Kramers Moyal equation

Taking the Taylor expansion in e, one obtains the backward and forward Kramers-Moyal equations ... 

Finally we note that the Master Equation, Eq. (15), may be formally expanded in the Kramers-Moyal series8... 

This is called the Kramers-Moyal expansion. ) Formally (2.6) is identical with the master equation itself and is therefore not easier to deal with, but it suggests that one may break off after a suitable number of terms. The Fokker-Planck approximation assumes that all terms after v = 2 are negligible. Kolmogorov s proof is based on the assumption that av = 0 for v>2. This, however, is never true in physical systems In the next chapter we shall therefore expand the M-equation systematically in powers of a small parameter and find that the successive orders do not simply correspond to the successive terms in the Kramers-Moyal expansion. 

Exercise. Write the Kramers-Moyal expansion for one-step processes using (5.2). Exercise. Construct the Fokker-Planck approximation for the M-equation (VI.9.12) and use it to find [Pg.209]

First it is clear that for each sample function L equation (4.3) uniquely determines y(t) when y(0) is given. Since the values of L at different times are stochastically independent, it follows that y is Markovian. Hence it obeys an M-equation, which may be written in the Kramers-Moyal form (VIII.2.6). We compute the successive coefficients (VIII.2.4). 

Incidentally, suppose one replaces the M-equation (3.4) by the naive Fokker-Planck approximation (VIII.5.3), obtained by breaking off the Kramers-Moyal expansion after the second term rather than by the systematic expansion of chapter X. This cannot be correct for small n and cannot therefore reproduce the evolution starting from small initial m. It is therefore not paradoxical that the absorbing site n = 0 does not translate into an absorbing boundary condition of the Fokker-Planck equation - as remarked in an Exercise of XII.5. 

It is of importance to point out that if the right-hand side is truncated after two terms (diffusion approximation), the last relation leads to an expression similar to the familiar Fokker-Planck equation (4.116). The approximation of a master equation of a birth-death process by a diffusion equation can lead to false results. Van Kampen has critically examined the Kramers-Moyal expansion and proposed a procedure based on the concept of system size expansion.135 It can be stated that any diffusion equation can be approximated by a one-step process, but the converse is not true. 

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

D.2. The Kramers-Moyal Expansion Coefficients for Nonlinear Langevin Equations... 

Stratonovich [12] finds the Kramers-Moyal [31] coefficients for an equation of this form to be... 

D.3. Evaluation of the Kramers-Moyal Coefficients for the Gilbert-Langevin Equation... 

We next evaluate the Kramers-Moyal coefficients for the Gilbert-Langevin equation using the Stratonovich method. Here we define G, =... 

D.4. Formulation of the Fokker-Planck Equation from the Kramers-Moyal Expansion Coefficients... 

For the absolute density function g the Kramers-Moyal-Stratonovich equation (Kramers, 1940 Moyal, 1949 Stratonovich, 1963) can be derived ... 

Our starting point is the Kramers-Moyal-Stratonovich equation (5.13). Neglecting the higher-than-first-order conditional moments the drift equation is obtained ... 

The Fokker-Planck equation, also referred to as the Smoluchowski equation or the generalized diffusion equation, neglects the moments of order larger than 2. Higher order terms appears in the Kramers-Moyal expansion [78]. 

Kramers-Moyal Expansion and the Fokker-Planck equation... 

Starting from the master equation (3.14) the stochastic equation and subsequently the Fokker-Planck equation - using the Kramers Moyal expansion - can be derived in a general and straight forward manner. 

The expansion in Eq. 13.65 is known as the Kramers-Moyal expansion. Assuming that D (x) = 0 for > 2, yields the Fokker-Planck equation... 

The linear treatment of the noise in equation (11.7) has cast some doubt on the validity of the fl expansion. Therefore, a different class of Fokker-Planck equations have been proposed that keep the nonlinearities of the master equation. Kramers and Moyal have treated the shifts nio 1, 1... 

---