Fokker-Planck-Smoluchowski Equation

As seen above, a solution of the Langevin equation (Equation 6.50) (which is a nonlinear partial differential equation with random noise) consists of constructing the correlation functions of f (t) from the equation and then averaging the expressions with the help of the properties of the noise r(t). An alternative method of solution is to find the probability distribution function P(x, t) for realizing a situation in which the random variable f (t) has the particular value X at time t. P(x, t) is an equivalent description of the stochastic process f (0 and is given by the Fokker-Planck equation (Chandrasekhar 1943, Gardiner 1985, Risken 1989, Redner 2001, Mazo 2002) 

The mathematical form of this equation is exactly the same as Equation 6.25, with J(x, t) being the flux of the probability distribution function, and the first and second terms on the right-hand side of Equation 6.57 being the drift and diffusion terms, respectively. For the general stochastic process given by 

As a specific example pertinent to oiu interests in polymer translocation, let us consider a Brownian particle undergoing diffusion in the presence of an external field arising from free energy barriers or/and an electric field, as described in Section 6.2.2. In view of Equations 6.53 through 6.55, we get 

Using the Einsteinian relation (Equation 6.43), we get from Equations 6.56 

This equation is also known as the Smoluchowski equation. If the force is a constant, then Df/ksT is the uniform velocity of the particle. Now, the above equation is identical to Equation 6.25 derived in Section 6.1 for a biased random walk. 

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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]. 

The motion of polymer molecules in a solution under chemical potential gradients or externally imposed electric fields is an example of the drift-diffusion stochastic processes. We have introduced several equivalent formalisms for studying these processes biased random walk, master equation, and Langevin equation of motion. In each of these lines of arguments, we have arrived at the Fokker-Planck-Smoluchowski equation. 

Equation (2.6) is called the Fokker-Planck equation (FPE) or forward Kolmogorov equation, because it contains time derivative of final moment of time t > to. This equation is also known as Smoluchowski equation. The second equation (2.7) is called the backward Kolmogorov equation, because it contains the time derivative of the initial moment of time to < t. These names are associated with the fact that the first equation used Fokker (1914) [44] and Planck (1917) [45] for the description of Brownian motion, but Kolmogorov [46] was the first to give rigorous mathematical argumentation for Eq. (2.6) and he was first to derive Eq. (2.7). The derivation of the FPE may be found, for example, in textbooks [2,15,17,18],... 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

Another derivation has been given by Resibois and De Leener. In principle, eqn. (287) can be applied to describe chemical reactions in solution and it should provide a better description than the diffusion (or Smoluchowski) equation [3]. Reaction would be described by a spatial- and velocity-dependent term on the right-hand side, — i(r, u) W Sitarski has followed such an analysis, but a major difficulty appears [446]. Not only is the spatial dependence of the reactive sink term unknown (see Chap. 8, Sect. 2,4), but the velocity dependence is also unknown. Nevertheless, small but significant effects are observed. Harris [523a] has developed a solution of the Fokker—Planck equation to describe reaction between Brownian particles. He found that the rate coefficient was substantially less than that predicted from the diffusion equation for aerosol particles, but substantially the same as predicted by the diffusion equation for molecular-scale reactive Brownian particles. 

It has recently been pointed out by Gordon1 that the root-mean-square fluctuations in the sampled values of the autocorrelation function of a dynamical variable do not necessarily relax to their equilibrium values at the same rate as the autocorrelation function itself relaxes. It is the purpose of this paper to investigate the relative rates of relaxation of autocorrelation functions and their fluctuations in certain systems that can be described by Smoluchowski equations,2 i.e., Fokker-Planck equations in coordinate space. We exhibit the fluctuation and autocorrelation functions for several simple systems, and show that they usually relax at different rates. 

As is well known, dynamic properties of polymer molecules in dilute solution are usually treated theoretically by Brownian motion methods. Tn particular, the standard approach is to use a Fokker-Planck (or Smoluchowski) equation for diffusion of the distribution function of the polymer molecule in its configuration space. 

Now we present the standard derivation of the Fokker-Planck equation for polymers in solution. (Terminology can often be confusing in the present instance, the equation of interest is also called the Smoluchowski equation, and may be regarded as a limiting case of a more general Fokker-Planck equation, or a Kramers equation.)... 

We shall meet more general Fokker-Planck equations the special form (1.1) is also called Smoluchowski equation , generalized diffusion equation , or second Kolmogorov equation . The first term on the right-hand side has been called transport term , convection term , or drift term the second one diffusion term or fluctuation term . Of course, these names should not prejudge their physical interpretation. Some authors distinguish between Fokker-Planck equations and master equations, reserving the latter name to the jump processes considered hitherto. 

A one-dimensional Fokker-Planck equation was used by Smoluchowski [19], and the bivariate Fokker-Planck equation in phase space was investigated by Klein [21] and Kramers [22], Note that, in essence, the Rayleigh equation [23] is a monovariate Fokker-Planck equation in velocity space. Physically, the Fokker-Planck equation describes the temporal change of the pdf of a particle subjected to diffusive motion and an external drift, manifest in the second- and first-order spatial derivatives, respectively. Mathematically, it is a linear second-order parabolic partial differential equation, and it is also referred to as a forward Kolmogorov equation. The most comprehensive reference for Fokker-Planck equations is probably Risken s monograph [14]. 

