Fokker-Planck and Langevin Equations

Velocity relaxation effects can be accounted for in an approximate fashion by going to a phase-space description in terms of Fokker-Planck or Langevin equations. Perhaps the best known study of this type is due to Kramers, who studied the escape of particles over potential barriers as a model for certain types of isomerization or dissociation reaction. 

Suppose the particle moves in the one-dimensional potential shown schematically in Fig. 3.2. Kramers assumes that the time evolution of the phase-space distribution function F q, v, t) is given by the Fokker-Planck equation 

In the intermediate to high friction limit, the steady-state Fokker-Planck equation can be solved with the potential in (3.19) to obtain the following more general result  

For high friction, f//n 2w, the equation reduces to (3.21), whereas for intermediate friction, f//n 2w, (3.22) reduces to the transition state theoiy (TST) result.  

Since Kramers s paper there have been a large number of studies devoted to the problem of passage over a barrier, focusing especially on the transition between the low and high friction regimes. We discuss some of the more recent developments later (cf. Section XII). 

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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. 

At this stage it is of importance to examine the relationship between the Fokker-Planck and Langevin equations. 

Internal noise is described by a master equation. When this equation cannot be solved exactly it is necessary to have a systematic approximation method - rather than the naive Fokker-Planck and Langevin approximations. Such a method will now be developed in the form of a power series expansion in a parameter Q. In lowest order it reproduces the macroscopic equation and thereby demonstrates how a deterministic equation emerges from the stochastic description. 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

Several techniques have been used to derive from (2.11), or generalization thereof, generalized Fokker-Planck and master equations as well as Langevin and transport equations [2.12,14,15]. 

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]. 

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. 

Brownian dynamics is nothing but the numerical solution of the Smoluchowski equation. The method exploits the mathematical equivalence between a Fokker-Planck type of equation and the corresponding Langevin... 

The dynamical behavior of particles whose mass and size are much larger than those of the solvent particles can be explained by the theory of Brownian motion. Two approaches, Fokker-Planck and Chandrasekhar, have generally been used to solve the Langevin equation to describe Brownian motion. The Fokker-Planck differential equation is the diffusion equation in velocity space, while the Chandrasekhar equation is... 

In this section, we first introduce the most frequently used definitions of static properties, which will be used throughout this chapter and in some later chapters, and then discuss stochastic processes in the motion of macromolecular chains that is. Brownian motion that leads to the well-known Fokker-Planck equation, which further reduces to the Smoluchowski equation and Langevin equation. These two equations play a very important role in describing the motion of macromolecular chains. Owing to the limited space available here, we do not present rigorous derivations of various expressions. 

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). 

In a stochastic approach, one replaces the difficult mechanical equations by stochastic equations, such as a diffusion equation, Langevin equation, master equation, or Fokker-Planck equations.5 These stochastic equations have fewer variables and are generally much easier to solve than the mechanical equations, One then hopes that the stochastic equations include the significant aspects of the physical equations of motion, so that their solutions will display the relevant features of the physical motion. 

In a purely formal sense, Langevin and Fokker-Planck approaches to a problem are equivalent but, as is often the case, one approach or the other may be preferable for practical reasons. In the Fokker-Planck method, one has to solve a partial differential equation in many variables. In the Langevin method, one has to solve coupled equations of motion for the same variables, under the influence of a random force. It is likely that this second approach will be most useful for performing computer experiments to simulate the actual motion of individual polymer molecules. 

The plan of the article is as follows. First, we discuss the phenomenon of hydrodynamic interaction in general terms, and at the same time, we present some convenient notation. Then, we give the usual argument leading to the Fokker-Planck equation. After that we derive the Langevin equation that is formally equivalent to the Fokker-Planck equation, together with a statistical description of the fluctuating force. 

The Langevin equation associated with the preceding Fokker-Planck equation can be found by standard methods. A fine review has been given by Lax5 we use his procedure and notation. 

The Langevin approach is widely used for the purpose of finding the effect of fluctuations in macroscopically known systems. The fluctuations are introduced by adding random terms to the equations of motion, called noise sources . This approach is popular because it gives a more concrete picture than the Fokker-Planck equation, but it is mathematically equivalent to it. In nonlinear cases it is subject to the same difficulties, and some additional ones. 

The equivalence of the Langevin equation (1.1) to the Fokker-Planck equation (VIII.4.6) for the velocity distribution of our Brownian particle now follows simply by inspection. The solution of (VIII.4.6) was also a Gaussian process, see (VIII.4.10), and its moments (VIII.4.7) and (VIII.4.8) are the same as the present (1.5) and (1.6). Hence the autocorrelation function (1.8) also applies to both, so that both solutions are the same process. Q.E.D. 

E is an electric field and V the interaction potential. When Ll,L2 are independent Langevin forces, write the corresponding Fokker-Planck equation. Find the equilibrium solution and determine and r2. ... 

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