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 dim ension of the system, and surface effects are avoided by periodic boun dary conditions. We assume that upon lowering the temperature the syst em discontinuously goes to one of q ordered states, such as it e.g. ha ppens for the Potts model in d = 3 for q greater-than-or-equal-to 3, w ith the correlation length xi of order parameter fluctuation staying f inite at the transition. We then describe each of these q ordered phas es and the disordered phase for L much greater than xi by a properly w eighted Gaussian. From this phenomenological ansatz for the total dist ribution of the order parameter, all moments of interest are calculate d 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-proportion al-to L-d and the value of the cumulant at the minimum (g (Tin)) behav ing 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 poin t 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 search ing for such a behavior in numerical simulation data, the first order character of a phase transition can be asserted. The usefulness of thi s approach is shown using data for the q = 3, d = 3 Potts ferromagnet.