Diffusion can be considered as a stochastic or random process and described by the so-called Fokker-Planck equation adapted to Brownian motion. This equation is also known as the Smoluchowski equation. We consider the description of stochastic processes and Brownian motion in more detail in Section 11.1 and Appendix H. 

It is important to notice that both the original and the modified Fokker-Planck equations give the probability distribution of a particle as a function of time, position and velocity. However, if we are interested in time intervals large enough compared to jS 1, the Fokker-Planck equation, equation (3), can be reduced to a diffusional equation for the distribution function w, frequently called the Smoluchowski equation (Chandrasekhar, 1943) ... 

In order to establish the validity condition of a diffusion like equation for the probability of escape of a particle over a potential barrier, the solution of the modified Fokker-Planck equation is compared to the solution of the modified Smoluchowski equation. Since the main contribution to the determination of the escape probability comes from the neighborhood of the maximum in the potential energy (x = x J, the potential energy function was approximated by a parabolic function and the original Fokker-Planck equation was approximated at the vicinity of xmax by (Chandrasekhar, 1943) ... 

The excellent review of Chandrasekhar provides a detailed account of the history of the subject, to which both Smoluchowski and Einstein made fundamental contributions. It is worth mentioning the well-known paper of Kramers, who provided a rigorous derivation of the Smoluchowski equation from the complete Fokker-Planck equation of a Brownian particle in an external potential. This problem allows us to explain what we mean by a systematic version of the AEP. We can state the problem as follows. Let us consider the motion of a free Brownian particle described by the one-dimensional counterpart of Eq. (1.2),... 

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 product J Eg)coQ is equal to Eg for a harmonic oscillator potential truncated at = Eg, and to 2Eg for a Morse potential with dissociation energy equal to Eg. Equation (2.41) is the low-friction limit result of Kramers. There are other methods to derive the results obtained in the previous section. One is to look for the eigenvalue with smallest positive real part of the ojjerator L defined so that dP/dt = — LP is the relevant Fokker-Planck or Smoluchowski equation. Under the usual condition of time scale separation this smallest real part is the escapie rate for a single well potential. Another way uses the concept of mean passage time. For the one-dimensional Fokker-Planck equation of the form... 

The diffusion equation (485), which is usually known as the Smoluchowski equation (a particular form of the Fokker-Planck equation), can be obtained using the equation of continuity in one dimension... 

This form is useful in the generalization of Eq. (485) to fractional diffusion. The investigation of the diffusion equation (485) began when Louis Bachellier (Jules Poincare s student) wrote his thesis in 1900. It was called The Theory of Speculations and was devoted to the evolution of the stock market. Many of the most famous scientists have contributed to our knowledge of diffusion processes, amongst them Einstein, Langevin, Smoluchowski, Fokker, Planck, Levy, and others. 

In order to demonstrate how the anomalous relaxation behavior described by the hitherto empirical Eqs. (9)—(11) may be obtained from our fractional generalizations of the Fokker-Planck equation in configuration space (in effect, fractional Smoluchowski equations), Eq. (101), we first consider the fractional rotational motion of a fixed axis rotator [1], which for the normal diffusion is the first Debye model (see Section II.C). The orientation of the dipole is specified by the angular coordinate 4> (the azimuth) constituting a system of one rotational degree of freedom. Electrical interactions between the dipoles are ignored. 

We suppose that a small probing held Fj, having been applied to the assembly of dipoles in the distant past (f = —oo) so that equilibrium conditions have been attained at time t = 0, is switched off at t = 0. Our starting point is the fractional Smoluchowski equation (172) for the evolution of the probability density function W(i), cp, t) for normal diffusion of dipole moment orientations on the unit sphere in configuration space (d and (p are the polar and azimuthal angles of the dipole, respectively), where the Fokker-Planck operator LFP for normal rotational diffusion in Eq. (8) is given by l j p — l j /> T L where... 

This concept, which is based on a random walk with a well-defined characteristic time and which applies when collisions are frequent but weak [13], leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 71, due to Fiirth), we obtain the Fokker Planck equation for the evolution of the distribution function in phase space which describes normal diffusion. 

As is well known, the considered microscopic Langevin equations, are equivalent to the reformulation of (2) as a Smoluchowski equation it is a variant of a Fokker-Planck equation [47]. It describes the temporal evolution of the distribution function F( r , f) of the particle positions... 

The essence of the calculation that leads to the desired Fokker-Planck equation, known in this limit as the Smoluchowski equation, is a coarse-grained average of the time evolution (8.131) over the fast variation of 7 (Z). This procedure, described in Appendix 8B, leads to... 

The physical manifestation of friction is the relaxation of velocity. In the high friction limit velocity relaxes on a timescale much faster than any relevant observation time, and can therefore be removed from the dynamical equation, leading to a solvable equation in the position variable only, as discussed in Section 8.4.4. The Fokker-Planck or Kramers equation (14.41) then takes its simpler, Smoluchowski form, Eq. (8.132)... 

The simplest form of the Fokker-Planck equation is that for diffusion in configuration space, namely the Smoluchowski equation, which was originally derived by Einstein [13] in 1905 in the context of the theory of the Brownian movement of a particle in one dimension under no external forces. We have seen earlier that this equation is (Eq. 2.50)... 

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