Discrete dynamics and metastability: mean first passage times and escape rates

Journal Article · Wed Jul 01 04:00:00 EDT 1987 · J. Stat. Phys.; (United States)

DOI:https://doi.org/10.1007/BF01010408· OSTI ID:5372779

Talkner, P; Haenggi, P; Freidkin, E; Trautmann, D

The problem of escape from a domain of attraction is applied to the case of discrete dynamical systems possessing stable and unstable fixed points. In the presence of noise, the otherwise stable fixed point of a nonlinear map becomes metastable, due to the noise-induced hopping events, which eventually pass the unstable fixed point. Exact integral equations for the moments of the first passage time variable are derived, as well as an upper bound for the first moment. In the limit of weak noise, the integral equation for the first moment, i.e., the mean first passage time (MFPT), is treated, both numerically and analytically. The exponential leading part of the MFPT is given by the ratio of the noise-induced invariant probability at the stable fixed point and unstable fixed point, respectively. The evaluation of the prefactor is more subtle: It is characterized by a jump at the exit boundaries, which is the result of a discontinuous boundary layer function obeying an inhomogeneous integral equation. The jump at the boundary is shown to be always less than one-half of the maximum value of the MFPT. On the basis of a clear-cut separation of time scales, the MFPT is related to the escape rate to leave the domain of attraction and other transport coefficients, such as the diffusion coefficient. Alternatively, the rate can also be obtained if one evaluates the current-carrying flux that results if particles are continuously injected into the domain of attraction and captured beyond the exit boundaries. The two methods are shown to yield identical results for the escape rate of the weak noise result for the MFPT, respectively. As a byproduct of this study, they obtain general analytic expressions for the invariant probability of noisy maps with a small amount of nonlinearity.

• Research Organization: Polytechnic Institute of New York, Brooklyn (USA)
• OSTI ID: 5372779
• Journal Information: J. Stat. Phys.; (United States), Journal Name: J. Stat. Phys.; (United States) Vol. 48:1/2; ISSN JSTPB
• Country of Publication: United States
• Language: English

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71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
CAPTURE
CHARGE CARRIERS
CRYSTAL MODELS
CRYSTALS
CURRENTS
DIFFUSION
DYNAMICS
ELECTRIC CURRENTS
ENERGY LEVELS
EQUATIONS
EXCITED STATES
INTEGRAL EQUATIONS
MAPPING
MATHEMATICAL MANIFOLDS
MATHEMATICAL MODELS
MECHANICS
METASTABLE STATES
NOISE
NONLINEAR PROBLEMS
STATISTICAL MECHANICS
TOPOLOGICAL MAPPING
TRANSFORMATIONS
TRANSPORT THEORY