Similarity of ensembles of trajectories of reversible and irreversible growth processes

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.

Figure 1

Phase diagrams of the stable dynamic attractors of the (a) reversible and (b) irreversible models of growth, for $h=0$. U and M indicate unmagnetized and magnetized regions, respectively, with the latter admitting two stable attractors. The solid line in (a) and dotted line in (b) denote continuous phase transitions. To the left of the dotted line in (a), we have no growth.

Figure 2

Dynamic growth trajectories in the space of magnetization $m$ and system size $N$ for reversible (top) and irreversible (bottom) models. We show trajectories (a),(b) in the magnetized regime, with attractors marked as dotted lines, and (c),(d) at a dynamic critical point, where the attractor lies at zero magnetization. Parameters: (a) $J=2.5$ and $c=2$, (b) $J=1.25$, (c) $J=2$ and $c=2$, (d) $J=1$.

Figure 3

Trajectory-to-trajectory (a),(b) averages and (c),(d) variance of magnetization taken over ensembles of ${10}^5$ trajectories, at various values of $J$, for (a),(c) reversible ($c=2$) and (b),(d) irreversible growth. The two types of processes display similar phase transitions at the level of trajectory averages. Numerical simulations are overlaid on the analytic results (a) (6) and (b) (3). In (a), we also show the results of simulations done in the presence of a system size constraint, overlaid on (5) (see Fig. 4).

Figure 4

The evolution of (a) $\langle N\rangle$ and (b) $\langle \left|m\right|\rangle$ for size-limited reversible growth shows that trajectories fall under the influence of the dynamic attractor while growth persists, and evolve to the equilibrium attractor when the size constraint is reached. Lines denote averages over 500 trajectories. Parameters: $J=1.5,c=2$. The line labeled “constrained” in Fig. 3 shows the results of similarly constrained simulations for several values of $J$.