The Ornstein-Uhlenbeck equation as a limiting case of a successive interactions model

Daniel M. Packwood

    Research output: Contribution to journalArticlepeer-review

    3 Citations (Scopus)

    Abstract

    A model for the motion of a test particle through a medium composed of many other particles is presented. It supposes that the test particle interacts with another particle up to a random time U1, and then another particle up to a random time U2, and so on. The test particle-particle interactions end when either another particle interrupts the interaction or the impulse of the force due to the current interaction exceeds a given bound. The 'random walk on polynomials' process is introduced in this paper and is used to model the total impulse of the force arising from these interactions. It is shown that as the duration of these interactions approaches zero, the velocity of the test particle becomes a solution of an Ornstein-Uhlenbeck equation. The renormalization required for this convergence shows that the limiting dynamics involve interactions of short duration and relatively large forces, as well as a relatively massive test particle.

    Original languageEnglish
    Article number465001
    JournalJournal of Physics A: Mathematical and Theoretical
    Volume43
    Issue number46
    DOIs
    Publication statusPublished - 2010 Nov 19

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Statistics and Probability
    • Modelling and Simulation
    • Mathematical Physics
    • Physics and Astronomy(all)

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