Stochastic, equations processes

Equation (57) is the stochastic equation of motion for r(/), in which the matrix element 2(/) is a random process. This is similar to Eq. (2). This may be written as a stochastic Liouville equation in the form... 

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... 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

The theory of Brownian motion is a particular example of an application of the general theory of random or stochastic processes [2]. Since Kramers approach is based on a more general stochastic equation than the Langevin equation, we have reviewed some of the fundamental ideas and methods of the theory of stochastic processes in Appendix H. 

The theory of relaxation processes for a macromolecular coil is based, mainly, on the phenomenological approach to the Brownian motion of particles. Each bead of the chain is likened to a spherical Brownian particle, so that a set of the equation for motion of the macromolecule can be written as a set of coupled stochastic equations for coupled Brownian particles... 

The random process in the last stochastic equation from set (3.37) is related to the above introduced random process by equation... 

The establishment of stochastic equations frequently results from the evolution of the analyzed process. In this case, it is necessary to make a local balance (space and time) for the probability of existence of a process state. This balance is similar to the balance of one property. It means that the probability that one event occurs can be considered as a kind of property. Some specific rules come from the fact that the field of existence, the domains of values and the calculation rules for the probability of the individual states of processes are placed together in one or more systems with complete connections or in Markov chains. 

A careful observation of Eqs. (4.79), (4.80), (4.100) and their respective theoretical basis [4.44, 4.45], allows one to conclude that the probability density distribution that describes the fact that the particle is in position x at t time, when the medium is moving according to one stochastic diffusion process (see relation (4.62) for the analogous discontinuous process), is given by Eq. (4.111). This relation is known as the Eokker-Planck-Kolmogorov equation. 

Both equations give good results for the description of mass and heat transport without forced flow. Here, it is important to notice that the Fokker-Plank-Kolmo-gorov equation corresponds to a Markov process for a stochastic connection. Consequently, it can be observed as a solution to the stochastic equations written below ... 

The diffusion model can usually be used for the description of many stochastic distorted models. The equivalent transformation of a stochastic model to its associated diffusion model is fashioned by means of some limit theorems. The first class of limit theorems show the asymptotic transformation of stochastic models based on polystochastic chains the second class is oriented for the transformation of stochastic models based on a polystochastic process and the third class is carried out for models based on differential stochastic equations. 

Another very important feature of the stochastic equations considered here, when they are subjected to RMT analysis, is their resemblance to the general formalism arrived at in the thermodynamics of nonequilibrium processes this suggests an analogy between the effects of multiplicative noise and the continuous flux of energy which maintains the systems far from equilibrium. This is considered the main characteristic of self-organizing living systems and means that multiplicative stochastic models could take on a new and fundamentally important role. 

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. 

A third approach for treating trapping was developed by Burshtein ) and is based on treating the transfer rate as a random variable in a stochastic hopping process. The host luminescence emission is then governed by the equation... 

The stochastic equation for ji [Eq. (8.A.1)] is used along with suitable statistical assumptions to evaluate the moments A and B. We assume that the random force in Eq. (8.A.1) is a Gaussian random process with the average values... 

If we further assume that dynamics of n is a Markov process, then the "diffusional" chemical kinetic equation, Eq. (4) or (6), acquires the form of the chemical stochastic equation [54,55] (CSE),... 

Summarizing, we place a probability function for the position steps the position with a stochastic process. While in the deterministic case a fixed increment is added, in the stochastic case the generating function is multiplied by an increment. The assigned equation for the probability generating function to the stochastic equation X (t + At) = KAt + X (t) is therefore... 

In diluted polymer solution macromolecules are in random environment, influencing on the process of either aggregation or degradation (more rarely on both of them) that results in the stochastic equation instead of the Eq. (6) [13] ... 

In paper [102] the theoretical treatment of a cluster-cluster aggregation accounting for the existence of coalescing of particles or of clusters (monomers or macromolecular coils) to aggregate in real polymerization processes, and their disconnection (destruction) was reported. Macromolecules are in a random environment, influencing the processes of aggregation or destmction in dilnte polymeric solution that is described by the stochastic equation [101] ... 

To have a stochastic differential equation model for both the internal and external fluctuations in complex chemical reactions would have advantages other than providing the framework for estimation. As it turned out from the analysis of Chapter 1 it would be necessary to have a common model of macroscopic phenomena that does include stochasticity, spatial processes (such as diffusion) and sources and sinks (such as reactions). These topics will be further analysed in Section 6.3. 

Set (7.1) consists of two vector and one scalar stochastic equation. This set is helpful in examining statistical properties of two vector (v and w ) and one scalar (p ) random unknown variable as functions of 1) the statistical properties of random variable 2) the macroscopic characteristics of suspension flow, and 3) physical parameters. Since Equations 7.1 are linear, it is natural to use the correlation theory of random processes when investigating these vector and scalar variables in terms of those functions [35], Particulars of necessary calculation are described at considerable length in reference [14,25). Here, we confine ourselves to only a brief enumeration of the major logical steps of this calculation. 

Instead of solving the evolution equation in terms of the orientation tensor, one can simulate the stochastic equation such as Eq. 5.7 for the orientation vector p without the need of closure approximations, using the numerical technique for the simulation of stochastic processes (Ottinger 1996) known as the Brownian dynamics simulation. Once trajectories for aU fibers are obtained, the orientation tensor can be calculated in terms of the ensemble average of the discrete form ... 

Abstract We numerically study the pair-annihilation process of magnetic vortices in two-dimensional type-II superconductors. The dynamics for interacting vortices is described in terms of the Langevin-type stochastic equation. Carrying out the Langevin dynamics simulation, we find that the power-law behavior of the mean distance among vortices exhibits a crossover phenomenon as... 

On the basis of the definition of mutation frequency given in equation (6), nonlinearities in frequency curves would be expected to arise only from the existence of multilesion mutation-induction processes. However, nonlinearities in mutation-frequency curves can arise if mutation and killing are not stochastically independent processes as assumed in Section 3 and in writing equation (6). We can generalize the formalism of Section 3 and 4 to allow for stochastic dependence of mutation and killing in the following way ... 

III.3.2 Biasing the Reptation Model The reptation process is fully defined by the four elements present in the basic stochastic equations (8) and (9). These are the probabilities that the next Jump will be made towards the end of the tube, the time duration and the length 0 =0 of this Jump, and finally the vector c( ) giving the properties of the new tube section created by this Jump. The fact that p =i, x =XBrown (c(0>=0 in absence of a field reflects the fact that we then have Brownian motion. The tube concept and the assumption of a constant pore size a are implicit in Eq.(8). 

Consider the stochastic translocation process given by Equation 10.39. Let mo be the number of monomers, which have been nucleated in the receiver compartment, to begin with. Starting from this initial condition, the probability distribution function of the first passage time r is given by Equation 6.86. Let us now consider the key results for the boundary conditions BCl and BC2 (Table 6.1). The results for the radiation boundary condition BCi can be obtained similarly by looking up the results in Section 6.7.4. 

The stochastic equations of motion of the dynamic system subjected to a Poisson impulse process excitation are... 

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