The Fokker-Planck equation

The relaxation to thermal equilibrium is seen to be exponential, with a rate given by (8.104). It is interesting to note that in the infinite temperature limit, where ky = 0, Eq. (8.102) describes a constant heating rate of the oscillator. It is also interesting to compare the result (8.106) to the result (9.65) of the very different quantum formalism presented in Section 9.4 see the discussion at the end of Section 9.4 of this point. 

For completeness we also cite from the same paper (see footnote 11) the expression for the variance cr(Z) = 

The result (8.107) shows that in the course of the relaxation process the width of the energy level distribution increases (due to the last term in (8.107)) before decreasing again. This effect is more pronounced for larger [( )o — (n)eq]. that is, when the initial excitation energy is much larger than hgT. 

In many practical situations the random process under observation is continuous in the sense that (1) the space of possible states is continuous (or it can be transformed to a continuous-like representation by a coarse-graining procedure), and (2) the change in the system state during a small time interval is small, that is, if the system is found in state x at time t then the probability to find it in state y x at time t + St vanishes when St 0. When these, and some other conditions detailed below, are satisfied, we can derive a partial differential equation for the probability distribution, the Fokker-Planck equation, which is discussed in this Section. 

In fact we will require that this probability vanishes faster than St when St 

This is an important integral equation for the transition probability and is often taken as the definition of a Markov process. It is called the Chapman-Kolgomorov equation or sometimes the Smoluchowski equation. The physical interpretation of this equation is the probability of a transition from y at t to ys at ts can be calculated by taking the product of the probability of a transition to some value ys at an intermediate time t2 and the probability of a transition from that value to the final one at ts and summing over all possible intermediate values. Note that nothing is said about the choice of f 2, only that it should be an intermediate time. 

In the physical applications that will be of interest to us, the transition probability and probability density do not depend on the times t and t2 at which transitions occur but only on the time interval t2—t. This is the condition of stationarity, which means that the statistics of the process is invariant to a change of the origin of time or to a translation in time. Equation (H.6) may then be written 

With this in mind, we may write the Markov integral equation in Eq. (H.7) for a stationary process as 

This equation can be said to relate the W2 function at two slightly separated time instants, both at finite time, with the limiting form of the function for very short times r. This function is only non-zero for very small changes in y due to the small 

The minus sign in Eq. (H.12) has been absorbed in an inversion of the limits of integration in Eq. (H.13). Then comes a somewhat tricky point. The second w2 function in the integrand is the limiting form of the function for short times. It expresses the probability that y will undergo a transition Ay in a time interval r starting from y — Ay. It is plausible that w2 in the short time interval limit is a function of Ay, so in order to emphasize this it is customary to introduce a particular notation, namely 

In case of Gaussian white noise the probability density obeys a diffusion equation with a drift. In particular, the probability density is the conditioned average [50] 

Both the drift term Ki x) and the diffusion coefficient K ix) are nonlinear functions of the state variable x. They are defined as the moments of the conditioned increments per unit time, i.e. 

For multiplicative noise the determination of these moments requires a more detailed consideration of the stochastic integral since white noise is too irregular for Riemann integrals to be applied. Application of Stieltjes integration yields a dependence of the moments on how the limit to white noise is taken. If t) is the limit of the Ornstein-Uhlenbeck -process with r — 0 (Stratonovich sense) the coefficients read [50] 

Excitable systems have more than one dynamic variables. Hence the probability density will depend on these variables, i.e. P x,y, In 

For colored noise sources the derivation of evolution equations for the probability densities is more difficult. In a Markovian embedding, i.e. if the Ornstein-Uhlenbeck process is defined via white noise (cf. chapter 1.3.2) and v t) is part of the phase space one again gets a Fokker-Planck equation for the density P x,y, Similarly, one finds in case of the telegraph 

Consider now a collection of stochastic processes X(t) that are the solutions of the SDE dX = fl(X)df + fe(X)dW. If the initial conditions are drawn from a distribution with given density p(x, 0) we would like to derive an equation for the corresponding density at time t, p(x, t). We assume that p is C°° for all t and that the density tends to zero exponentially fast as x oo, whereas a and b are C°° functions which grow at most polynomially fast in these limits, that is, there are constants pa, r]b such that 

This condition of polynomial growth of the coefficients is a common assumption in the analysis of SDEs. For future reference we define the space of functions which are smooth and satisfy the polynomial growth restriction by Cg  

For any ip e Cf , the expectation with respect to p(x, t) is well defined and we have 

Taking the differential, we obtain an equation for Ed which must be equal to (6.37)  

Using the assumptions given above regarding the exponential rate of decay of the density, it is possible to recast the right hand side as an integral of a product with 4 . First, we find 

This effectively states that the probability of the final state (left-hand side) is equal to that of all the initial states transforming to the final state (with probability P). Chandrasekhar expanded out the infinitesimal velocity and time changes of these quantities as Taylor series and used the Langevin equation to relate 5u and 5f. He showed that if the probability of changing velocity and position is given by a Gaussian distribution, then the probability, W(u, r, t) that a Brownian particle has a velocity u at a position r and at time t is 

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. 

More detailed derivation and analysis of the Fokker—Planck equation have been given by Resibois and De Leener [490], Forster [453], Wang and Uhlenbeck [511], Rice and Gray [513], and Levine and Berrondo [523b]. 

