Properties Of Diffusion Processes Research Articles

Non-Gaussian diffusion of mixed origins

The properties of diffusion processes are drastically affected by heterogeneities of the medium that can induce non-Gaussian behavior of the propagator in contrast with the idealized realm of Brownian motion. In this paper we analyze the diffusion propagator when distinct origins of heterogeneity (e.g. time-fractional diffusion, diffusing diffusivity, distributed diffusivity across a population) are combined. These combinations allow one to describe new classes of strongly heterogeneous processes relevant to biology. Based on a combined subordination technique, we obtain the exact propagator for different instructive examples. This approach is then used to calculate analytically the first-passage time statistics (on half-real line and in any bounded domain) for a particle undergoing non-Gaussian diffusion of mixed origins.

Spatial parity violation and the turbulent magnetic Prandtl number

Using the field theory renormalization-group technique in the two-loop approximation, we study the influence of helicity (spatial parity violation) on the turbulent magnetic Prandtl number in the model of kinematic magnetohydrodynamic turbulence, where the magnetic field behaves as a passive vector quantity advected by the helical turbulent environment given by the stochastic Navier-Stokes equation. We show that the presence of helicity decreases the value of the turbulent magnetic Prandtl number and that the two-loop helical contribution to the turbulent magnetic Prandtl number is up to 4.2% of its nonhelical value. This result demonstrates the strong stability of the properties of diffusion processes of the magnetic field in turbulent environments with spatial parity violation compared with the corresponding systems without the helicity.

Turbulent magnetic Prandtl number in helical kinematic magnetohydrodynamic turbulence: Two-loop renormalization group result

Using the field theoretic renormalization group technique, the influence of helicity (spatial parity violation) on the turbulent magnetic Prandtl number in the kinematic magnetohydrodynamic turbulence is investigated in the two-loop approximation. It is shown that the presence of helicity decreases the value of the turbulent magnetic Prandtl number and, at the same time, the two-loop helical contribution to the turbulent magnetic Prandtl number is at most 4.2% (in the case with the maximal helicity) of its nonhelical value. These results demonstrate, on one hand, the potential importance of the presence of asymmetries in processes in turbulent environments and, on the other hand, the rather strong stability of the properties of diffusion processes of the magnetic field in the conductive turbulent environment with the spatial parity violation in comparison to the corresponding systems without the spatial parity violation. In addition, obtained results are compared to the corresponding results found for the two-loop turbulent Prandtl number in the model of passively advected scalar field. It is shown that the turbulent Prandtl number and the turbulent magnetic Prandtl number, which are the same in fully symmetric isotropic turbulent systems, are essentially different when one considers the spatial parity violation. It means that the properties of the diffusion processes in the turbulent systems with a given symmetry breaking can considerably depend on the internal tensor structure of advected quantities.

A probability-based method for calculating effective diffusion coefficients of composite media

Abstract A probability-based method is proposed for calculating the concentration of chloride at specified points of deterministic and random heterogeneous microstructures. The method, referred to as local method, is used to calculate effective diffusion coefficients of composite media. The effective diffusion coefficient of a heterogeneous material specimen is defined as the diffusion coefficient of a fictitious homogeneous specimen with the same global properties as those of the original specimen. The local method is based on properties of diffusion processes and Ito’s formula; it is simple to code, always stable, accurate and suitable for parallel computing. We use the method to calculate effective diffusion coefficients of a material specimen with a deterministic diffusivity field and a random heterogeneous mortar specimen calibrated to an experiment reported in Ref. Care (2003) [11] .

Dissipativity of diffusion Itô processes with Markovain switching and problems of robust stabilization

Consideration is given to a class of systems described by a finite set of controlled diffusion Ito processes that are control-affine, with jump transitions between them, and are defined by the evolution of a uniform Markovian chain (Markovian switching). Each state of this chain corresponds to a certain system mode. A stochastic version of the notion dissipativity by Willems is introduced, and properties of diffusion processes with Markovian switching are studied. The relationship between passivity and stabilizability in the process of output-feedback control is established. The obtained results are applied to the problem of robust simultaneous stabilization for the set of nonlinear systems with undetermined parameters. As a partial case, a problem of robust simultaneous stabilization for the set of linear systems where final results are obtained in terms of linear matrix inequalities.

Local solution for a class of mixed boundary value problems

A local method is developed for solving locally partial differential equations with mixed boundary conditions. The method is based on a heuristic idea, properties of diffusion processes, stopping times and the Ito formula for semimartingales. According to the heuristic idea, the diffusion process used for solving locally a partial differential with mixed boundary conditions is stopped when it reaches a Neumann boundary and then restarted inside the domain of definition of this equation at a point depending on the Neumann conditions. The proposed method is illustrated and its accuracy assessed by two simple numerical examples solving locally mixed boundary value problems in one and two space dimensions.

A local solution of the Schrödinger equation

A local method is developed for solving the Schrodinger equation. The method is local in the sense that it can determine the value of the solution of the Schrodinger equation at an arbitrary point directly rather than extracting this value from the field solution. The method is based on properties of diffusion processes, the Ito formula, and Monte Carlo simulation. Simplicity, accuracy, and generality are the main features of the proposed local solution. The extension of the proposed method to solve the stochastic version of the Schrodinger equation is elementary. Two examples with Dirichlet and Neumann boundary conditions are presented to demonstrate the application and evaluate the accuracy of the proposed local solution.

Local Solutions of Laplace, Heat, and Other Equations by Itô Processes

A method is presented for finding the solution of deterministic partial differential equations at an arbitrary point of the domain of definition of these equations referred to as the local solution. The method is based on the Ito calculus, properties of diffusion processes, and Monte Carlo simulation. The theoretical background of the proposed method is relatively difficult. However, the method has attractive features for applications. For example, the numerical algorithms based on the proposed method are simple, stable, accurate, local, and ideal for parallel computation. Numerical examples from mechanics are presented to demonstrate the use and the accuracy of the proposed method.

Vlastnosti difusních procesů určených obecnou laterální podmínkou

Vlastnosti difusních procesů určených obecnou laterální podmínkou