Browsing by Author "Tupper, P.F."

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Constructing stationary Gaussian processes from deterministic processes with random initial conditions
    (2002-06) Tupper, P.F.
    We consider a family of stationary Gaussian processes that includes the stationary Ornstein-Uhlenbeck process. We show that processes in this family can be attained as the limit of a sequence of deterministic processes with random initial conditions. Weak convergence in the supremum norm on finite time-intervals is shown. We also establish the convergence of a wide variety of long-term statistics. Our construction provides a rigorous example of how macroscopic stochastic dynamics can be derived from microscopic deterministic dynamics.
  • Thumbnail Image
    Item
    The motion of a tracer particle in a one-dimensional system: Analysis and simulation
    (2002-07) Tupper, P.F.
    Our goal is to obtain a test system for the evaluation of time-stepping methods in molecular dynamics. We consider a family of deterministic systems consisting of a finite number of particles interacting on a compact interval. The particles are given random initial conditions and interact through instantaneous energy- and momentum-conserving collisions. As the number of particles, the particle density, and the mean particle speed go to infinity, the trajectory of a tracer particle is shown to converge to a stationary Gaussian process. We simulate the system with two numerical methods, one symplectic, the other energy-conserving, and assess the methods' ability to recapture the system's limiting statistics.