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 − ∑ ∇ V (r_i − r_j)dt + σ dB_i                             j≠1 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.

Citation

Brillinger, D. R.; Preisler, H. K.; Wisdom, M. J. 2011. Modelling particles moving in a potential field with pairwise interactions and an application. Brazilian Journal of Probability and Statistics. 25(3): 421-436.