Brazilian Journal of Probability and Statistics

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

D. R. Brillinger, H. K. Preisler, and M. J. Wisdom

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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, ri(t) at times ti. A second, V, depends on particle i’s vector distances from others, ri(t)−rj(t). This function introduces pairwise interactions. A third, W, depends on the Euclidian distances, ‖ri(t)−rj(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

dri=−∇U(ri) dt−∑jiV(rirj) dt+σdBi

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.

Article information

Braz. J. Probab. Stat., Volume 25, Number 3 (2011), 421-436.

First available in Project Euclid: 22 August 2011

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Zentralblatt MATH identifier

Elk gradient system particle process potential function


Brillinger, D. R.; Preisler, H. K.; Wisdom, M. J. Modelling particles moving in a potential field with pairwise interactions and an application. Braz. J. Probab. Stat. 25 (2011), no. 3, 421--436. doi:10.1214/11-BJPS153.

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