Optimal annealing schedules for two-, three-, and four-level systems using a genetic algorithm approach

Abstract

An annealing schedule, T(t), is the temperature as function of time whose goal is to bring a system from some initial low-order state to a final high-order state. We use the probability in the lowest energy level as the order parameter, so that an ideally annealed system would have all its population in its ground-state. We consider a model system comprised of discrete energy levels separated by activation barriers. We have carried out annealing calculations on this system for a range of system parameters. In particular, we considered the schedule as a function of the energy level spacing, of the height of the activation barriers, and, in some cases, as a function of degeneracies of the levels. For a given set of physical parameters, and maximum available time, tm, we were able to obtain the optimal schedule by using a genetic algorithm (GA) approach. For the two-level system, analytic solutions are available, and were compared with the GA-optimized results. The agreement was essentially exact. We were able to identify systematic behaviors of the schedules and trends in final probabilities as a function of parameters. We have also carried out Metropolis Monte Carlo (MMC) calculations on simple potential energy functions using the optimal schedules available from the model calculations. Agreement between the model and MMC calculations was excellent.