Fokker-Planck equation reliability

Chapter II addresses another fundamental problem under what physical conditions can the Fokker-Planck equation provide a reliable picture of fluctuation-dissipation processes Aware as they are of the technical and conceptual difidculties involved in nonlinear statistics, the authors share Zwanzig s optimistic view that use of the Fokker-Planck equation is practicable and advantageous. Chapter II contains a brief description of rules to construct, via a suitable procedure, equations of Fokker-Planck type for the slow variables of the system under study. This theoretical method eliminates explicit analytic dependence on fast variables and thereby produces a significant simplification of the problem under discussion. 

The numerical solution of both the fractional Fokker-Planck equation in terms of the Griinwald-Letnikov scheme used to find a discretized approximation to the fractional Riesz operator exhibits reliable convergence, as corroborated by direct solution of the corresponding Langevin equation. 

The upward-directed electric field accelerates the ambient thermal energy electrons of mean energy = 1.5 kT to a new distribution fimction that depends upon the local E field and neutral composition and density. The connection between the spatial E field and the electron energy distribution function is made through solution of either the Boltzmann equation (Pitchford et al., 1981 Phelps and Pitchford, 1985) or the derived Fokker-Planck equation (Milikh et al., 1998a). In either case, a full database of cross sections for electron-molecule (N2, O2) excitation, ionization (both direct and dissociative), and attachment (for O2) is needed for reliable solutions. Electron-ion and electron-atom (N, O) scattering are usually neglected because of the small product of electron and ion or atom densities. 

The AEP should allow us to derive a reliable Fokker-Planck type of equation for set a alone. The solution of this type of equation still involves formidable problems. In the monodimensional case (i.e., when a consists of one variable x), the AEP usually results in the following type of equation ... 

ABSTRACT The paper presents a probabilistic method to assess lifetimes of devices/components that operate under conditions typical of ageing processes. It has been assumed that the random rate of the component s wear is of the form taken by the failure rate function for the Weibull distribution, or approximately follows the linear pattern. From the point of view of mathematics, it has been based on the difference equation that, after some rearrangements, results in a partial differential equation of the Fokker-Planck type. From the particular solution to this equation one gets density function of the wear-and-tear in the form of normal distribution. Having found the density function of the wear-and-tear, one can formulate a relationship for reliability for the assumed permissible value of the wear-and-tear. With the normal distribution normalized and the required level of reliabUity reached, one can then compute the lifetime of a device or component under consideration. 

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