The Use of Hamiltonian Mechanics in Systems Driven by Colored Noise

Abstract

The evaluation of the path-integral representation for stochastic processes in the weak-noise limit shows that these systems are governed by a set of equations which are those of a classical dynamics. We show that, even when the noise is colored, these may be put into a Hamiltonian form which leads to better insights and improved numerical treatments. We concentrate on solving Hamilton's equations over an infinite time interval, in order to determine the leading order contribution to the mean escape time for a bistable potential. The paths may be oscillatory and inherently unstable, in which case one must use a multiple shooting numerical technique over a truncated time period in order to calculate the infinite time optimal paths to a given accuracy. We look at two systems in some detail: the underdamped Langevin equation driven by external exponentially correlated noise and the overdamped Langevin equation driven by external quasi-monochromatic noise. We deduce that caustics, focusing and bifurcation of the optimal path are general features of all but the simplest stochastic processes.

25 pages with 5 encapsulted postscript figures appended (need epsf)

References in corpus (3)

• The Escape Problem for Irreversible Systems
• Observable and hidden singular features of large fluctuations in nonequilibrium systems
• The Effect of Focusing and Caustics on Exit Phenomena in Systems Lacking Detailed Balance

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• Most Probable Dynamics of Stochastic Dynamical Systems with Exponentially Light Jump Fluctuations
• Escape rate of metastable states in a driven NbN superconducting microwave resonator
• A ratchet driven by quasimonochromatic noise
• Investigation on the properties of Sine-Wiener noise and its induced escape in the particular limit case $D\to \infty$