Markov equations

The Markovian approach is twofold. If the electrolyzer is viewed as a Markov processor, Eqs. (27) and (28) are replaced by their Kolmogorov-equivalents, Eqs. (8) and (9) the solution is obtained in terms of probabilities for state, V (species concentration A), and state. s 2 (species concentration 8). It is more inspiring (elegant ) to regard the electrolyzer as a Markov-chain generator with Markov equations... 

Q + kmAe)IV and dc21dt = ju2ci, ju2 = QIV2, the Markov equations are obtained as... 

We notice also that the Levy scaling of Eq. (110) is obtained by turning Eq. (133) into a Markov equation, through a procedure adopted by the authors of Ref. 60. Let us refer to this procedure as delta trick. This name suggests... 

The rationale for this equation has been widely discussed in the past [156]. According to the spirit of the Reduced Model Theory (RMT) [156], the system of interest is assumed to interact with a set of auxiliary variables whose pdf obey a conventional Markov equation of motion. Thus, in the case here under discussion, the auxiliary variable is one-dimensional, and it coincides with X itself. We assume that its pdf, p(7.. f), obeys the equation of motion dp(X,t)/dt = Rp(X,t). Thus, Eq. (286) corresponds to the Liouville-like equation of motion of the whole Universe, the variable of interest, y, and the variable X. The first two terms on the right-hand side of Eq. (286) describe the motion of y under the influence of a given X that is, they have to do with the interaction between v and X. The third term is responsible for the fluctuations of the variable X. 

One can see that a particle located at point (where a is the label of the macromolecule to which the Brownian particle belongs, and a is the label of the particle in the macromolecule) is dragged with mean velocity vf = where Uik is the tensor of velocity gradients. The dynamics of each macromolecule can be described by equation which has the form of Eq. (25). We ought to add the forces of interaction with particles of the other macromolecules to the right-hand side of Eq. (25), so that the collective motion of the entire set of macromolecules (= set of Brownian particles) is described by a set of stochastic Markov equations which, for very slow motion, can be written in the form... 

While the Smoliichowski equation is necessary for a Markov process, in general it is not sufficient, but known counter-examples are always non-Gaiissian as well. 

Using W2 = 17jP2, (A3.2.81 and (A3.2.9) may be used to satisfy the Smoluchowski equation, (A3.2.2). another necessary property for a stationary process. Thus u(t) is an example of a stationary Gaussian-Markov... 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

Onsager s theory can also be used to detemiine the fomi of the flucUiations for the Boltzmaim equation [15]. Since hydrodynamics can be derived from the Boltzmaim equation as a contracted description, a contraction of the flucUiating Boltzmann equation detemiines fluctuations for hydrodynamics. In general, a contraction of the description creates a new description which is non-Markovian, i.e. has memory. The Markov... 

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

The Boltzmaim weight appears implicitly in the way the states are chosen. The fomi of the above equation is like a time average as calculated in MD. The MC method involves designing a stochastic algorithm for stepping from one state of the system to the next, generating a trajectory. This will take the fomi of a Markov chain, specified by transition probabilities which are independent of the prior history of the system. 

Electro-conductivity of molten salts is a kinetic property that depends on the nature of the mobile ions and ionic interactions. The interaction that leads to the formation of complex ions has a varying influence on the electroconductivity of the melts, depending on the nature of the initial components. When the initial components are purely ionic, forming of complexes leads to a decrease in conductivity, whereas associated initial compounds result in an increase in conductivity compared to the behavior of an ideal system. Since electro-conductivity is never an additive property, the calculation of the conductivity for an ideal system is performed using the well-known equation proposed by Markov and Shumina (Markov s Equation) [315]. 

Such a model of the melt structure does not contradict conductivity data [324], if plotted against the composition of the KF - TaF5 system. Fig. 63 presents isotherms of molar conductivity, in which molar conductivity of the ideal system was calculated using Markov s Equation [315], and extrapolation... 

Equation (3-325), along with the fact that Y(t) has zero mean and is gaussian, completely specifies Y(t) as a random process. Detailed expressions for the characteristic function of the finite order distributions of Y(t) can be calculated by means of Eq. (3-271). A straightforward, although somewhat tedious, calculation of the characteristic function of the finite-order distributions of the gaussian Markov process defined by Eq. (3-218) now shows that these two processes are in fact identical, thus proving our assertion. 

While other programs require modification of the actual code in changing the polymer, spectra, or model, only changes in the user database is required here. Changes in the program since a brief report (22) in 1985 include improvement of the menu structure, added utilities for spectral manipulations, institution of demo spectra and database. Inclusion of Markov statistics, and automation for generation of the coefficients in Equation 1. Current limitations are that only three models (Bernoul llan, and first- and second-order Markov) can be applied, and manual input Is required for the N. A. S. L.. 

Because the dependence of probability P Uk on x should be established by means of the theory of Markov chains, in order to make such an averaging it is necessary to know how the monomer mixture composition drifts with conversion. This kind of information is available [2,27] from the solution of the following set of differential equations ... 

Here Jta(x) denotes the a-th component of the stationary vector x of the Markov chain with transition matrix Q whose elements depend on the monomer mixture composition in microreactor x according to formula (8). To have the set of Eq. (24) closed it is necessary to determine the dependence of x on X in the thermodynamic equilibrium, i.e. to solve the problem of equilibrium partitioning of monomers between microreactors and their environment. This thermodynamic problem has been solved within the framework of the mean-field Flory approximation [48] for copolymerization of any number of monomers and solvents. The dependencies xa=Fa(X)(a=l,...,m) found there in combination with Eqs. (24) constitute a closed set of dynamic equations whose solution permits the determination of the evolution of the composition of macroradical X(Z) with the growth of its length Z, as well as the corresponding change in the monomer mixture composition in the microreactor. 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

The last two results are rather similar to the quadratic forms given by Fox and Uhlenbeck for the transition probability for a stationary Gaussian-Markov process, their Eqs. (20) and (22) [82]. Although they did not identify the parity relationships of the matrices or obtain their time dependence explicitly, the Langevin equation that emerges from their analysis and the Doob formula, their Eq. (25), is essentially equivalent to the most likely terminal position in the intermediate regime obtained next. 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

The hydrodynamic equations can be derived from the MPC Markov chain dynamics using projection operator methods analogous to those used to obtain... 

Markov process. In the next few sections we will briefly introduce properties of Markov processes as well as equations describing Markov processes. 

In the most general case the diffusive Markov process (which in physical interpretation corresponds to Brownian motion in a field of force) is described by simple dynamic equation with noise source ... 

The transition probability density of continuous Markov process satisfies to the following partial differential equations (WXo(x, t) = W(x, t xo, to)) ... 

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