Introduction to Volume 2

Summary

While in Volume 1 the treatments are confined exclusively to Fokker–Planck systems, a range of contemporary problems, indicative of the rich diversity of applications of noise driven dynamics, are reviewed in this volume. Though most problems currently treated are classical, recent work on dissipative quantum tunnelling has focused attention on quantum mechanical applications of stochastic dynamics. Chapter 1 builds on the theory of the Anderson–Kubo oscillator, generalized to account for colored noise effects, with applications to molecular spectroscopy. Helpful comparisons with the classical theory of Brownian motion are drawn. Chapter 2 reviews the important and highly topical problem of the crossover from classical Kramers-type thermal activation to the quantum tunnelling of particles out of a metastable state. Having set the stage with a discussion of classical Brownian motion in a periodic potential, the theory is applied to the diffusion of an impurity hopping among interstitial sites in a crystal lattice. An interesting non-Markovian application follows wherein the impurity is driven by a random force weighted around a characteristic frequency. Enhanced activation rates are observed when the noise correlation becomes comparable to the inverse characteristic frequency. This review concludes with a discussion of the formidable problem of quantum tunnelling in the low damping regime. Even classical treatments of this topic are fraught with difficulties, as the following review in Chapter 3 demonstrates. Building on Kramers' original treatment, the escape rate of particles from a metastable state is discussed in the limits of low and moderately high damping.