Ecient Monte Carlo sampling by direct ̄attening of free energy barriers

Systems undergoing ®rst-order phase transitions are accompanied by free energy barriers which separate the free energy minima characterizing the co-existing phases. These barriers grows with increasing system size. With conventional Monte Carlo simulation methods the characteristic time for crossing the barriers grows exponentially with system size and the system will necessarily get trapped in one of the free energy minima. In order to escape from this trapping, various novel simulation schemes (e.g., multicanonical/multimagnetical sampling, entropic sampling and simulated tempering) have been proposed and successfully applied to model systems. All these methods combine an iterative scheme with histogram reweighting techniques. We apply here another variant of these methods, which involves the use of shape functions, which are added to the model Hamiltonian in order to level out the free energy barriers. One of the virtues of this approach is the transparent formulation of the common philosophy underlying all the di€erent so-called `non-Boltzmann' simulational schemes devised to overcome free energy barriers. The basic principles of the method are presented. The easy adaption of the method to di€erent model systems is demonstrated by application to two case studies, a multi-state lattice model for phase equilibria in a binary lipid bilayer, and a twodimensional lattice gas model which exhibits interfacial melting, which are known to be notoriously dicult to study by conventional Monte Carlo methods. The practical aspects of the implementation of the method are discussed. The results demonstrate the eciency and versatility of the shape function method. 02.70.Lq; 05.70.Fh; 64.60.Cu; 87.15.Da Monte Carlo simulation; Multicanonical sampling; Non-Boltzmann sampling; Histograms; Spectral free energies; Shape functions; Asymmetric ®rst-order phase transitions; Binary lipid bilayers; Interfacial melting