An interacting particle model with compact hierarchical structure
In the case of solutions for some problems in statistical physics and population biology we have the situation where there is a large number interacting individuals which are spatially distributed. Such models are described by a deterministic master equation, however, usually when the spatially localized interactions between individuals appear they do not lead to a closed hierarchy of reduced distribution functions (by analogy like in the case of BBGKY hierarchy) -- the evolution equation for the single particle-density function involves the pair correlation function, and the evolution equation for the pair correlation function involves the triplet function, and so on. This complicates models of biological populations. The authors of this paper formulate a model that has no such a drawback. Their model of an interacting particle system with a population of fixed size allows particles wandering randomly in space and pairs can interact at a rate determined by a reaction kernel with finite range. An assembly of \(N\) organisms is moving through a continuous space and time via diffusion. When one organism kills another a new individual is instantly born but the birth-death interaction occurs only between pairs that are sufficiently close. The proposed model is similar to the other, so far presented in the literature, however this one assumes that the interaction encoded by \(v(r)\) has a finite range \(a\) whereas in other models there is \(a\rightarrow\infty\). The authors were able to show that the basic statistical properties of such a model behavior can be found analytically. The paper consists of six sections and two appendixes. In Section 2 the reader can find a mathematical formulation of the problem which is complemented in Section 3 where there are the considerations about one-dimensional dynamics in periodic domains starting with the problem of the initial-value. Considerations are based on event-driven Monte Carlo simulations basing on the algorithm of Gillespie. In the next section we have the expansion of the so far presented approach -- cluster dynamics is considered in the case of infinite domains. In Section 5 the reader can find a description of the cluster interaction. Its solution is given basing on Laplace-transform solution and also by appropriate simulations. The paper is crowned in Section 6. In the appendixes we can find details of the numerical algorithm that was used and also the analytical solutions (by Laplace transform) for single-cluster problem in Section 4 and for two-cluster solution in Section 5.
scientific article; zbMATH DE number 5843257