Overcoming the limitation of finite size in simulations: From the phase transition of the ising model to polymers, spin glasses, etc.

Monte Carlo simulations always deal with systems of finite size, e.g. a d-dimensional Ising lattice of linear dimension L (with periodic boundary conditions) is treated. However, phase transitions occur only in the thermodynamic limit, L --> infinity, for finite L the transition is rounded and shifted. Hence it is a problem to locate precisely a phase transition with Monte Carlo methods, distinguish whether the transition is of first order or of second order, and characterize its properties accurately. A brief review is given how this problem is solved, both in principle and practically, by the application of finite size scaling concepts, both for critical points and for first order transitions. It will be argued that the fourth order cumulant of the "order parameter" of the transition is a particular convenient tool for locating the transition, and examples given will include systems such as unmixing of polymer blends, and Ising spin glasses. However, problems still remain for systems with phases exhibiting a power law decay of the order parameter correlations, such as the hexatic phase (if it exists) at the liquid-solid transition of the hard disk systems. Unsolved problems also occur for systems such as Potts glasses, where a first order transition without latent heat is theoretically predicted to occur. Interesting finite size effects also occur when one simulates phase coexistence: long wavelength capillary wave-type fluctuations of interfaces cause interfacial widths to depend in an intricate way on linear dimensions parallel and perpendicular to the interface(s); simulating liquid droplets coexisting with surrounding supersaturated gas in a finite volume an unconventional droplet evaporation transition with a (rounded) jump of the supersaturation occurs; etc. These phenomena still are the subject of current research.