Stochastic process, description

Under current treatment of statistical method a set of the states of the Markovian stochastic process describing the ensemble of macromolecules with labeled units can be not only discrete but also continuous. So, for instance, when the description of the products of living anionic copolymerization is performed within the framework of a terminal model the role of the label characterizing the state of a monomeric unit is played by the moment when this unit forms in the course of a macroradical growth [25]. 

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 ... 

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 begin the description of Brownian motion in terms of stochastic process. Here, we establish the link between stochastic processes and diffusion equations by giving expressions for the drift velocity and diffusivity of a stochastic process whose probability distribution obeys a desired diffusion equation. The drift velocity vector and diffusivity tensor are defined here as statistical properties of a stochastic process, which are proportional to the first and second moments of random changes in coordinates over a short time period, respectively. In Section VILA, we describe Brownian motion as a random walk of the soft generalized coordinates, and in Section VII.B as a constrained random walk of the Cartesian bead positions. 

The role in physics of probability and stochastic methods is the subject of many a profound study. We here merely make a few down-to-earth remarks about the way stochastic processes enter into the physical description of nature. 

It should be emphasized that this way of including fluctuations has no other justification than that it is convenient and bypasses a description of the noise sources, compare IX.4. It may provide some qualitative insight into the effect of noise, but does not describe its actual mechanism. For instance, fluctuations in the pumping should give rise to randomness in the coefficient a, rather than to an additive term. Yet the equation (7.6) has been the subject of extensive study and it is famous in statistical mechanics under the name of generalized Ginzburg-Landau equation. It may well serve us as an illustration for a stochastic process.510... 

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. 

In Section V the reorientation mechanism (A) was investigated in terms of the only (hat curved) potential well. Correspondingly, the only stochastic process characterized by the Debye relaxation time rD was discussed there. This restriction has led to a poor description of the submillimeter (10-100 cm-1) spectrum of water, since it is the second stochastic process which determines the frequency dependence (v) in this frequency range. The specific vibration mechanism (B) is applied for investigation of the submillimetre and the far-infrared spectrum in water. Here we shall demonstrate that if the harmonic oscillator model is applied, the small isotope shift of the R-band could be interpreted as a result of a small difference of the masses of the water isotopes. 

In order to calculate ensemble averages the explicit time-dependence of the exciton Hamiltonian is replaced by stochastic processes. If drastic changes of Jmn appear due to CC conformational transitions it is hard to apply this approach (Refs. [33] and [34] introduced a dichotomically fluctuating transfer coupling to cover such large conformational transitions). Instead, as it will be demonstrated here, it is more appropriate to directly generate the time-dependence of the exciton parameters Em and Jmn by MD simulations. Then, a microscopic account for solvent effects as well as a detailed description of solvent induced conformational transitions is possible. 

Diffusion can be considered as a stochastic or random process and described by the so-called Fokker-Planck equation adapted to Brownian motion. This equation is also known as the Smoluchowski equation. We consider the description of stochastic processes and Brownian motion in more detail in Section 11.1 and Appendix H. 

One can consider equations (3.37), (3.39) and (3.41) to be a basic system of equations for description of dynamics of entangled systems. The system can be investigated analytically in linear approximation as will be demonstrated in the ensuing chapters. However, to study these non-linear equations in complete form, one has to use numerical methods of simulation of the stochastic processes for the particle coordinates. 

The mathematical description of the modelled process uses a combination of one or more stochastic cores and phenomenological parts related to non-stochastic process components. 

After this description, we can appreciate the evolution of the liquid element in a MWPB through a continuous stochastic process. So, when the liquid element evolves through an i state, the probability of skipping to the j type evolution is written as PijaAt. Consequently, we express the probability describing the possibility for the liquid element to keep a type I evolution as ... 

If we combine all the aspects above with the descriptions of basic stochastic processes, then we can conclude that we have the case of a stochastic process with complete and random connections (see Section 4.4.1.1). 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

With turbulent combustion viewed as a random (or stochastic) process, mathematical bases are available for addressing the subject. A number of textbooks provide introductions to stochastic processes (for example, [55]). In turbulence, any stochastic variable, such as a component of velocity, temperature, or the concentration of a chemical species, which we might call v, is a function of the continuous variables of space x and time t and is, therefore, a stochastic function. A complete statistical description of a stochastic function would be provided by a probability-density functional, tf, defined by stating that the probability of finding the function in a small range i (x, t) about a particular function v(x, t) is [t (x, t)]<3t (x, t) ... 

As an example for a stochastic process consider a system evolving according to the Langevin equation in the high friction limit where inertial effects can be neglected and momenta are not required for the description of the system ... 

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view of the fact that the physical equations of motion are deterministic and this issue was discussed in Section 1.5.1. Random functions, ordered sequences of random variable, were discussed in Section 1.5.3. The focus of this chapter is a particular class of random functions, stochastic processes, for which the ordering parameter is time. Time is a continuous ordering parameter, however in many practical situations observations of the random function z(Z) are made at discrete time 0 < Zi < t2, , < tn < T. In this case the sequence z(iz) is a discrete sample of the stochastic process z(i). 

While Markovian stochastic processes play important role in modeling molecular dynamics in condensed phases, their applicability is limited to processes that involve relatively slow degrees of freedom. Most intramolecular degrees of freedom are characterized by timescales that are comparable or faster than characteristic environmental times, so that the inequality (7.53) often does not hold. Another class of stochastic processes that are amenable to analytic descriptions also in non-Markovian situations is discussed next. 

The function (Z) describes the effects of random collisions between our subsystem (henceforth referred to as system ), that may sometimes be a single particle or a single degree of freedom, and the molecules of the thermal environment ( bath ). This force is obviously a stochastic process, and a full stochastic description of our system is obtained once we define its statistical nature. 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

In a very general way, we can say that the AES is mathematically described by a stochastic process [22, 23], i.e., a random function depending on some parameter. In our case, the adsorptive energy is a random function of the position on the surface, E R), where the symbol (A) indicates a random quantity and R is the position vector on the surface whose components are (X, T). A particular realization of the stochastic process E R) is the function (X, V) represented in Fig. 10.1 (we can drop the dependence with Zq). The statistical description of such a stochastic process could be very complex. However, some simplifying assumptions, based on physical grounds, may greatly reduce this complexity. In fact, it is reasonable to assume that the surface is statistically... 

MCMC methods are essentially Monte Carlo numerical integration that is wrapped around a purpose built Markov chain. Both Markov chains and Monte Carlo integration may exist without reference to the other. A Markov chain is any chain where the current state of the chain is conditional on the immediate past state only—this is a so-called first-order Markov chain higher order chains are also possible. The chain refers to a sequence of realizations from a stochastic process. The nature of the Markov process is illustrated in the description of the MH algorithm (see Section 5.1.3.1). 

A series of probable transitions between states can be described with the Markov chain. A Markovian stochastic process is memoryless, and this is illustrated subsequently. We generate a sequence of random variables, (yo, yi, yi, ), so that each time t > 0, the next state yt+i would be sampled from a distribution P(y,+ily,), which would depend only on the current state of the chain, y,. Thus, given y, the next state y,+i would not depend additionally on the history of the chain (yo, yi, yi,---, y i). The name Markov chain is used to describe this sequence, and the transition kernel of the chain is i (.l.) does not depend on t if we assume that the chain is time homogeneous. A detail description of the Markov model is provided in Chapter 26. 

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