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    Author(s): D. R. Brillinger; Haiganoush Preisler; M. J. Wisdom
    Date: 2011
    Source: Brazilian Journal of Probability and Statistics
    Publication Series: Scientific Journal (JRNL)
    Station: Pacific Southwest Research Station
    PDF: Download Publication  (2.0 MB)


    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 − V (rirj)dt + σ dB
    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.

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    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.


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    Elk, gradient system, particle process, potential function

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