Kolmogorov operator

In the special case of Langevin dynamics (6.32)-(6.33) for a system with H-configuration variables, the density is a function of the vectors of positions q, momentap, and time, and the formula for the Kolmogorov operator becomes... 

It is an easy exercise to show that if Pn satisfies the Kolmogorov consistency conditions (equations 5.68) for all blocks Bj of size j < N, then T[N- N+LPN) satisfies the Kolmogorov consistency conditions for blocks Bj of size j < N + 1. Given a block probability function P, therefore, we can generate a set of block probability functions Pj for arbitrary j > N hy successive applications of the operator TTN-tN+i, this set is called the Bayesian extension of Pn-... 

The Local Structure Operator By the Kolmogorov consistency theorem, we can use the Bayesian extension of Pn to define a measure on F. This measure -called the finite-block measure, /i f, where N denotes the order of the block probability function from which it is derived by Bayesian extension - is defined by assigning t.o each cylinder c Bj) = 5 G F cti = 6i, 0 2 = 62, , ( j — bj a value equal to the probability of its associated block ... 

The essence of the LST for one-dimensional lattices resides in the fact that an operator TtN->N+i could be constructed (equation 5.71), mapping iV-block probability functions to [N -f l)-block probabilities in a manner which satisfies the Kolmogorov consistency conditions (equation 5.68). A sequence of repeated applications of this operator allows us to define a set of Bayesian extended probability functions Pm, M > N, and thus a shift-invariant measure on the set of all one-dimensional configurations, F. Unfortunately, a simple generalization of this procedure to lattices with more than one dimension, does not, in general, produce a set of consistent block probability functions. Extensions must instead be made by using some other, approximate, method. We briefly sketch a heuristic outline of one approach below (details are worked out in [guto87b]). 

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)... 

The traditional derivation of the Fokker-Planck equation (1.5) or (VIII. 1.1) is based on Kolmogorov s mathematical proof, which assumes infinitely many infinitely small jumps. In nature, however, all jumps are of some finite size. Consequently W is never a differential operator, but always of the type (V.1.1). Usually it also has a suitable expansion parameter and has the canonical form (X.2.3). If it then happens that (1.1) holds, the expansion leads to the nonlinear Fokker-Planck equation (1.5) as the lowest approximation. There is no justification for attributing a more fundamental meaning to Fokker-Planck and Langevin equations than in this approximate sense. 

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers N as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65], Describing the state of a system by a vector N = TV),..., Ns, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P(N, t)... 

The applications of the stochastic theory in chemical engineering have been very large and significant [4.5-4.7, 4.49-4.59, 4.69-4.78]. Generally speaking, we can assert that each chemical engineering operation can be characterized vdth stochastic models. If we observe the property transport equation, we can notice that the convection and diffusion terms practically correspond with the movement and diffusion terms of the Fokker-Plank-Kolmogorov equation (see for instance Section 4.5) [4.79]. 

According to equation (1.6) the size of the material balls for a relatively intensive mixing operation of P/pV = 1 W/kg in an aqueous liquid (v = 10 m /s) is 32 pm and in a liquid with the viscosity of pure glycerine at room temperature (v = 10 m /s) is already 5.6 mm (see Table 1.1). This shows that viscous systems will always remain to a certain extent segregated, since the Kolmogorov micro-scale a can be comparatively little influenced by the mixing power 2 cc... 

Note that some authors have used these operators interchangeably for the description of the mesoscopic transport process. It is clear that if L is self-adjoint, then it can be used as a transport operator and the function u(x, t) can represent the particle density. For example, the one-dimensional Brownian motion B t) has the infinitesimal generator L = 9 /9x which is self-adjoint. A symmetric a-stable Levy process on R has the generator L = 9 /9 x , which is self-adjoint too. In the next section we obtain L and L from the Chapman-Kolmogorov equation. 

From the forward Kolmogorov equations for matrices and considering that the initial macro-state is 0, that is, X(0) = 0, aU the components are operational, we have,... 

When a large number N of equally active (p = 1) screw dislocations distributed randomly over the surface are operating, a Kolmogorov-Avrami type of analysis can be made for the evaluation of the current transient ... 

To relate the Kolmogorov scale, q, to operating variables, we need to get a measure of the rate of dissipation of tmbulence kinetic energy per unit mass, e. The easiest way to do this is via scaling arguments and the use of characteristic length and velocity scales. These scales are an important tool in engineering fluid mechanics and deserve some explanation. 

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