Fokker-Planck equation calculations

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... 

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... 

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 ... 

Use of the stochastic differential equation (2.2.2) as the equation of motion instead of equation (2.1.1) results in the treatment of the reaction kinetics as a continuous Markov process. Calculations of stochastic differentials, perfectly presented by Gardiner [26], allow us to solve equation (2.2.2). On the other hand, an averaged concentration given by this equation could be obtained making use of the distribution function / = f(c, ..., cs t). The latter is nothing but solution of the Fokker-Planck equation [26, 34]... 

Numerical calculations on Eq. (5.6) can be compared to classical mechanics by computing the classical dynamics of the phase-space density pc (x,y,px,pY), which is obtained as the solution to the Fokker-Planck equation ... 

The relative Brownian motion between the constituents of doublets consisting of sufficiently small equal-size aerosol particles is described by a one-dimensional Fokker-Planck equation in the particle energy space. A first passage time approach is employed for the calculation of the average lifetime of the doublets. This calculation is based on the assumption that the initial distribution of tire energy of the relative motion of the constituent particles is Maxwellian. The average dissociation time of doublets, in air at 1 atm and 298 K, for a Hamaker constant of 10 12 erg has been calculated for different sizes of the constituent particles. The calculations are found to be consistent with the assumption that the... 

The escape rate for the process described by the Fokker-Planck equation, Eq. (86), has been studied in ref. 73. We choose (/) = (x(/))/(x(0)) as the observable of interest, (Jc(oo)> = 0. Then we apply the approach described in the Section IV.A to evaluate the escape rate A as the area below the curve (t) k = (0)", where (0) is the Laplace transform of (r) at Tsto frequency. To make the convergent of the computer calculations faster, the CFP algorithm has been applied by taking... 

We perform concrete calculations in the complex P-representation [Drummond 1980 McNeil 1983] in the frame of both probability distribution functions and stochastic equations for the complex c-number variables. We follow the standard procedures of quantum optics to eliminate the reservoir operators and to obtain a master equation for the density operator of the modes. The master equation is then transformed into a Fokker-Planck equation for the P-quasiprobability distribution function. In particular, for an ordinary NOPO and in the case of high cavity losses for the pump mode (73 7), if in the operational regime the pump depletion effects are involved, this approach yields... 

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... 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

The latter condition means that the conditional probability ceases to be conditional as t approaches infinity. P2 is the transition probability. It is evident that if we know F, we can calculate all the desired properties of the system. We may now derive the Fokker-Planck equation as follows. 

In the Fokker-Planck equation these quantities (which express the fact that in small times in the process under consideration the space coordinate can only change by a small amount, which is the central idea underlying the theory of the Brownian motion) are to be calculated from the Langevin equation. Thus that equation is the basic equation of the theory of the Brownian movement. We have assumed in writing down the... 

In this article we shall not utilize the generalized Fokker-Planck equations 6 which have been successfully used to calculate coefficients of viscosity and thermal conductivity.13 14 Rather, we shall find it more convenient to proceed directly from the Liouville equation. To obtain an expression for the contribution of the intermolecular forces to the heat flux, we shall postulate a plausible generalization of the usual phenomenological equations of the thermodynamics of irreversible processes to the space of molecular pairs. Although we shall not prove it here, it may be shown that the same results can also be obtained (with greater labor) from the Fokker-Planck equations ... 

The use of Eq. IV.l in Eq. III.5 neglects quantities (negligible in liquids) of the order of the nonequilibrium terms in the kinetic contribution to the stress tensor.13 More exact calculations using the generalized Fokker-Planck equations show that the neglected terms are actually equal to zero. 

Equation (11-4) is an accurate description of the stochastic dynamics represented by the scheme in Figure 11.1. We will derive approximations like Fokker-Planck equations to calculate escape time characteristics from this master equation. 

The study of Fokker-Planck equations instead of master equations is often motivated by easier treatment. That holds in particular in higher dimensions, because a broader spectrum of tools is available for Fokker-Planck equations than for master equations [27] and even analytical calculations may be possible as in the case of equation (11.12). That constitutes one of the reasons for the derivations in section 11.4. However, Fokker-Planck equations always represent approximations. The only way to test their quality is a comparison with results obtained from a master equation. 

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