Markov stochastic process

Since these characteristics are time-dependent, let us assume particle birth-death and migration to be the Markov stochastic processes. Note that making use of the stochastic models, we discuss below in detail, does not contradict the deterministic equations employed for these processes. Say, the equations for nv t), Xu(r,t), Y(r,t) given in Section 2.3.1 are deterministic since both the concentrations and joint correlation functions are defined by equations (2.3.2), (2.3.4) just as ensemble average quantities. Note that the... 

If we consider the evolution of the liquid element together with the state of probabilities of elementary evolutions, we can observe that we have a continuous Markov stochastic process. If we apply the model given in Eq. (4.68), Pj(z, t) is the probability of having the liquid element at position x and time t evolving by means of a type 1 elementary process (displacement with a d-v flow rate along a positive direction of x). This probability can be described through three independent events ... 

In this section we remind the reader of the Kolmogorov forward and backward equations, infinitesimal generators, stochastic differential equations, and functional integrals and then consider how the basic transport equations are related to underlying Markov stochastic processes [141, 142],... 

This set of assumptions on the statistical properties of f(t) determines the statistical properties of the solution v(0 of the stochastic differaitial equation in Equation 1.1, which are summarized saying that v(0 is a Gaussian stationary Markov stochastic process, that is, it is generally not delta-correlated. The specific results that follow from this simple mathanatical model regarding propo ties such as the velocity autocorrelation function, msd, and so on, are reviewed in standard statistical physics textbooks [48]. 

This means that for a Markov stochastic process the probability of each outcome depends only on the immediately previous outcome. 

In the framework of this ultimate model [33] there are m2 constants of the rate of the chain propagation kap describing the addition of monomer to the radical Ra whose reactivity is controlled solely by the type a of its terminal unit. Elementary reactions of chain termination due to chemical interaction of radicals Ra and R is characterized by m2 kinetic parameters k f . The stochastic process describing macromolecules, formed at any moment in time t, is a Markov chain with transition matrix whose elements are expressed through the concentrations Ra and Ma of radicals and monomers at this particular moment in the following way [1,34] ... 

This is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

In order to obtain the expression for the components of the vector of instantaneous copolymer composition it is necessary, according to general algorithm, to firstly determine the stationary vector ji of the extended Markov chain with the matrix of transitions (13) which describes the stochastic process of conventional movement along macromolecules with labeled units and then to erase the labels. In this particular case such a procedure reduces to the summation ... 

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 aim of this chapter is to describe approaches of obtaining exact time characteristics of diffusion stochastic processes (Markov processes) that are in fact a generalization of FPT approach and are based on the definition of characteristic timescale of evolution of an observable as integral relaxation time [5,6,30—41]. These approaches allow us to express the required timescales and to obtain almost exactly the evolution of probability and averages of stochastic processes in really wide range of parameters. We will not present the comparison of these methods because all of them lead to the same result due to the utilization of the same basic definition of the characteristic timescales, but we will describe these approaches in detail and outline their advantages in comparison with the FPT approach. 

For many synthetic copolymers, it becomes possible to calculate all desired statistical characteristics of their primary structure, provided the sequence is described by a Markov chain. Although stochastic process 31 in the case of proteinlike copolymers is not a Markov chain, an exhaustive statistic description of their chemical structure can be performed by means of an auxiliary stochastic process 3iib whose states correspond to labeled monomeric units. As a label for unit M , it was suggested [23] to use its distance r from the center of the globule. The state of this stationary stochastic process 31 is a pair of numbers, (a, r), the first of which belongs to a discrete set while the second one corresponds to a continuous set. Stochastic process ib is remarkable for being stationary and Markovian. The probability of the transition from state a, r ) to state (/i, r") for the process of conventional movement along a heteropolymer macromolecule is described by the matrix-function of transition intensities... 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

In both the Ito and Stratonovich formulations, the randomness in a set of SDEs is generated by an auxiliary set of statistically independent Wiener processes [12,16]. The solution of an SDE is defined by a hmiting process (which is different in different interpretations) that yields a unique solution to any stochastic initial value problem for each possible reahzation of this underlying set of Wiener processes. A Wiener process W t) is a Gaussian Markov diffusion process for which the change in value W t) — W(t ) between any two times t and t has a mean and variance... 

