Diffusion as a Stochastic Process

The retardation of the protein movement has been discussed qualitatively in terms of a sieving mechanism rather than a frictional resistance37). Ogston et al.39) have theoretically described the diffusion as a stochastic process in which the particles move by unit displacements and in which the decrease in the rate of diffusion in a polymer network depends on the probability that a particle finds a hole in the network into which it can move. The relationship derived from this approach is in close agreement with Eq. (35). 

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. 

Ion transfer in solid electrolytes can also be approached as a stochastic process. The transfer of carriers occurs by their continuous hopping in different lattice positions. In the case of a three-dimensional isotropic crystal, carriers move randomly. The diffusion coefficient is related to the hopping distance r and the hopping frequency o. 

Markov process A stochastic process is a random process in which the evolution from a state X(t ) to X(t +i) is indeterminate (i.e. governed by the laws of probability) and can be expressed by a probability distribution function. Diffusion can be classified as a stochastic process in a continuous state space (r) possessing the Markov property as... 

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. 

Here, X. is the stochastic state vector, B(r,X.j) is a vector describing the contribution of the diffusion to the stochastic process and W. is a vector with the same dimensions as X. and B(t,X.j). After Eqs. (4.94) and (4.95), the W,. vector is a Wiener process (we recall that this process is stochastic with a mean value equal to zero and a gaussian probability distribution) with the same dimensions as D(t,X,) ... 

This stochastic model of the flow with multiple velocity states cannot be solved with a parabolic model where the diffusion of species cannot depend on the species concentration as has been frequently reported in experimental studies. Indeed, for these more complicated situations, we need a much more complete model for which the evolution of flow inside of system accepts a dependency not only on the actual process state. So we must have a stochastic process with more complex relationships between the elementary states of the investigated process. This is the stochastic model of motion with complete connections. This stochastic model can be explained through the following example we need to design some flowing liquid trajectories inside a regular porous structure as is shown in Fig. 4.33. The porous structure is initially filled with a fluid, which is non-miscible with a second fluid, itself in contact with one surface of the porous body. At the... 

The limitations associated with (7) are essentially a consequence of the stochastic nature of atmospheric transport and diffusion. Because the wind velocities are random functions of space and time, the airborne pollutant concentrations are random variables in space and time. Thus, the determination of the Cj, in the sense of being a specified quantity at any time, is not possible, but we can at best derive the probability density functions satisfied by the c. The complete specification of the probability density function for a stochastic process as complex as atmospheric diffusion is almost never possible. Instead, we must adopt a less desirable but more feasible approach, the determination of certain statisical moments of Ci, notably its mean, . (The mean concentration can be... 

In order to understand and model the effect of absorbed moisture on adhesives we need to investigate moisture transport mechanisms and how they can be represented mathematically. Mass (or molecular) diffusion describes the transport of molecules from a region of higher concentration to one of lower concentration. This is a stochastic process, driven by the random motion of the molecules. This results in a time-dependent mixing of material, with eventual complete mixing as equilibrium is reached. Mass diffusion is analogous to other types of diffusion, such as heat diffusion, which describes conduction of heat in a solid material, and hence similar mathematical representations can be made. 

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. 

An aimless path resulting from a set of successive steps (or excursions) whose individual lengths and directions are only randomly related to each other. As with any stochastic process, the preceding step has no influence at all on any of the subsequent steps. See Diffusion Persistence Time Twiddling... 

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. 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

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