistical time-reversible invariance of this equilibrium state can be seen in Fig. 17.1, in which the direction of time is ambiguous (compare it with Figs. 16.1 and 16.4).

The equilibrium state is a state of maximum entropy, and therefore the particles follow a completely random (Poisson) distribution. This is characterized by an exponential IPDF, peq(x) = ceqexp(−ceqx). Alternatively, the state may be described by the lack of correlation among the occupation probabilities of different sites: each site is occupied with probability ceq Δx, independent of other sites.