Kramers diffusion equation

The rate of reaction is given by the diffusion current over the potential barrier, and the energy distribution of the reacting species along the reaction coordinate is given through the density distribution in momentum space. The calculation rests, as remarked by Kramers, on the construction and solution of the equation of diffusion obeyed by a density distribution of particles in phase space. A very clear presentation of the Kramers diffusion equation has been given by Chandrasekhar.1,2... 

Exercise. In the Kramers equation (VIII.7.4) allow y and T to depend on the position x. The expansion for large y is again possible and produces the diffusion equation for inhomogeneous temperature510... 

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. 

Converting the Fokker-Planck equation (4.160) into a diffusion equation for the energy, Kramers has obtained the following approximate relation for the rate kQ 16 (see also Eqs. 2.8 and 2.9) ... 

When y is very small, the thermal relaxation in the well is not fast relative to the escape rate, and the assumption that the distribution within the well can be represented by the equilibrium Boltzmann distribution no longer holds. On the other hand we can make use of the fact that the total energy E varies on a time scale much longer than either x or u (it is conserved for y = 0). Thus changing variables from (x, v) to ( , ) and eliminating the fast phase variable (f> leads to a Smoluchowski (diffusion) equation for E. [Kramers gave the equivalent equation in terms of the action variable J( ).]... 

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). 

It is not difficult to show that this function ip(R) satisfies the diffusion equation in Eq. (3.8). This establishes then a connection between the method of Kramers (an equivalent fictitious equilibrium problem in phase space) and the method of Kirkwood used here (a nonequilibrium problem in configuration space). 

When considering large spatial and temporal scales the transport of a concentration field by advection and molecular diffusion can be be approximately described by a diffusion equation with an effective diffusion coefficient. The main question then is to find an expression for the effective diffusivity as a function of the flow parameters and molecular diffusivity. A range of this type of problems are discussed in the review by Majda and Kramer (1999). Here we consider two simple examples of this problem in the case of steady two-dimensional flows with open and closed streamlines, respectively. 

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 and Albreda applied the mixing cell model for studying the frequency response of continuous flow in a packed tube by combining the convection/ diffusion equations to obtain ... 

The Curtiss-Bird theory for concentrated systems is structured quite differently, taking as the starting point the general phase-space formalism as shown in Figure 8. The polymer molecule is modeled throughout as a Kramers chain, and the restricted motion of the molecule is described by means of an anisotropic Stokes law expression for the links and an anisotropic Brownian motion. Despite the great differences between the Doi-Edwards and Curtiss-Bird theories, some of the key results are very similar. For example, both theories lead to a diffusion equation for a segment of the polymer chain as follows ... 

Sun Y-P and Saltiel J 1989 Application of the Kramers equation to stiibene photoisomerization in / -alkanes using translational diffusion coefficients to define microviscosity J. Phys. Chem. 93 8310-16... 

The first paper that was devoted to the escape problem in the context of the kinetics of chemical reactions and that presented approximate, but complete, analytic results was the paper by Kramers [11]. Kramers considered the mechanism of the transition process as noise-assisted reaction and used the Fokker-Planck equation for the probability density of Brownian particles to obtain several approximate expressions for the desired transition rates. The main approach of the Kramers method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact timescales and probability densities, it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the problem of investigating diffusion transition processes. 

So far we studied the first passage of Markov processes such as described by the Smoluchowski equation (1.9). On a finer time scale, diffusion is described by the Kramers equation (VIII.7.4) for the joint probability of the position X and the velocity V. One may still ask for the time at which X reaches for the first time a given value R, but X by itself is not Markovian. That causes two complications, which make it necessary to specify the first-passage problem in more detail than for diffusion. 

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