FINITE SIZE SCALING AND LONG-RANGE INTERACTIONS (SPHERICAL MODELS, EPSILON EXPANSIONS)

A study of the effects of the presence of long range interactions on finite systems near a critical point is carried out. Away from the critical point, convergence to the thermodynamic limit is qualitatively different than in short range systems. Using general arguments, it is shown thermodynamic functions converge to the bulk limit as a power of the system size, as opposed to exponentially in the system size as in short range systems. This power is predicted to be the power of the decay of interactions in the case of the susceptibility. The problem of defining a correlation length suitable for measurement in computer simulations is discussed, and the second moment of the correlation function, which diverges at all temperatures, is proposed as a useful quantity to measure. A detailed calculation of the spherical model with long ranges interactions is described. The model is solvable in arbitrary dimension with ferromagnetic interaction decaying as an arbitrary power of spin separation. Closed form expressions for the susceptibility, free energy, and correlation function are derived. Finite size scaling is obeyed in this model. Mean field theory of finite systems is verified by the interaction independence of the scaling functions above the upper critical dimensions. The convergence to the bulk properties is power law in the system size; the predicted power for the susceptibility is verified and the power for the free energy is found. It is argued that the long range interaction introduces corrections to scaling which are not present in short range systems. However, these corrections are higher order than the leading ones. A method of renormalization group calculation is developed for finite systems in which the exponentiation of logarithms to produce scaling variables is explicit. A lowest order calculation using this method is described for long ranged, Ising-like systems. Finite size scaling is verified to lowest order, and anomalous corrections similar to those found in the spherical model are present. Arguments are given in support of the power law convergence of thermodynamic functions away from the critical point.