A mechanical model of a particle immersed in a heat bath is studied, in which a distinguished particle interacts via linear springs with a collection of n particles with variable masses and random initial conditions; the jth particle oscillates with frequency jp, where p is a parameter. For p > 1/2 the sequence of random processes that describe the trajectory of the distinguished particle tends almost surely, as n → ∞, to the solution of an integro-differential equation with a random driving term; the mean convergence rate is 1/np-1/2. We further investigate whether the motion of the distinguished particle can be well approximated by an integration scheme - the symplectic Euler scheme - when the product of time step h and highest frequency np is of order 1, that is, when high frequencies are underresolved. For 1/2 < p < 1 the numerical solution is found to converge to the exact solution at a reduced rate of |log h|h 2-1/p. These results shed light on existing numerical data.
Bibliographical noteFunding Information:
We thank Profs. A. Chorin, A. Majda, Z. Schuss, A. Stuart, and B. Weiss for helpful discussions. This work was supported in part by the Director, Office of Science, Office of Advanced Scientific Computing Research, Mathematical, Information, and Computational Sciences Division, U.S. Department of Energy under Contract DE-AC03-76SF00098. RK was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, and by the Alon Fellowship.
- Generalized Langevin equation
- Heat bath
- Order reduction
- Stiff oscillatory systems
- Symplectic Euler scheme
- Volterra equation