Fokker-Planck equation boundary conditions

For obtaining the solution of the Fokker-Planck equation, besides the initial condition one should know boundary conditions. Boundary conditions may be quite diverse and determined by the essence of the task. The reader may find enough complete representation of boundary conditions in Ref. 15. 

This is the so-called natural boundary condition to the Fokker-Planck equation (Gardiner 1990). 

In closing this section, we note that the stochastic master equation, Eq. (16), can be used to study the effect of boundary conditions on transport equations. If a(x, y, t) is sufficiently peaked as a function of x — y, that is if transitions occur from y to states in the near neighborhood of y, only, then the master equation can be approximated by a Fokker-Planck equation. The effects of the boundary on the master equation all appear in the properties of a(x, y, t). However, in the transition to the Fokker-Planck differential equation, these boundary effects appear as boundary conditions on the differential equation.7 These effects are prototypes for the study of how molecular boundary conditions imposed on the Liouville equation are reflected in the macroscopic boundary conditions imposed on the hydrodynamic equations. 

Special examples involving these boundary conditions have been worked out and it appeared that a systematic expansion in O 1/2 again led to the Fokker-Planck equation with higher order corrections.16 However, a general theory has not yet been developed. 

Exercise. The one-step process (VI.4.8) has at n = 0 a natural boundary in the sense of VI.5. However, the corresponding Fokker-Planck equation (VIII.5.3) has a regular boundary (in the present sense). Show that the reflecting condition has to be imposed to obtain agreement with the discrete M-equation. [Compare XIII.3.]... 

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. 

Risken, Vollmer, and Mdrsch studied the Kramers equation, that is, the Fokker-Planck equation (1.9), by expanding the distribution function p(x, o /) in Hermitian polynomials (velocity part) and in another complete set satisfying boundary conditions (position part). The Laplace transform of the initial value problem was obtained in terms of continued fractions. An inverse friction expansion of the matrix continued fraction was then used to show that the first Hermitian expansion coefficient may be determined by a generalized Smoluchowski equation. This provides results correcting the standard Smoluchowski equation with terms of increasing power in 1/y. They evaluated explicit expressions up to order y . ... 

The initial condition states that dreflecting boundary condition at a = d in the adjoint Fokker-Planck equation is expressed by a no-flux boundary condition. Setting G = 0 at the right boundary corresponds to an absorbing boundary. We solve equation (11.15) with the ansatz G x,t) = exp(—Xt)u x), A > 0 so that it reduces to the ordinary differential equation... 

The above equations are known as the forward and backward Fokker-Planck equations. For a particular stochastic process (namely, prescribed A x) and p(x,t xo,0) needs to be calculated with appropriate boundary conditions (Gardiner 1985). Substituting this result in Equation 6.86, g xo, r) is then obtained. 

The probability distribution of the first passage time and its moments can be obtained from the Fokker-Planck equation for general situations of F x) and i/f (x), by following the standard procedures given in Section 6.6. In the present context, the average translocation time is the mean first passage time, which can be calculated by choosing the appropriate boundary conditions for P(x,t). 

As discussed in the introduction it is quite often sufficient for the solution -except in the case of bifurcation - to estabhsh the paths of the mean values of relevant variables. It is easy to derive equations for (x), and (x ), from the Fokker-Planck equation (2.26) using the definition (2.32). Multiplying (2.26) with X and x respectively, after partial integration over the interval [-1,1] and taking into account the boundary condition (2.29) the exact equations are obtained as... 

The first term on the right-hand side represents the force due to the parabolic barrier, the second term is the frictional force with C the solvent friction experienced during the motion, and the final term F (t) is a random force which causes the fluctuations in the position and velocity. The mass m must be interpreted as the reduced mass related to the reactive mode in nuclear motion. This equation is equivalent to the Fokker-Planck equation used by Kramers, but it is much easier to solve. The solution of this equation is straightforward, and from the solution the current correlation function in the integral (Eq. 9.2) over the barrier can be calculated with the given boundary conditions. Some standard mathematics, well, actually quite a bit, leads to the result aheady derived by Kramers for the rate ... 

So far the Fokker-Planck approximation has only been formulated for cases where there is no boundary, or where the boundary is too far away to bother about it. The question now is how a boundary with certain physical properties is to be translated into a boundary condition for the differential equation. In the case of a reflecting boundary the answer is clear the probability flow (1.3) has to vanish, as in (3.6),... 

The above specific examples are only for illustrative purposes. These examples show how to implement the technology of Fokker-Planck formalism and how to apply the derived equations for a given situation. Variations in the shape of the pore, entropic barrier, pore-polymer interaction, electrical forces inside and outside the pore, hydrodynamic flows, electroosmotic flows, pressure gradients, polymer sequence, boundary conditions, etc., can be readily addressed by performing calculations analogous to those presented above. 

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