Gaussian ensemble: An alternate Monte Carlo scheme

The recently introduced Gaussian ensemble involves a sample (of size N) thermally connected to a finite heat bath (of size N’) with specific properties. Treating N’ as a parameter, we use a leading-order analysis of the β (inverse temperature) -versus-E (energy of sample) curves to show how static properties of finite samples become ensemble dependent. Inflection points in β(E) at phase transitions, however, appear as nontrivial fixed points with respect to N’ and are defined as the transition temperature of the sample. By developing a fluctuation relation for the heat capacity C we show that, for small N’, states with C<0 are accessible at first-order transitions resulting in van der Waals loops in β(E). Monte Carlo studies of phase transitions in Potts models on two- and three-dimensional lattices confirm the finite-N’ and finite-N effects. We find that the method significantly reduces computer time (sometimes by a factor of 100) compared with canonical-ensemble simulations and is effective in diagnosing the order of phase transition. Specific-heat data at second-order transitions reveal a new phenomenon; the peak in C sharpens as N’ becomes smaller, leading us to speculate on sharp transitions in finite samples.