Markovian processes equations

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

The non-equivalence of the statistical and kinetic methods Is given by the fact that the statistical generation Is always a Markovian process yielding a Markovian distribution, e.g. In case of a blfunc-tlonal monomer the most probable or pseudo-most probable distributions. The kinetic generation Is described by deterministic differential equations. Although the Individual addition steps can be Markovian, the resulting distribution can be non-Markovian. An Initiated step polyaddltlon can be taken as an example the distribution Is determined by the memory characterized by the relative rate of the Initiation step ( ). ... 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

Northrup and Hynes [103] solved these equations for the case of a diffusion model and found the same results as Collins and Kimball [4] of eqn. (25). This case is reasonably easy to solve because the diffusion and reaction of the pair can be separated. When the motion of the pair involves a non-Markovian process, that is the reactants recall which direction they were moving a moment before (i.e. have a memory ) and the process is not diffusional, this elegant separation becomes very difficult or impossible to effect. Under these circumstances, eqn. (368) can only be solved approximately for the pair probability. The initial condition term, l(t), is non-zero if the initial distribution p(0) is other than peq. 

The stochastic equation of motion of v(t), Eq. (77), can be transformed into a stochastic Liouville equation of the type Eq. (7) if a Markovian process can be properly defined to generate the process of H(t). Then we again obtain Eq. (63) for the conditional expectation V(t) defined by Eq. (60). The line shape function is then given by... 

One can derive Eqn. (12,12) in a more fundamental way by starting the statistical approach with the (Markovian) master equation, assuming that the jump probabilities obey Boltzmann statistics on the activation saddle points. Salje [E. Salje (1988)] has discussed the following general form of a kinetic equation for solid state processes... 

First of all, we define the transition rates for our stochastic model using an ansatz of Kawasaki [39, 40]. In the following we use the abbreviation X for an initial state (07 for mono- and oion for bimolecular steps), Y for a final state (ct[ for mono- and a[a n for bimolecular steps) and Z for the states of the neighbourhood ( cr f 1 for mono- and a -1 a -1 for bimolecular steps). If we study the system in which the neighbourhood is fixed we observe a relaxation process in a very small area. We introduce the normalized probability W(X) and the corresponding rates 8.(X —tY Z). For this (reversible) process we write down the following Markovian master equation... 

Dekker has studied multiplicative stochastic processes. In his work the stochastic Liouville equation was solved explicitly through first order in an expansion in terms of correlation times of the multiplicative Gaussian colored noise for a general multidimensional weakly non-Markovian process. He followed the suggestions of refs. 17 and 18 and applied, Novikov s theorem. In the general multidimensional case, however, he improved the earlier work by San Miguel and Sancho. ... 

Equations (5.7) were introduced so as to treat the non-Markovian process of Eq. (5.8) in the frame of the time-independent Fokker-Planck formalism. The equivalence has been shown to require that the fluctuation-dissipation relationship (5.10) holds the white noise limit can then be recovered by making t vanish for a fixed value of D. If we substitute Eq. (5.10) into Eqs. 

For this (reversible) process we write down the following Markovian master equation... 

Clearly, this approach is straightforward only for ultimate group or terminal group copolymerization. If more than penultimate effects are required, the equations become unwieldy. Using the concepts and mathematics of Markovian processes, Ham (9) generalized an extended form of the selectivity equation. Later, Price (20, 21) formalized the theory of Markov chains as applied to... 

Equation (19) describes a non-Markovian process in the CV space. In fact, the forces acting on the CVs depend explicitly on their history. Due to this non-Markovian nature, it is not clear if, and in which sense, the system can reach a stationary state under the action of this dynamics. In [32] we introduced a formalism that allows to map this history-dependent evolution into a Markovian process in the original variable and in an auxiliary field that keeps track of the visited configurations. Defining... 

Markov processes have no memory of earlier information. Newton equations describe deterministic Markovian processes by this definition, since knowledge of system state (all positions and momenta) at a given time is sufficient in order to determine it at any later time. The random walk problem discussed in Section 7.3 is an example of a stochastic Markov process. 

It is important to point out that this does not imply that Markovian stochastic equations cannot be used in descriptions of condensed phase molecular processes. On the contrary, such equations are often applied successfully. The recipe for a successful application is to be aware of what can and what cannot be described with such approach. Recall that stochastic dynamics emerge when seeking coarsegrained or reduced descriptions of physical processes. The message from the timescales comparison made above is that Markovian descriptions are valid for molecular processes that are slow relative to environmental relaxation rates. Thus, with Markovian equations of motion we cannot describe molecular nuclear motions in detail, because vibrational periods (10 " s) are short relative to environmental relaxation rates, but we should be able to describe vibrational relaxation processes that are often much slower, as is shown in Section 8.3.3. 

The starting point of the Kramers theory of activated rate processes is the onedimensional Markovian Langevin equation, Eq. (8.13)... 

The Metropolis MC [41] was originally developed as a method suited to electronic computers for calculating statistical mechanical configurational integrals. Since the MC sampling is a Markovian process, if we introduce a time scale t that actually labels the order of subsequent configurations X, the dynamic evolution of the probability distribution function P(X, t) is governed by the master equation... 

The memory kernel in (2.59), recall that v] represents a nonlocal-in-time integral operator, is a clear indication that subdiffusive transport is non-Markovian. Incorporating kinetic terms into a non-Markovian transport equation requires great care and is best carried out at the mesoscopic level. We show in Sect. 3.4 how to proceed directly at the level of the mesoscopic balance equations for non-Markovian CTRWs. Here we pursue a different approach. As stated above, if all processes are Markovian, then contributions from different processes are indeed separable and simply additive. As is well known, processes often become Markovian if a sufficiently large and appropriate state space is chosen. For the case of reactions and subdiffusion, the goal of a Markovian description can be achieved by taking the age structure of the system explicitly into account as done by Vlad and Ross [460,461]. This approach is equivalent to Model B, see Sect. 3.4. 

An immediate application of those properties is constituted by the equation Chapman-Komogorov-Smoluchowski (CKS), in feet the equation on defining the Markovian processes. For example, it can be immediately write that ... 

In the polymerization of 1,5-HD, 1,2-addition and cyclization reactions can be described by Equations 1.1 through 1.3 (where r = the rate of reaction), on the condition that the 1,2-addition obeys a first-order Markovian process and the cyclization follows a Bernoullian process... 

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 ... 

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