Equilibrium return times of small fluctuating clusters and vacancies

Abstract

The expected return time of a fluctuating two-dimensional cluster or vacancy to a given configuration is studied in thermodynamic equilibrium. We define a family of bond-breaking models that preserve the number of particles. This family includes edge diffusion and surface diffusion inside vacancies in the limit of fast particle diffusion and slow attachment-detachment kinetics. Within the frame of these bond-breaking models, the expected return time is found to depend on the energies of the configurations and on the energies of the excited states formed by removing a single particle from the cluster. High and low temperature regimes are studied. We clarify the conditions under which the return time is a non-monotonous function of temperature: a minimum is found when the energy obtained by the average over the excited states of the configuration weighted by their attachment probabilities is lower than the energy averaged over all states. In addition, we show that the optimal temperature at which the return time is minimum is shifted to a higher temperature as compared to the temperature at which the equilibrium probability is maximum. This shift is influenced by the average curvature of the cluster edge, and is therefore larger for vacancies.I.

Temperature transitions and degeneracy in the control of small clusters with a macroscopic field

“…As a first remark, τ φ (s, s) is always increasing when decreasing temperature (as long as F 0 < 2 for the energy barriers to be positive as discussed in section 2). Such a behavior is expected for small clusters as discussed in [13,15].…”

“…The temperature dependence of the expected return time to target τ r φ (s) was discussed in our previous paper with or without force [13,15]. In this section, we briefly report the main features of this temperature-dependence.…”

“…There are four possible moves that take the system from s to s. This is actually a specific property of dimers. Indeed, for larger clusters with N > 2, there is always a unique move to go from one state to another state [13,15]. The moves are shown in figure 3.…”

“…The energy cost for breaking each bond is denoted as J. This contribution is compatible with detailed balance in the absence of forces [15]. Hence, the system can reach thermodynamic equilibrium in the absence of force, with an edge energy that is identical to that of the low-temperature Ising model [10,[16][17][18].…”