Scaling studies of percolation phenomena in systems of dimensionality two to seven. II. Equation of state

Abstract

Cluster statistics obtained by the Monte Carlo method for percolation processes in systems of dimensionality two to seven are analysed for the percolation analogue of the thermodynamic equation of state, thus complementing the work of paper I on cluster numbers. In particular, we calculate the scaling functions for the analogues of the thermodynamic potentials and their derivatives, and investigate their dependence on dimension d. We are guided by the two exactly soluble limits of d = 1 and the Bethe lattice (d =a). The scaling region, where a good degree of data collapsing can be observed, is investigated in terms of the two 'thermodynamic' variables, one of which is analogous to the temperature and the other to the magnetic field. This region is found to be comparatively large and symmetrical in two dimensions, but considerably smaller in higher dimensions. In addition, we find that the characteristic forms of the scaling functions are closely related to the 'thermodynamic' stability conditions. Finally, we analyse the logarithmic corrections to the scaled equation of state at the upper marginal dimension, d, = 6, and a numerical demonstration of the significance of the logarithmic corrections is presented in terms of data collapsing.