Brazilian Journal of Probability and Statistics
- Braz. J. Probab. Stat.
- Volume 25, Number 3 (2011), 421-436.
Modelling particles moving in a potential field with pairwise interactions and an application
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−∑j≠i∇V(ri−rj) dt+σ dBi
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.
Braz. J. Probab. Stat., Volume 25, Number 3 (2011), 421-436.
First available in Project Euclid: 22 August 2011
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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. https://projecteuclid.org/euclid.bjps/1313973402