Master diffusion equation approximation

It is of importance to point out that if the right-hand side is truncated after two terms (diffusion approximation), the last relation leads to an expression similar to the familiar Fokker-Planck equation (4.116). The approximation of a master equation of a birth-death process by a diffusion equation can lead to false results. Van Kampen has critically examined the Kramers-Moyal expansion and proposed a procedure based on the concept of system size expansion.135 It can be stated that any diffusion equation can be approximated by a one-step process, but the converse is not true. 

Abundant spins with a high gyromagnetic ratio, e.g., protons, lead in the absence of line-narrowing methods to unresolved spectra described by the Hamiltonian of Equations (4.1) and (4.2). The only relevant observable is the spatial distribution of the polarization, P(r t). Using the master-equation approach (Equation (4.8)), the spatial evolution of the polarization is described by a random walk on a grid that can be approximated by a diffusion equation... 

An approximate but efficient solution is achieved by assuming a time scale separation between the fast jiggering motions within the potential wells and the slow conformational jumps. Under this assumption, projection of the diffusion operator onto a set of site functions, in the same number of the potential minima, can be performed to convert the diffusion equation into a master equation for jumps between discrete sites ... 

The theoretical method developed here provides a rigorous approach to the description of the internal dynamics of flexible aliphatic tails. The treatment is able to link the master equations used in connection with the RIS approximation to the multivariate Fokker Planck or diffusive equations, avoiding loosely defined phenomenological parameters. 

The diffusion approximation (1.5) is the nonlinear Fokker-Planck equation (VIII.2.5). In fact, we have now justified the derivation in VIII.2 by demonstrating that it is actually the first term of a systematic expansion in Q 1 for those master equations that have the property (1.1). Only under that condition is it true that the two coefficients... 

Master equations of diffusion type were characterized by the property that the lowest non-vanishing term in their -expansion is not a macroscopic deterministic equation but a Fokker-Planck equation. One may ask whether it is still possible to obtain an approximation in the form of a deterministic equation, although Q is no longer available as an expansion parameter. The naive device of omitting from the Fokker-Planck equation the term involving the second order derivatives is, of course, wrong the result would depend on which of the various equivalent forms (4.1), (4.7), (4.17), (4.18) one chooses to mutilate in this way. 

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

Section III is devoted to illustrating the first theoretical tool under discussion in this review, the GME derived from the Liouville equation, classical or quantum, through the contraction over the irrelevant degrees of freedom. In Section III.A we illustrate Zwanzig s projection method. Then, in Section III.B, we show how to use this method to derive a GME from Anderson s tight binding Hamiltonian The second-order approximation yields the Pauli master equation. This proves that the adoption of GME derived from a Hamiltonian picture requires, in principle, an infinite-order treatment. The case of a vanishing diffusion coefficient must be considered as a case of anomalous diffusion, and the second-order treatment is compatible only with the condition of ordinary... 

A great amount of stochastic physics investigates the approximation of jump processes by diffusion processes, i.e. of the master equation by a Fokker-Planck equation, since the latter is easier to solve. The rationale behind this procedure is the fact that the usual deterministic (CCD) and stochastic (CDS) models differ from each other in two aspects. The CDS model offers a stochastic description with a discrete state space. In most applications, where the number of particles is large and may approach Avogadro s number, the discreteness should be of minor importance. Since the CCD model adopts a continuous state-space, it is quite natural to adopt CCS model as an approximation for fluctuations. 

Contents A Historical Introduction. - Probability Concepts. -Markov Processes. - The Ito Calculus and Stochastic Differential Equations. - The Fokker-Planck Equatioa - Approximation Methods for Diffusion Processes. - Master Equations and Jump Processes. - Spatially Distributed Systems. - Bistability, Metastability, and Escape Problems. - Quantum Mechanical tokov Processes. - References. - Bibliogr hy. - Symbol Index. - Author Index. - Subject Index. 

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