Equation Chapman-Kolmogorov

1 Chapman-Kolmogorov equation Let us integrate Eq. 13.49 over all possible X2 to obtain 

Dividing by Pi(xi, ti) we obtain the Chapman-Kolmogorov equation, which is obeyed by any Markov process, 

The Chapman-Kolmogorov equation makes obvious sense a process starting with a value xi at time h can reach a value X3 at a later time 3 through any one of the possible values at intermediate times, t2 (Fig. 13.4). 

---

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. 

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). 

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... 

Indeed, from the Chapman-Kolmogorov equation and the relations (4.38)-(4.40), one obtains... 

If the Poisson process of enrichment is assumed (the gas is completely mixed and randomly enriched), then, using the Chapman-Kolmogorov equation, we have... 

The procedure based on a generalized Chapman-Kolmogorov equation in phase space proposed by Metzler and Klafter [7,85,86] then leads, assuming the diffusion limit, to the following generalization of the Klein-Kramers... 

The differential form of the Chapman—Kolmogorov equation [11]. 3That is, we consider the overdamped case. 

Another version of the Kolmogorov equation is obtained by considering the transition Sj Sj S - Eq.(2-30) gives the Chapman-Kolmogorov equation,... 

E(Z2t2 Izoto) = dziP(z2t2 ziti)P(ziti Izoto) This is the Chapman- Kolmogorov equation. 

The derivation of the Fokker-Planck (FP) equation described above is far from rigorous since the conditions for neglecting higher-order terms in the expansion of exp( 9/9x) were not established. Appendix 8A outlines a rigorous derivation of the FP equation for a Markov process that starts from the Chapman-Kolmogorov equation... 

---