17 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
Analytical and numerical investigation of escape rate for a noise driven bath
We consider a system-reservoir model where the reservoir is modulated by an
external noise. Both the internal noise of the reservoir and the external noise
are stationary, Gaussian and are characterized by arbitrary decaying
correlation functions. Based on a relation between the dissipation of the
system and the response function of the reservoir driven by external noise we
numerically examine the model using a full bistable potential to show that one
can recover the turn-over features of the usual Kramers' dynamics when the
external noise modulates the reservoir rather than the system directly. We
derive the generalized Kramers' rate for this nonequilibrium open system. The
theoretical results are verified by numerical simulation.Comment: Revtex, 25 pages, 5 figures. To appear in Phys. Rev.
The large deviation approach to statistical mechanics
The theory of large deviations is concerned with the exponential decay of
probabilities of large fluctuations in random systems. These probabilities are
important in many fields of study, including statistics, finance, and
engineering, as they often yield valuable information about the large
fluctuations of a random system around its most probable state or trajectory.
In the context of equilibrium statistical mechanics, the theory of large
deviations provides exponential-order estimates of probabilities that refine
and generalize Einstein's theory of fluctuations. This review explores this and
other connections between large deviation theory and statistical mechanics, in
an effort to show that the mathematical language of statistical mechanics is
the language of large deviation theory. The first part of the review presents
the basics of large deviation theory, and works out many of its classical
applications related to sums of random variables and Markov processes. The
second part goes through many problems and results of statistical mechanics,
and shows how these can be formulated and derived within the context of large
deviation theory. The problems and results treated cover a wide range of
physical systems, including equilibrium many-particle systems, noise-perturbed
dynamics, nonequilibrium systems, as well as multifractals, disordered systems,
and chaotic systems. This review also covers many fundamental aspects of
statistical mechanics, such as the derivation of variational principles
characterizing equilibrium and nonequilibrium states, the breaking of the
Legendre transform for nonconcave entropies, and the characterization of
nonequilibrium fluctuations through fluctuation relations.Comment: v1: 89 pages, 18 figures, pdflatex. v2: 95 pages, 20 figures, text,
figures and appendices added, many references cut, close to published versio
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
Optimum Paths for Systems Subject to Internal Noise
We formulate the stochastic dynamics of a particle subject to internal
non-white (coloured) noise in terms of path-integrals. In the simplest case,
where the noise is exponentially correlated, the weak-noise limit is
characterised by optimum paths which are given by third order differential
equations. In contrast to systems subject to white noise or external coloured
noise, the overdamped limit for these systems is singular. We analyse the
origin of this behaviour. The whole formalism is generalised to more general
noise processes and the essential features are shown to be similar to the
exponentially correlated case.Comment: 25 pages with 2 encapsulated postscript figures appended (need epsf