Modelling particles moving in a potential field with pairwise interactions and an application

Abstract

Motions of particles in fields characterized by real-valued potential functions, are considered. Three particular expressions for potential functions are studied. One, U, depends on the ith particle’s location, r_i(t) at times t_i. A second, V, depends on particle i’s vector distances from others, r_i(t)−r_j(t). This function introduces pairwise interactions. A third, W, depends on the Euclidian distances, ‖r_i(t)−r_j(t)‖ between particles at the same times, t. The functions are motivated by classical mechanics.

Taking the gradient of the potential function, and adding a Brownian term one, obtains the stochastic equation of motion

dr_i=−∇U(r_i) dt−∑_j≠i∇V(r_i−r_j) dt+σ dB_i

in the case that there are additive components U and V. The ∇ denotes the gradient operator. Under conditions the process will be Markov and a diffusion. By estimating U and V at the same time one could address the question of whether both components have an effect and, if yes, how, and in the case of a single particle, one can ask is the motion purely random?

An empirical example is presented based on data describing the motion of elk (Cervus elaphus) in a United States Forest Service reserve.