Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- Introduction to Volume 1
- 1 Noise-activated escape from metastable states: an historical view
- 2 Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems
- 3 Langevin equations with colored noise
- 4 First passage time problems for non-Markovian processes
- 5 The projection approach to the Fokker–Planck equation: applications to phenomenological stochastic equations with colored noises
- 6 Methods for solving Fokker–Planck equations with applications to bistable and periodic potentials
- 7 Macroscopic potentials, bifurcations and noise in dissipative systems
- 8 Transition phenomena in multidimensional systems – models of evolution
- 9 Colored noise in continuous dynamical systems: a functional calculus approach
- Appendix: On the statistical treatment of dynamical systems
- Index
9 - Colored noise in continuous dynamical systems: a functional calculus approach
Published online by Cambridge University Press: 05 January 2012
- Frontmatter
- Contents
- List of contributors
- Preface
- Introduction to Volume 1
- 1 Noise-activated escape from metastable states: an historical view
- 2 Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems
- 3 Langevin equations with colored noise
- 4 First passage time problems for non-Markovian processes
- 5 The projection approach to the Fokker–Planck equation: applications to phenomenological stochastic equations with colored noises
- 6 Methods for solving Fokker–Planck equations with applications to bistable and periodic potentials
- 7 Macroscopic potentials, bifurcations and noise in dissipative systems
- 8 Transition phenomena in multidimensional systems – models of evolution
- 9 Colored noise in continuous dynamical systems: a functional calculus approach
- Appendix: On the statistical treatment of dynamical systems
- Index
Summary
Introduction
Recent work on dye lasers (Fox and Roy, 1987; Jung and Risken, 1984; Lett, Short and Mandel, 1984; Roy, Yu and Zhu, 1985; Short, Mandel and Roy, 1982) and the optical ring laser gyroscope (Vogel et al., 1987a, b) has emphasized the physically important role of colored noise sources. A well-known classical situation in which strongly colored noise has an impact on the physics is the phenomenon of motional narrowing in magnetic resonance (Kubo, 1962). Kubo has shown that a fluctuating magnetic field with very short noise correlation time (almost white noise) does typically not manifestly affect the motion of spins; on the contrary, if the fluctuations are correlated over a long time scale (colored noise) the motion of the spin becomes greatly modified.
Another area where there has been much recent activity addresses escape problems. These are currently in the limelight both from the theoretical viewpoint (Grote and Hynes,1980; Hänggi, 1986; Hänggi and Mojtabai, 1982; Hänggi and Riseborough, 1983; Hänggi, Mroczkowski, Moss and McClintock, 1985) as well as from an experimental point of view (Devoret, Martinis, Esteve and Clarke, 1984; Fleming, Courtney and Balk, 1986; Hänggi et al., 1985; Maneke, Schroeder, Troe and Voss, 1985). In this latter case, a frequency-dependent friction, or noise of finite correlation time, can considerably modify the classical barrier transmission. Except for two-state noise (Hänggi and Talkner, 1985; Masoliver, Lindenberg and West, 1986; Rodriguez and Pesquera, 1986; Van den Broeck and Hänggi, 1984) there exist no exact analytic methods for truly nonlinear systems, being driven by correlated noise.
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- Noise in Nonlinear Dynamical Systems , pp. 307 - 328Publisher: Cambridge University PressPrint publication year: 1989
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