FINITE SIZE EFFECTS AT THERMALLY-DRIVEN 1ST-ORDER PHASE-TRANSITIONS - A PHENOMENOLOGICAL THEORY OF THE ORDER PARAMETER DISTRIBUTION

We consider the rounding and shifting of a first-order transition in a finite d-dimensional hypercubic L(d) geometry, L being the linear dimension of the system, and surface effects are avoided by periodic boundary conditions. We assume that upon lowering the temperature the system discontinuously goes to one of q ordered states, such as it e.g. happens for the Potts model in d = 3 for q greater-than-or-equal-to 3, with the correlation length xi of order parameter fluctuation staying finite at the transition. We then describe each of these q ordered phases and the disordered phase for L much greater than xi by a properly weighted Gaussian. From this phenomenological ansatz for the total distribution of the order parameter, all moments of interest are calculated straight-forwardly. In particular, it is shown that for L exceeding a characteristic minimum size L(min) the forth-order cumulant g(L) (T) exhibits a minimum at T(min) > T(c), with T(min) - T(c) is-proportional-to L-d and the value of the cumulant at the minimum (g (Tin)) behaving as g (T(min)) is-proportional-to L-d. All cumulants g(L)(T) for L much greater than xi approximately intersect at a common crossing point T(cross) is-proportional-to L-2d, with a universal value g (T(cross)) = 1 - n/2q, where n is the order parameter dimensionality. By searching for such a behavior in numerical simulation data, the first order character of a phase transition can be asserted. The usefulness of this approach is shown using data for the q = 3, d = 3 Potts ferromagnet.