Markovian-Fokker-Planck equation

Adelman [530] and Stillman and Freed [531] have discussed the reduction of the generalised Langevin equation to a generalised Fokker— Planck equation, which provides a description of the probability that a molecule has a velocity u at a position r at a time t, given certain initial conditions (see Sect. 3.2.). The generalised Fokker—Planck equation has important differences by comparison with the (Markovian) Fokker— Planck equation (287). However, it has not proved so convenient a vehicle for studies of chemical reactions in solution as the generalised Langevin equation (290). 

The perturbation reduction of the corresponding Markovian Fokker-Planck equation for the two-variable process (x t), ((t)) to an approximate one in x(t) has been carried out in Section V.A of Chapter II. For brevity we report only the approximate time-evolution equation for a x,t) up to order D-,... 

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]. 

Exercise. From the previous Exercise it follows that even for non-Markovian Gaussian processes the conditional probability P(x,t x0, t0) obeys an equation of type (6.16). However, this is not a master equation, as is betrayed by the fact that the coefficients depend on t0. Compare V.l. Indicate this dependence in the Fokker-Planck equation used by S.A. Adelman, J. Chem. Phys. 64, 124 (1976). 

Doob s theorem states that a Gaussian process is Markovian if and only if its time correlation function is exponential. It thus follows that V is a Gaussian-Markov Process. From this it follows that the probability distribution, P(V, t), in velocity space satisfies the Fokker-Planck equation,... 

A rigorous analysis of the same problem has been given by San Miguel and Sancho. They emphasized that the reduction process produces unavoidably non-Markovian statistics. Nevertheless they noted that the existence of Fokker-Planck equations does not conflict with the non-Markovian character of the stochastic process, since the corresponding solution, in harmony with ref. 19, is valid only to evaluate one-time averages and is of no use in multitime averages. 

In the Markovian case, where is the shortest time scale, it is usually found that (1) moments of the form <(AJ)"(A ) > with m -I- k > 2 are of order t", n > 2, and therefore do not contribute to Eq. (5.41), and (2) all the relevant terms (that is, terms of order t) which contribute to the first and second moments (m -H k = 1 or 2) are obtained at the second iteration stage. This leads to the standard Fokker-Planck equation. 

We skip the technical details, which are straightforward but very cumbersome, and note only that, as in the Markovian case, only first and second moments yield terms that are not negligible by these criteria. Unlike in the Markovian case, three iteration steps are needed to collect all relevant contributions to these moments. The final result is the Fokker-Planck equation for... 

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... 

For colored noise sources the derivation of evolution equations for the probability densities is more difficult. In a Markovian embedding, i.e. if the Ornstein-Uhlenbeck process is defined via white noise (cf. chapter 1.3.2) and v t) is part of the phase space one again gets a Fokker-Planck equation for the density P x,y, Similarly, one finds in case of the telegraph... 

The (5.259) distribution automatically allows also the calculation of the Markovian probability flow, abstracted from the rewritten of Fokker-Planck equation on the probability density level (5.239), in an appropriate hydrod5mamic form ... 

Equations (5.7) were introduced so as to treat the non-Markovian process of Eq. (5.8) in the frame of the time-independent Fokker-Planck formalism. The equivalence has been shown to require that the fluctuation-dissipation relationship (5.10) holds the white noise limit can then be recovered by making t vanish for a fixed value of D. If we substitute Eq. (5.10) into Eqs. 

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