Fokker-Planck equation operator

With the Laplace operator V. The diffusion coefficient defined in Eq. (62) has the dimension [cm /s]. (For correct derivation of the Fokker-Planck equation see [89].) If atoms are initially placed at one side of the box, they spread as ( x ) t, which follows from (62) or from (63). 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

It is also described by a Fokker-Planck equation defined by the evolution operator,... 

The Fokker-Planck equation is a master equation in which W is a differential operator of second order... 

The traditional derivation of the Fokker-Planck equation (1.5) or (VIII. 1.1) is based on Kolmogorov s mathematical proof, which assumes infinitely many infinitely small jumps. In nature, however, all jumps are of some finite size. Consequently W is never a differential operator, but always of the type (V.1.1). Usually it also has a suitable expansion parameter and has the canonical form (X.2.3). If it then happens that (1.1) holds, the expansion leads to the nonlinear Fokker-Planck equation (1.5) as the lowest approximation. There is no justification for attributing a more fundamental meaning to Fokker-Planck and Langevin equations than in this approximate sense. 

In the limit a —> 1, the Riemann-Liouville fractional integral oD a reduces to an ordinary integration so that lim i oL>,1- = JjJodr becomes the identity operator that is, Eqs. (15) and (19) simplify to the standard diffusion and Fokker-Planck equations, respectively. 

It is easily seen by inspection that the biorthogonal basis set definition (3.55) cmnddes with the definifion (3.18) ven in the discussion of the Lanczos method. We recall that the dynamics of operators (liouville equations) or probabilities (Fokker-Planck equations) have a mathematical structure similar to Eq. (3.29) and can thus be treated with the same techniques (see, e.g., Chapter 1) once an appropriate generalization of a scalar product is performed. For instance, this same formalism has been successfully adopted to model phonon thermal baths and to include, in principle, anharmonicity effects in the interesting aspects of lattice dynamics and atom-solid collisions. ... 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

In order to generalize the normal Fokker-Planck equation excluding inertial effects to fractional diffusion, we first recall the general form of that equation for normal diffusion in operator representation [49]... 

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. 

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 principle, the presence of slow stochastic torques directly affecting the solute reorientational motion can be dealt with in the framework of generalized stochastic Fokker-Planck equations including frequency-dependent frictional terms. However, the non-Markovian nature of the time evolution operator does not allow an easy treatment of this kind of model. Also, it may be difficult to justify the choice of frequency dependent terms on the basis of a sound physical model. One would like to take advantage of some knowledge of the physical system under... 

The above Fokker-Planck equation can be solved by linear algebraic methods, after making recourse to the matrix representation of the operator acting on the distribution function on the right hand side of eq. (23). For that purpose we made use of the same basis set as for solving the Schrodinger equation for the Hamiltonian in eq.(7). Due to the fact that all the operators (as well as the initial distribution) are even with respect to the inversion

initial distribution Po(q>) = P(< >,t = 0) should be well represented and ii) the results of the... 

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 Fokker-Planck equation is automatically obtained by cutting the expansion of the KM operator (5.263) in the second order (Pawula, 1967)... 

For revealing the adjoint Fokker-Planck equation and the associate operator one starts from the CKS equation (5.249) by taking the derivative of this equation respecting and with the direct KM equation (5.267) there results ... 

In case when the KM operator is limited to the adjoint FP operator, this one, on its turn, can be rewritten in function of the stationary solution (5.269) of the direct Fokker-Planck equation ... 

In order to obtain the transformation of Fokker-Planck equation into Schrodinger equation one starts employing the autoadjoint operator introduced in Eq. (5.277), which along (5.276) becomes ... 

Whit these quantities one can reveal the essential difference of the Fokker-Planck equation towards the Schrodinger one for the same type of external potential (harmonic, in here). Firstly, worth remarking the fact that, when it is consider eigen-value equation for the Hamiltonian operator specific to Schrodinger equation... 

From now on the eigen-function solutions of the Fokker-Planck equation in the Lfp operator immediately results through the relations (5.281) and... 

The additivity property of thermostats mentioned above holds also for stochastic thermostats, since the Fokker-Planck equation is constructed from linear operations on the vector fields involved, thus it is enough to require that the newly introduced terms satisfy appropriate fluctuation-dissipation relationships so that they are compatible with the extended Gibbs distribution of the deterministic part. That is, we require... 

It was argued (Horsthemke Brenig, 1971 Blomberg, 1981 Haenggi et al., 1984) that the nonlinear Fokker-Planck equations (derived in a slightly different way) also operate correctly in the critical point. 

Finally, we would like to elaborate the proposed protocol of the high-friction map, eqn (13.17). Its construction is based purely on the thermodynamic consideration, eqn (13.15), validated by the central limit theorem. Therefore it may offer a general rule to obtain the Smoluchowski limit to any phase-space dynamics under study. The protocol proposed in this chapter is based on the fact that the map is universal at formal level and is therefore obtainable with thermodynamic consideration. It means the Smoluchowski dynamics can be taken care of by the related Fokker-Planck equation, upon the universal map being carried out. It is worth pointing out that the resultant diffusion operator in eqn (13.18) clearly originates from only the Hamiltonian part of the... 

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