Markov diffusion

The connection between a diffusion equation and a corresponding Markov diffusion process may be established through expressions for drift velocities and diffusitivies. The drift velocity for both unconstrained and constrained systems may be expressed in an arbitrary system of coordinates in the generic form... 

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. 

Throughout this section, we will use the notation X (t),..., X t) to denote a unspecified set of L Markov diffusion processes when discussing mathematical properties that are unrelated to the physics of constrained Brownian motion, or that are not specific to a particular set of variables. The variables refer specifically to soft coordinates, generalized coordinates for a system of N point particles, and Cartesian particle positions, respectively. The generic variables X, ..., X will be indexed by integer variables a, p,... = 1,...,L. 

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... 

Also, under continuous CO oxidation conditions, alkaline media exhibit a much higher activity than acidic media. Markovic and co-workers observed a shift of about 150 mV of the main oxidation wave, and a pre-wave corresponding to CO oxidation at potentials as low as 0.2-0.3 V [Markovic et al., 2002]. Remarkably, the hysteresis that is so prominently observed in the diffusion-controlled CO oxidation wave in acidic media (see Fig. 6.9), is no longer present in alkaline media. Markovic and co-workers also attribute the high activity of alkaline media to a pH-dependent adsorption of OH ds at defect/step sites. 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

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. 

A continuous Markov process (also known as a diffusive process) is characterized by the fact that during any small period of time At some small (of the order of %/At) variation of state takes place. The process x(t) is called a Markov process if for any ordered n moments of time t < < t < conditional probability density depends only on the last fixed value ... 

In the most general case the diffusive Markov process (which in physical interpretation corresponds to Brownian motion in a field of force) is described by simple dynamic equation with noise source ... 

The first approach to obtain exact time characteristics of Markov processes with nonlinear drift coefficients was proposed in 1933 by Pontryagin, Andronov, and Vitt [19]. This approach allows one to obtain exact values of moments of the first passage time for arbitrary time constant potentials and arbitrary noise intensity moreover, the diffusion coefficient may be nonlinear function of coordinate. The only disadvantage of this method is that it requires an artificial introducing of absorbing boundaries, which change the process of diffusion in real smooth potentials. 

Statistical models based on data correlations and on Markov chains are being actively developed and their fidelity evaluated by several research groups. Photochemical-diffusion models based on deterministic equations are also being developed, but because of their complexity will probably be used only as research tools for some time. 

Brownian motion of a constrained system of N point particles may also be described by an equivalent Markov process of the Cartesian bead positions R (f),..., R (f). The constrained diffusion of the Cartesian coordinates may be characterized by a Cartesian drift velocity vector and diffusivity tensor... 

This algorithm differs from the Markov process used in Section IX to define a kinetic SDE only in that an explicitly predicted midstep position, rather than an implicitly defined midstep position, is used to calculate the midstep velocity for the final update. To the accuracy needed to calculate the drift velocity and diffusivity, the analyses of the explicit and implicit midstep schemes are identical. As a result, the preceding calculation of the diffusivity and drift for a kinetic SDE also applies to this algorithm. 

So far we studied the first passage of Markov processes such as described by the Smoluchowski equation (1.9). On a finer time scale, diffusion is described by the Kramers equation (VIII.7.4) for the joint probability of the position X and the velocity V. One may still ask for the time at which X reaches for the first time a given value R, but X by itself is not Markovian. That causes two complications, which make it necessary to specify the first-passage problem in more detail than for diffusion. 

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. 

Fig. 5.13 Typical Markov lattice used to model the SAXS patterns for the PE-PVCH diblock in Fig. 5.11 (Hamley et al. 1996b). The disorder within and between stacks of linear arrangements of crystallites (represented as points) leads to diffuse scattering.

A very important application of the Markov dynamics is random walk. In the special case of random walk/(x) = 0 and g(x) = 1, then the diffusion equation for a random walk in one dimension is... 

Fig. 1. An illustration of the potential governing molecular motion in a system. The nucleus may hop between different potential wells, and in addition oscillate within a well. The often-used Markov model assumes that nuclei hop between discrete sites and that the length of time taken to hop is very small compared with the residence time in each site. Such a model can only account effectively for hops between the minima of the potential wells in this illustration diffusion within wells cannot be treated properly.

The difference between the Markov model lineshapes and those from the Smoluchowski model is particularly pronounced when the diffusion coefficient is of the order of the quadrupole coupling constant. In the limit of large diffusion coefficients, the two models converge, and in the limit of low diffusion coefficients, the spectra are dominated by small-amplitude oscillations within potential wells, which can be approximately modelled by a suitable Markov model. This work strongly suggests that there could well be cases where analysis of powder pattern lineshapes with a Markov model leads to a fit between experimental and simulated spectra but where the fit model does not necessarily describe the true dynamics in the system. 

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