A stochastic particles system is truly fully characterized only when the infinite hierarchy of multiple-point density correlation functions - the probability of finding any given number of particles at some specified locations, simultaneously -is known. The IPDF method is capable of handling this complicated question, and, in fact, in several cases the complete exact solution may be thus obtained. Such studies reveal a peculiar property of “shielding”, particular to reversible coalescence, whereby a particle at the edge of the system seems to shield the rest of the particles from the imposed boundary conditions.

We begin with an analysis of inhomogeneous systems, when translational symmetry is broken, at the simple level of point densities (i.e., the particle concentration). An interesting application is to the study of Fisher waves, and the effect of internal fluctuations on this well-known mean-field model for invasion of an unstable phase by a stable phase.

Inhomogeneous initial conditions

Until now we have discussed only translationally symmetric systems. The method of interparticle distribution functions can be generalized to inhomogeneous situations (Doering et al, 1991). To this end, En(t) need simply be replaced by En,m(t) – the probability that the sites n, n + 1,…, m are empty at time t.