Modified Fisher Droplet Model
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MODIFIED FISHER DROPLET MODEL
J. MARRO AND R. TORAL Departamento de Fisica Te6rica, Universidad de Barcelona, Diagonal 647, Barcelona-28, Spain
ABSTRACT We propose a simple modification of the Fisher droplet model which, unlike classical nucleation theory, reproduces very well some Monte Carlo equilibrium cluster distributions Pl for the three-dimensional Ising or lattice gas model. It then follows that pI(h) /p (h=O) = Aexp(-VQl) where Q (,hl/Y , y O.45, when the magnetic field h is small enough, as suggested from the consideration of an effective cluster size lY , while seems rather proportional to h at larger values of the field, as implied by some exact results.
INTRODUCTION A simple cubic lattice of N vertices or sites with periodic boundary conditions, each site having two states, "occupied by a particle" and "unoccupied", with a constant attractive interaction V between nearestneighbor particles, is a familiar realization of the lattice gas model [1,21. The relevant (configurational) energy of the system in the configuration C is defined as E(C) = -V-j) (i,j)1
nin. ,
(1)
V>O.
Here ni is the occupation number of the ith site which, for a given configuration C , takes on the value 1 if site i is occupied and the value 0 if site i is unoccupied, C denotes the subset of the lattice comprising the f N occupied sites, r N=2Lini , and the sum goes over all pairs (i,j) of different sites, each pair being counted once only, which are nearestneighbors in the lattice. There are a total of 2 N possible configurations on the lattice. In a grand canonical ensemble at temperature T and fugacity z the probability of the configuration C is p(C)
= z
Nexp[-E(C) /kT]
ΒΆ
1
(z,T; N) I
(2)
where = C z IN exp (-E/ k T ) is the grand partition function [2]. Any configuration C can be partitioned into subsets Cr called clusters defined as sets of occupied sites connected by bonds. A bond is a pair of nearest-neighbor sites in the lattice. The size ir of a given cluster Cr is defined as the number of (occupied) sites which belong to Cr ; its energy sr is defined as the number of occupied-unoccupied bonds (including both, surface and interior ones) incident on Cr. The average degree of compactness of clusters may be measured, at least partially, in terms of sI, the average value of sr over all clusters of size 1 . This information is to be combined with the knowledge of p1 , the probability for the occurrence of a cluster of size 1 in the system, which is induced by the Gibbs probabilities (2).
sity
The description of the system configuration at temperature T and denf in terms of clusters via the quantities sl and p1 has both
Mat. Res. Soc. Symp. Proc. Vol. 21 (1984)
Elsevier Science Publishing Co., Inc.
14
physical and mathematical interest in the percolation problem, in the theory of metastable states and nucleation processes in a lattice gas or Ising spin system and in many other problems [3-6]. The physical relevance of the concept of clusters comes in part from the fact that they are expected to be related in s
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