9 research outputs found
The Use of Hamiltonian Mechanics in Systems Driven by Colored Noise
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.Comment: 25 pages with 5 encapsulted postscript figures appended (need epsf
Surmounting Oscillating Barriers
Thermally activated escape over a potential barrier in the presence of
periodic driving is considered. By means of novel time-dependent path-integral
methods we derive asymptotically exact weak-noise expressions for both the
instantaneous and the time-averaged escape rate. The agreement with accurate
numerical results is excellent over a wide range of driving strengths and
driving frequencies.Comment: 4 pages, 4 figure
Ratchet driven by quasimonochromatic noise.
The currents generated by noise-induced activation processes in a periodic potential are investigated analytically, by digital simulation and by performing analog experiments. The noise is taken to be quasimonochromatic and the potential to be a smoothed sawtooth. Two analytic approaches are studied. The first involves a perturbative expansion in inverse powers of the frequency characterizing quasimonochromatic noise and the second is a direct numerical integration of the deterministic differential equations obtained in the limit of weak noise. These results, together with the digital and analog experiments, show that the system does indeed give rise, in general, to a net transport of particles. All techniques also show that a current reversal exists for a particular value of the noise parameters
Experiments on critical phenomena in a noisy exit problem.
We consider a noise-driven exit from a domain of attraction in a two-dimensional bistable system lacking detailed balance. Through analog and digital stochastic simulations, we find a theoretically predicted bifurcation of the most probable exit path as the parameters of the system are changed, and a corresponding nonanalyticity of the generalized activation energy. We also investigate the extent to which the bifurcation is related to the local breaking of time-reversal invariance
Escape Rates in Bistable Systems Induced by Quasi-Monochromatic Noise
Path integral techniques are used to understand the behaviour of a particle
moving in a bistable potential well and acted upon by quasi-monochromatic
external noise. In the limit of small diffusion coefficient, a steepest descent
evaluation of the path integral enables mean first passage times and the
transition times from one well to the other to be computed. The results and
general approach are compared with computer simulations of the process. It is
found that the bandwidth parameter, , has a critical value above which
particle escape is by white-noise-like outbursts, but below which escape is by
oscillatory type behaviour.Comment: 20 pages (LaTex) + 4 figures (encapsulted postscript) upon reques