Semiclassical theory of underdamped nonlinear Brownian processes
In the limit of weak noise, the conditional probabilities of damped, noise-driven anharmonic oscillator can be expressed (at least formally) in terms of solutions to a nonlinear ordinary differential equation (''Onsager--Machlup'' equation) of fourth order in the time derivative. In the case of a one-dimensional oscillator, this equation is recast in the form of a mapping; and in the limit of long time and weak time-independent friction, it is shown how to reduce the iteration of this map to quadratures. This procedure is checked by using it to reproduce a number of standard results usually derived in other ways. In connection with the statistical mechanics, of electron--positron colliding-beam storage rings, and attempt is also made to apply this mapping formalism to the case of weak, periodically time-dependent friction. When no frequency in the Fourier decomposition of the friction coefficient is close to an (even, for symmetric potentials) integral multiple of the (amplitude-dependent) frequency of the oscillator, then results can be derived that parallel those obtained in the case of constant friction. Otherwise (''thermal resonance'') the situation is not clear.
- Research Organization:
- Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois 60510
- OSTI ID:
- 5668603
- Journal Information:
- Ann. Phys. (N.Y.); (United States), Vol. 160:1
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANHARMONIC OSCILLATORS
SEMICLASSICAL APPROXIMATION
STORAGE RINGS
STATISTICAL MECHANICS
BROWNIAN MOVEMENT
FOKKER-PLANCK EQUATION
FRICTION
HAMILTONIANS
MAPPING
NONLINEAR PROBLEMS
ONE-DIMENSIONAL CALCULATIONS
TIME DEPENDENCE
DIFFERENTIAL EQUATIONS
ELECTRONIC EQUIPMENT
EQUATIONS
EQUIPMENT
MATHEMATICAL OPERATORS
MECHANICS
OSCILLATORS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM OPERATORS
430400* - Particle Accelerators- Storage Rings
657006 - Theoretical Physics- Statistical Physics & Thermodynamics- (-1987)