Statistical mechanics of finite three-dimensional Ising models

The singularities observable at second-order phase transitions ( e.g. , the divergence of the specific heat) can occur only if the size of the considered system is very large (strictly speaking. infinite). Therefore some pronounced modifications of critical phenomena have to be expected in small systems. We investigate these modifications by numerical calculations for the case of the three-dimensional Ising model (the total number of spins in the considered systems being in the range from N ≈ 10 N ≈ 10 3 ; for N ⪕ 54 the exact zero-field partition functions are given, while we derive the properties of the larger systems using the Monte-Carlo technique). Assuming systems with free surfaces we find also a dependence on the shape of the system. Considering the behaviour of the energy, the specific heat, and the magnetization, one observes both a rounding of the critical anomalies and a shift to lower temperatures. This behaviour is in good quantitative agreement with the conclusions which can be drawn from the exactly soluble two-dimensional Ising model.