For a general definition of multidimensional Markov chains see, for example, N. Bailey, Elements of Stochastic Processes, Wiley, New York, 1964, Chap. 10. 

This chapter defines and describes the subclass of stochastic processes that have the Markov property. Such processes are by far the most important in physics and chemistry. The rest of this book will deal almost exclusively with Markov processes. 

The extraction of a homogeneous process from a stationary Markov process is a familiar procedure in the theory of linear response. As an example take a sample of a paramagnetic material placed in a constant external magnetic field B. The magnetization Y in the direction of the field is a stationary stochastic process with a macroscopic average value and small fluctuations around it. For the moment we assume that it is a Markov process. The function Px (y) is given by the canonical distribution... 

Owing to this reduction to a Markov process the model can again be treated in full detail. A stochastic process that can be made Markovian by means of one additional variable is said to be Markovian of the second degree , and if more variables are needed it is Markovian of some higher degree. 

Many stochastic processes are of a special type called birth-and-death processes or generation-recombination processes . We employ the less loaded name one-step processes . This type is defined as a continuous time Markov process whose range consists of integers n and whose transition matrix W permits only jumps between adjacent sites,... 

The stochastic function X(t) by itself is not Markovian. This is an example of the fact discussed in IV. 1 If one has an r-component Markov process and one ignores some of the components, the remaining sstochastic process but in general not Markovian. Conversely, it is often possible to study non-Marko-vian processes by regarding them as the projection of a Markov process with more components. We return to this point in IX.7. 

Conclusion. In classical statistical mechanics the evolution of a many-body system is described as a stochastic process. It reduces to a Markov process if one assumes coarse-graining of the phase space (and the repeated randomness assumption). Quantum mechanics gives rise to an additional fine-graining. However, these grains are so much smaller that they do not affect the classical derivation of the stochastic behavior. These statements have not been proved mathematically, but it is better to say something that is true although not proved, than to prove something that is not true. 

The knowledge of the two-minima energy surface is sufficient theoretically to determine the microscopic and static rate of reaction of a charge transfer in relation to a geometric variation of the molecule. In practice, the experimental study of the charge-transfer reactions in solution leads to a macroscopic reaction rate that characterizes the dynamics of the intramolecular motion of the solute molecule within the environment of the solvent molecules. Stochastic chemical reaction models restricted to the one-dimensional case are commonly used to establish the dynamical description. Therefore, it is of importance to recall (1) the fundamental properties of the stochastic processes under the Markov assumption that found the analysis of the unimolecular reaction dynamics and the Langevin-Fokker-Planck method, (2) the conditions of validity of the well-known Kramers results and their extension to the non-Markovian effects, and (3) the situation of a reaction in the absence of a potential barrier. 

A Markov process is a stochastic process, where the time dependence of the probability, P(x, t)dx, that a particle position at time, t, lies between x and x+dx depends only on the fact that x=x(l at t = t0, and not on the entire history of the particle movement. In this regard, the Fokker-Planck equation [11]... 

Markov process A stochastic process in which the next state of a system depends solely on the previous state. 

We begin the discussion by referring to the stochastic model given by relation (4.58), which is rewritten here as shown in relation (4.120). Here for a finite Markov connection process we must consider the constant time values for all the elements of the matrix P = pij i k-... 

Some restrictions are imposed when we start the application of limit theorems to the transformation of a stochastic model into its asymptotic form. The most important restriction is given by the rule where the past and future of the stochastic processes are mixed. In this rule it is considered that the probability that a fact or event C occurs will depend on the difference between the current process (P(C) = P(X(t)e A/V(X(t))) and the preceding process (P (C/e)). Indeed, if, for the values of the group (x,e), we compute = max[P (C/e) — P(C)], then we have a measure of the influence of the process history on the future of the process evolution. Here, t defines the beginning of a new random process evolution and tIt- gives the combination between the past and the future of the investigated process. If a Markov connection process is homogenous with respect to time, we have = 1 or Tt O after an exponential evolution. If Tt O when t increases, the influence of the history on the process evolution decreases rapidly and then we can apply the first type limit theorems to transform the model into an asymptotic... 

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