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

• Published in: Brazilian journal of probability and statistics 2011-11, Vol.25 (3), p.421-436
• Main Authors: Brillinger, D. R., Preisler, H. K., Wisdom, M. J.
• Format: Article
• Language: English
• Subjects: Approximation; Forest service; Harmonic functions; Least squares; Modeling; Particle interactions; Particle motion; Quantum statistics; Trajectories; Wisdom
• ISSN: 0103-0752; 2317-6199
• DOI: 10.1214/11-BJPS153

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, ǁ ri(t) — rj(t)ǁ 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 $d{r_r} = - \nabla U({r_i})dt - \sum \nabla V({r_i} - {r_j})dt + \sigma d{B_i}$ 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.