Path integrals and non-Markov processes. II. Escape rates and stationary distributions in the weak-noise limit

TY - JOUR

T1 - Path integrals and non-Markov processes. II. Escape rates and stationary distributions in the weak-noise limit

AU - Bray, A. J.

AU - McKane, A. J.

AU - Newman, T. J.

PY - 1990/1

Y1 - 1990/1

N2 - The path-integral formalism developed in the preceding paper [McKane, Luckock, and Bray, Phys. Rev. A 41, 644 (1990)] is used to calculate, in the weak-noise limit, the rate of escape of a particle over a one-dimensional potential barrier, for exponentially correlated noise (t)(t) =(D/)exp{-t-t/}. For small D, a steepest-descent evaluation of the appropriate path integral yields exp(-S/D), where S is the action associated with the dominant (instanton) path. Analytical results for S are obtained for small and large , and (essentially exact) numerical results for intermediate. The stationary joint probability density for the position and velocity of the particle is also calculated for small D: it has the form Pst (x,xI)exp[-S(x,xI)/D]. Results are presented for the marginal probability density Pst(x) for the position of the particle.

AB - The path-integral formalism developed in the preceding paper [McKane, Luckock, and Bray, Phys. Rev. A 41, 644 (1990)] is used to calculate, in the weak-noise limit, the rate of escape of a particle over a one-dimensional potential barrier, for exponentially correlated noise (t)(t) =(D/)exp{-t-t/}. For small D, a steepest-descent evaluation of the appropriate path integral yields exp(-S/D), where S is the action associated with the dominant (instanton) path. Analytical results for S are obtained for small and large , and (essentially exact) numerical results for intermediate. The stationary joint probability density for the position and velocity of the particle is also calculated for small D: it has the form Pst (x,xI)exp[-S(x,xI)/D]. Results are presented for the marginal probability density Pst(x) for the position of the particle.

UR - https://www.scopus.com/pages/publications/25944451974

U2 - 10.1103/PhysRevA.41.657

DO - 10.1103/PhysRevA.41.657

M3 - Article

AN - SCOPUS:25944451974

SN - 1050-2947

VL - 41

SP - 657

EP - 667

JO - Physical Review A

JF - Physical Review A

IS - 2

ER -