Projection approach to the Fokker-Planck equation. I. Colored Gaussian noise

Journal Article · Mon Aug 01 04:00:00 EDT 1988 · J. Stat. Phys.; (United States)

DOI:https://doi.org/10.1007/BF01019735· OSTI ID:6059733

Faetti, S; Fronzoni, L; Grigolini, P; Mannella, R

It is shown that the Fokker-Planck operator can be derived via a projection-perturbation approach, using the repartition of a more detailed operator L into a perturbation L/sub 1/ and an unperturbed part L/sub 0/. The standard Fokker-Planck structure is recovered at the second order in L/sub 1/, whereas the perturbation terms of higher order are shown to provoke the breakdown of this structure. To get rid of these higher order terms, a key approximation, local linearization (LL), is made. In general, to evaluate at the second order in L/sub 1/ the exact expression of the diffusion coefficient which simulates the influence of a Gaussian noise with a finite correlation time /tau/, a resummation up to infinite order in /tau/ must be carried out, leading to what other authors call the best Fokker-Planck approximation (BFPA). It is shown that, due to the role of terms of higher order in L/sub 1/, the BFPA leads to predictions on the equilibrium distributions that are reliable only up to the first order in /tau/. The LL, on the contrary, in addition to making the influence of terms of higher order in L/sub 1/ vanish, results in a simple analytical expression for the term of second order that is formally coincident with the complete resummation over all the orders in /tau/ provided by the Fox theory. The corresponding diffusion coefficient in turn is shown to lead in the limiting case /tau/ ..-->.. infinity to exact results for the steady-state distributions. Therefore, over the whole range 0 less than or equal to /tau/ less than or equal to infinity the LL turns out to be an approximation much more accurate than the global linearization proposed by other authors for the same purpose of making the terms of higher order in L/sub 1/ vanish. In the short-/tau/ region the LL leads to results virtually coincident with those of the BFPA. In the large-/tau/ region the LL is a more accurate approximation than the BFPA itself.

• Research Organization: Universita' di Pisa and Gruppo Nazionale di Struttura della Materia del CNR (France)
• OSTI ID: 6059733
• Journal Information: J. Stat. Phys.; (United States), Journal Name: J. Stat. Phys.; (United States) Vol. 52:3-4; ISSN JSTPB
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGORITHMS
ANALOG COMPUTERS
COMPARATIVE EVALUATIONS
COMPUTERIZED SIMULATION
COMPUTERS
DIFFERENTIAL EQUATIONS
DIFFUSION
DIGITAL COMPUTERS
EQUATIONS
FOKKER-PLANCK EQUATION
GAUSSIAN PROCESSES
MATHEMATICAL LOGIC
MATHEMATICAL OPERATORS
MECHANICS
NOISE
PARTIAL DIFFERENTIAL EQUATIONS
PERTURBATION THEORY
PROJECTION OPERATORS
SERIES EXPANSION
SIMULATION
STATISTICAL MECHANICS