Kolmogorov equation

L. E. Levine, K. Lakshmi Narayan, K. F. Kelton. Finite size corrections for the Johnson-Mehl-Avrami-Kolmogorov equation. J Mater Res 72 124, 1997. 

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 the solution of real tasks, depending on the concrete setup of the problem, either the forward or the backward Kolmogorov equation may be used. If the one-dimensional probability density with known initial distribution deserves needs to be determined, then it is natural to use the forward Kolmogorov equation. Contrariwise, if it is necessary to calculate the distribution of the mean first passage time as a function of initial state xo, then one should use the backward Kolmogorov equation. Let us now focus at the time on Eq. (2.6) as much widely used than (2.7) and discuss boundary conditions and methods of solution of this equation. 

We will not present here how to derive the first Pontryagin s equation for the probability Q(t, x0) or P(f,x0). The interested reader can see it in Ref. 19 or in Refs. 15 and 18. We only mention that the first Pontryagin s equation may be obtained either via transformation of the backward Kolmogorov equation (2.7) or by simple decomposition of the probability P(t, xq) into Taylor expansion in the vicinity of xo at different moments t and t + t, some transformations and limiting transition to r — 0 [18]. 

If the incoming call distribution is Poisson, and the time to service a call has an exponential distribution (Eq. 11 with density parameter p), then the transitional probabilities related to states s0 (the server is free) and Si (the server is engaged) are Pio= 1 - exp(-pt) poi = 1 -pw> pu= 1 -P o, Pooandpu are found by solving the Kolmogorov equations (Eq. 8)... 

Karty et al. [21] pointed out that the value of the reaction order r and the dependence of k on pressure and temperature in the JMAK (Johnson-Mehl-Avrami-Kolmogorov) equation (Sect. 1.4.1.2), and perhaps on other variables such as particle size, are what define the rate-limiting process. Table 2.3 shows the summary of the dependence of p on growth dimensionality, rate-limiting process, and nucleation behavior as reported by Karty et al. [21]. 

We only assume that q is a Markov process, so that its probability distribution obeys the Chapman-Kolmogorov equation... 

Exercise. The Chapman-Kolmogorov equation (2.1) expresses the fact that a process starting at with value yx reaches y3 at t3 via any one of the possible values y2 at the intermediate time t2. Where does the Markov property enter into this argument ... 

Exercise. Suppose one knows a solution of the Chapman-Kolmogorov equation and wants to use it for constructing a Markov process. How can that be done and how much freedom does one still have ... 

Exercise. Write (2.3) as a 2 x 2 matrix and formulate the Chapman-Kolmogorov equation as a property of that matrix. 

The other four have zero probability. Show that this process obeys the Chapman-Kolmogorov equation but is not Markovian. ... 

Exercise. It has been remarked in 1 that a Markov process with time reversal is again a Markov process. Construct the hierarchy of distribution functions for the reversed Wiener process and verify that its transition probability obeys the Chapman-Kolmogorov equation. 

These are the so-called forward and backward Kolmogorov equations for the Ornstein-Uhlenbeck process. Their paramount importance will appear in VIII.4 under the more familiar name of Fokker-Planck equation. 

The master equation is an equivalent form of the Chapman-Kolmogorov equation for Markov processes, but it is easier to handle and more directly related to physical concepts. It will be the pivot of most of the work in this book. 

Consider a Markov process, which for convenience we take to be homogeneous, so that we may write Tx for the transition probability. The Chapman-Kolmogorov equation (IV.3.2) for Tx is a functional relation, which is not easy to handle in actual applications. The master equation is a more convenient version of the same equation it is a differential equation obtained by going to the limit of vanishing time difference t. For this purpose it is necessary first to ascertain how Tx> behaves as x tends to zero. In the previous section it was found that TX (y2 yl) for small x has the form ... 

Now in the Chapman-Kolmogorov equation (IV.3.2) insert this expression for TZ 9... 

This differential form of the Chapman-Kolmogorov equation is called the master equation. 

Not only is the master equation more convenient for mathematical operations than the original Chapman-Kolmogorov equation, it also has a more direct physical interpretation. The quantities W(y y ) At or Wnn> At are the probabilities for a transition during a short time At. They can therefore be computed, for a given system, by means of any available approximation method that is valid for short times. The best known one is time-dependent perturbation theory, leading to Fermi s Golden Rule f)... 

This interpretation of the master equation means that is has an entirely different role than the Chapman-Kolmogorov equation. The latter is a nonlinear equation, which results from the Markov character, but contains no specific information about any particular Markov process. In the master equation, however, one considers the transformation probabilities as given by the specific system, and then has a linear equation for the probabilities which determine the (mesoscopic) state of that system. 

Exercise. Derive (9.8) in a similar way as the M-equation was derived in V.l starting from the Chapman-Kolmogorov equation (IV.2.1). 

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]. 

In the usual derivations of the Klein-Kramers equation, the moments of the velocity increments, Eq. (68), are taken as expansion coefficients in the Chapman-Kolmogorov equation [9]. Generalizations of this procedure start off with the assumption of a memory integral in the Langevin equation to finally produce a Fokker-Planck equation with time-dependent coefficients [67]. We are now going to describe an alternative approach based on the Langevin equation (67) which leads to a fractional IGein-Kramers equation— that is, a temporally nonlocal behavior. 

Fractional dynamics emerges as the macroscopic limit of the combination of the Langevin and the trapping processes. After straightforward calculations based on the continuous-time version of the Chapman-Kolmogorov equation [75, 114] which are valid in the long-time limit t max r, t, one obtains the fractional Klein-Kramers equation... 

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