Complex Fokker-Planck equation

Thus we have demonstrated how the empirical Havriliak-Negami equation [Eq. (11)] can be obtained from a microscopic model, namely, the fractional Fokker-Planck equation [Eq. (101)] applied to noninteracting rotators. This model can explain the anomalous relaxation of complex dipolar systems, where the anomalous exponents ct and v differ from unity (corresponding to the classical Debye theory of dielectric relaxation) that is, the relaxation process is characterized by a broad distribution of relaxation times. Hence, the empirical Havriliak-Negami equation of anomalous dielectric relaxation which has been... 

We perform concrete calculations in the complex P-representation [Drummond 1980 McNeil 1983] in the frame of both probability distribution functions and stochastic equations for the complex c-number variables. We follow the standard procedures of quantum optics to eliminate the reservoir operators and to obtain a master equation for the density operator of the modes. The master equation is then transformed into a Fokker-Planck equation for the P-quasiprobability distribution function. In particular, for an ordinary NOPO and in the case of high cavity losses for the pump mode (73 7), if in the operational regime the pump depletion effects are involved, this approach yields... 

In Section I.B we discuss how to devise a general MFPKE to describe complex liquids. A three-body model will be presented as a description of a system in which at least two significant additional sets of solvent degrees of freedom are introduced. In Section I.C we show the relation between some of the previously cited approaches and particular cases of our model. In particular, augmented Fokker-Planck equations (AFPE) of Stillman and Freed are seen to be directly related to the MFPK formal-... 

In Sect. 3, the Noyes approach to analysing reaction rates based on the molecular pair approach is discussed [5]. Both this and the diffusion equation analysis are identical under conditions where the diffusion equation is valid and when the appropriate recombination reaction rate for a molecular pair is based on the diffusion equation. Some comments by Naqvi et al. [38] and Stevens [455] have obscured this identity. The diffusion equation is a valid approximation to molecular motion when the details of motion in a cage are no longer of importance. This time is typically a few picoseconds in a mobile liquid. When extrapolating the diffusion equation back to such times, it should be recalled that the diffusion is a continuum form of random walk [271]. While random walks can be described with both a distribution of jump frequencies and distances, nevertheless, the diffusion equation would not describe a random walk satisfactorily over times less than about five jump periods (typically 10 ps in mobile liquids). Even with a distribution of jump distances and frequencies, the random walk model of molecular motion does not represent such motion adequately well as these times (nor will the telegrapher s or Fokker-Planck equation be much better). It is therefore inappropriate to compare either the diffusion equation or random walk analysis with that of the molecular pair over such times. Finally, because of the inherent complexity of molecular motion, it is doubtful whether it can be described adequately in terms of average jump distances and frequencies. These jump characteristics are only operational terms for very complex quantities which derive from the detailed molecular motion of the liquid. For this very reason, the identification of the diffusion coefficient with a specific jump formula (e.g. D = has been avoided. 

The probability density flow (5.260), or the one of conditioned probability density (5.252), stay on the basis of analytically representation of the no equilibrium dynamic for the electro-reactive chains in a temporary scale which cover, but also it overcome the one of the activated chemical complex. For the complexity of this study, at least from the perspective of the involving the path integrals in the dynamic of the respective equilibrium and no equilibrium, becomes extremely instmctive the solution of Fokker-Planck equation in the conditioned probability density form (5.250), from where, the calculation of the probability density as well as of the associated currents are immediate. 

By using the same cmicepts, a very large niun-ber of other problems may be solved. Such an example the probability density function of a random variable may be obtained with the same technique here used for representing cross-correlations in terms of FSMs. It follows that Fokker-Planck equation, Kolmogorov-Feller equation, Einstein-Smoluchowski equation, and path integral solution (Cottone et al. 2008) may be solved in terms of FSM. Moreover, wavelet transform and classical or fractional differential equations may be easily solved by using fractional calculus and Mellin transform in complex domain. 

Di Matteo A, Di Paola M, Pirrotta A (2014) Probabilistic characterization of nonlinear systems under Poisson white noise via complex fractional moments. Nonlinear Dyn. doi 10.10071sl 1071-014-1333-1 Di Paola M (2014) Fokker Planck equation solved in terms of complex fractional moments. Probab Eng Mech 38 70-76... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

In the first section we have discussed a general methodology for the theoretical description of rotational dynamics of rigid solute molecules in complex solvents. Many-body Fokker-Planck-Kramers equations (MFPKE), including collective solvent degrees of freedom (either rotational ones, i.e., rigid bodies, or translational ones, i.e., vector fields), and their conjugate momenta, have been described as convenient tools to reproduce (or simulate) the complexity of an actual liquid system. 

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