Similarity of ensembles of trajectories of reversible and irreversible growth processes

Abstract

Models of bacterial growth tend to be "irreversible," allowing for the number of bacteria in a colony to increase but not to decrease. By contrast, models of molecular self-assembly are usually "reversible," allowing for the addition and removal of particles to a structure. Such processes differ in a fundamental way because only reversible processes possess an equilibrium. Here we show at the mean-field level that dynamic trajectories of reversible and irreversible growth processes are similar in that both feel the influence of attractors, at which growth proceeds without limit but the intensive properties of the system are invariant. Attractors of both processes undergo nonequilibrium phase transitions as model parameters are varied, suggesting a unified way of describing typical properties of reversible and irreversible growth. We also establish a connection at the mean-field level between an irreversible model of growth (the magnetic Eden model) and the equilibrium Ising model, supporting the findings made by other authors using numerical simulations.

Large deviations in models of growing clusters with symmetry-breaking transitions

“…This Section defines the models considered in this work, which were introduced by Klymko, Garrahan and Whitelam [23]. We also describe the method used by KGGW to obtain bounds on the probabilities of rare events in these models [11].…”

“…For large k one finds m k+1 m k m k + tanh(Jm k ) − m k k . (5) This means that steady state values of the magnetisation must solve m = tanh Jm [23]. This equation is familiar from mean-field models of ferromagnets: there is only one solution if J ≤ 1 but for J > 1 there are three solutions.…”

“…The steady states of this model are described in [23]. There are symmetry-breaking transitions, similar to the irreversible model, but with some additional complexity.…”

“…In this work, we consider systems where clusters of particles grow, as a function of time [23,24]. The clusters contain two kinds of particles, which can be distinguished either by their spins (up or down), or their colors (red or blue).…”

“…In this paper, Sec. II describes the models and the main methods of [11,23]; this includes a model of irreversible cluster growth, and one where the cluster grows reversibly, so that the number of particles can both increase or decrease. Sec.…”