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A linear dependence approximately describes the results in a range of extraction times between 1 ps and 50 ps, and this extrapolates to a value of Ws not far from that observed for the 100 ps extractions. However, for the simulations with extraction times, tg > 50 ps, the work decreases more rapidly with l/tg, which indicates that the 100 ps extractions still have a significant frictional contribution. As additional evidence for this, we cite the statistical error in the set of extractions from different starting points (Fig. 2). As was shown by one of us in the context of free energy calculations[12], and more recently again by others specifically for the extraction process [1], the statistical error in the work and the frictional component of the work, Wp are related. For a simple system obeying the Fokker-Planck equation, both friction and mean square deviation are proportional to the rate, and... 

H. Risken, The Fokker-Planck equation Methods of solution and applications , Springer- Verlag, Berling, 1984, chapter 3. 

With the Laplace operator V. The diffusion coefficient defined in Eq. (62) has the dimension [cm /s]. (For correct derivation of the Fokker-Planck equation see [89].) If atoms are initially placed at one side of the box, they spread as ( x ) t, which follows from (62) or from (63). 

H. Risken. The Fokker-Planck Equation. Berlin Springer, 1989. 

Keilson J., Storer J. E. On a Brownian motion, Boltzmann equation and the Fokker-Planck equation, Quart. Appl. Math., 10, 243-53 (1952). 

Barcilon, V, Singular Perturbation Analysis of the Fokker-Planck Equation Kramer s Underdamped Problem, SIAM Journal of Applied Mathematics 56, 446, 1996. 

The Fokker-Planck equation is essentially a diffusion equation in phase space. Sano and Mozumder (SM) s model is phenomenological in the sense that they identify the energy-loss mechanism of the subvibrational electron with that of the quasi-free electron slightly heated by the external field, without delineating the physical cause of either. Here, we will briefly describe the physical aspects of this model. The reader is referred to the original article for mathematical and other details. SM start with the Fokker-Planck equation for the probability density W of the electron in the phase space written as follows ... 

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. 

The Fokker-Planck equation is a partial differential equation. In most cases, its time-dependent solution is not known analytically. Also, if the Fokker-Planck equation has more than one state variable, exact stationary solutions are... 

Equation (2.5) is a stochastic differential equation. Some required characteristics of stochastic process may be obtained even from this equation either by cumulant analysis technique [43] or by other methods, presented in detail in Ref. 15. But the most powerful methods of obtaining the required characteristics of stochastic processes are associated with the use of the Fokker-Planck equation for the transition probability density. 

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],... 

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. 

First of all we should mention that the Fokker-Planck equation may be represented as a continuity equation ... 

To calculate the mean escape time over a potential barrier, let us apply the Fokker-Planck equation, which, for a constant diffusion coefficient D = 2kT/h, may be also presented in the form... 

Let us calculate the relaxation time of particles in this potential (escape time over a barrier) which agrees with inverse of the lowest nonvanishing eigenvalue Yj. Using the method of eigenfunction analysis as presented in detail in Refs. 2, 15, 17, and 18 we search for the solution of the Fokker-Planck equation in the... 

To our knowledge, the first paper devoted to obtaining characteristic time scales of different observables governed by the Fokker-Planck equation in systems having steady states was written by Nadler and Schulten [30]. Their approach is based on the generalized moment expansion of observables and, thus, called the generalized moment approximation (GMA). 

In order to achieve the most simple presentation of the calculations, we shall restrict ourselves to a one-dimensional state space in the case of constant diffusion coefficient D = 2kT/h and consider the MFPT (the extension of the method to a multidimensional state space is given in the Appendix of Ref. 41). Thus the underlying probability density diffusion equation is again the Fokker-Planck equation (2.6) that for the case of constant diffusion coefficient we present in the form ... 

It is convenient to present the Fokker-Planck equation in the following dimensionless form ... 

When comparing (5.106) with (5.105), it becomes evident that these expressions coincide to make the interchange xo . This fact demonstrates the so-called reciprocity principle In any linear system, some effect does not vary if the source (at position x = xo) and the observation point (x = ) will be interchanged. The linearity of our system is represented by the linearity of the Fokker-Planck equation (5.72). 

Risken, H. The Fokker-Planck equation, Springer-Verlag, Berlin, 1989... 

Let us now suppose that each ion in the electrolyte solution may be described by a distribution function — (R, u4 t) which obeys the Fokker-Planck equation (203). 

Equation (241) is precisely the Fokker-Planck equation deduced above on a purely phenomenological basis it was first derived... 

The diffusion term in this expression differs from the Fokker-Planck equation. This difference leads to a fT-dependent term in the drift term of the corresponding stochastic differential equation (6.177), p. 294. 

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

Furthermore, since (6.159) does not depend on y, if the notional particles are uniformly distributed the Fokker-Planck equation for f is... 

The term A2Pr is a direct result of employing the IEM model. If a different mixing model were used, then additional terms would result. For example, with the FP model the right-hand side would have the form /3 = A3PC + A2Pr + AsPra, where rd results from the diffusion term in the Fokker-Planck equation. 

H. Kohn, Numerical integration of the Fokker-Planck equation and the evolution of stars clusters, Astrophys. J 234, 1036 (1979). 

Analysis of a physical problem involving Brownian motion can normally determine only the values of the coefficients and that appear in the Fokker-Planck equation. The matrix of coefficients S "" is required only to satisfy Eq. (2.229), which is generally not sufficient to determine a unique value for B ". For L > 1 and M = L, there are generally an infinite number of ways of... 

The dependence of the electron ion recombination rate constant on the mean free path for electron scattering has also been analyzed on the basis of the Fokker Planck equation [40] and in terms of the fractal theory [24,25,41]. In the fractal approach, it was postulated that even when the fractal dimension of particle trajectories is not equal to 2, the motion of particles is still described by difihsion but with a distance-dependent effective diffusion coefficient. However, when the fractal dimension of trajectories is not equal to 2, the motion of particles is not described by orthodox diffusion. For the... 

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