Diffusion Matrix Research Articles

Generalized result with a unified proof of global existence to a class of reaction‐diffusion systems

We prove in this paper a generalized result with a unified proof of global existence in time of classical solutions to a class of a reaction diffusion system with triangular diffusion matrix on a bounded domain in . The system in question is ut=aΔu − f(x,t,u,v), vt=cΔu + dΔv + ρf(x,t,u,v), , t > 0 with f(x,t,0,η) = 0 and f(x,t,ξ,η)≤Kφ(ξ)eση, for all x∈Ω, t > 0, ξ≥0, η≥0; where ρ, K and σ are real positive constants. Copyright © 2016 John Wiley & Sons, Ltd.

Near-wall diffusion tensor of an axisymmetric colloidal particle.

Hydrodynamic interactions with confining boundaries often lead to drastic changes in the diffusive behaviour of microparticles in suspensions. For axially symmetric particles, earlier numerical studies have suggested a simple form of the near-wall diffusion matrix which depends on the distance and orientation of the particle with respect to the wall, which is usually calculated numerically. In this work, we derive explicit analytical formulae for the dominant correction to the bulk diffusion tensor of an axially symmetric colloidal particle due to the presence of a nearby no-slip wall. The relative correction scales as powers of inverse wall-particle distance and its angular structure is represented by simple functions in sines and cosines of the particle's inclination angle to the wall. We analyse the correction for translational and rotational motion, as well as the translation-rotation coupling. Our findings provide a simple approximation to the anisotropic diffusion tensor near a wall, which completes and corrects relations known from earlier numerical and theoretical findings.

Numerical Study of Contaminant Transport in Fractured Porous Rock with Distance Dependent Dispersivity

This paper presents solute concentration profiles with constant, linear and exponential distance-dependent dispersion models for solute transport through fractured porous rock matrices. A numerical analysis has been developed for solution of the equations governing the dispersion model using the hybrid finite volume method. The results of the numerical analysis have been used to investigate the effect of the matrix diffusion coefficient, fracture velocity, and matrix and fracture retardation factors on concentration profiles. It is found that in the presence of higher matrix diffusion coefficients, the behaviour of concentration profiles is different for constant and distance-dependent dispersion models. However, the behaviour of concentration profiles is observed to be nonlinear during the small transport time period, and the matrix diffusion is found to control the spreading of solute in the fracture in the presence of constant and distance-dependent dispersion models.

Matrix coefficient identification in an elliptic equation with the convex energy functional method

In this paper we study the inverse problem of identifying the diffusion matrix in an elliptic PDE from measurements. The convex energy functional method with Tikhonov regularization is applied to tackle this problem. For the discretization we use the variational discretization concept, where the PDE is discretized with piecewise linear, continuous finite elements. We show the convergence of approximations. Using a suitable source condition, we prove an error bound for discrete solutions. For the numerical solution we propose a gradient-projection algorithm and prove the strong convergence of its iterates to a solution of the identification problem. Finally, we present a numerical experiment which illustrates our theoretical results.

Curl flux induced drift in stochastic differential equations in the zero-mass limit

Abstract We consider the nonlinear stochastic dynamics of dissipative Hamiltonian systems with state-dependent friction and diffusion connected by the fluctuation–dissipation relation in high dimensions. The system under study has a close connection to Ao’s framework in constructing a dynamical potential for non-equilibrium processes without detailed balance. We study the limiting case where the mass approaches zero and give a new and complete derivation of effective stochastic differential equations. Using the Ito stochastic integral convention, we show that the limiting effective Langevin equations have a new drift term. This extra term happens to be identical to the corresponding anti-Ito (or isothermal) integral (requiring constant temperature) in one dimension. We, however, cannot obtain this additional drift term using conventional stochastic integrals in high dimension. It is interesting to note that in a high-dimensional system, a curl flux induced drift may appear even if the diffusion matrix is constant. Our findings are supported by numerical simulations. We further analyze and discuss the role of this new drift term in calculating the classic escape time. For the first time, to our knowledge, the relation between the escape rate and the anti-Ito integral is presented. We also demonstrate that the derived diffusion equations give a new sampling algorithm which can increase convergence speed in a simple two-dimensional example.

Multiscale modelling of transport phenomena for materials with n-layered embedded fibres. Part I: Analytical and numerical-based approaches

• The overall diffusivity tensor of a multi-layered fibre-reinforced material is given. • Diffusivity is derived from a Morphological Representative Pattern-based approach. • Closed-form analytical relations are proposed depending on morphological parameters. • Diffusivity effects of interphase areas are investigated. Numerical and analytical modelling approaches are often competing to describe a desired phenomenon. However, there is no reason to look for a single method and complementarities between distinct modelling ways are always profitable. This paper, the first of a set of two consecutive papers published in this volume, provides such a modelling methods reconciliation in the case of transport phenomena acting in transversely isotropic multi-phased materials. Microstructural Finite Element Modelling (FEM) helps supplying the analytical implementation for a better description of interactions between the constituents. As an application example, the case of molecular diffusion within a unidirectional composite medium is investigated without any restriction for considering any other transport mechanism. Based on a rational microstructure description, this study proposes a model able to predict the overall diffusion tensor of the composite and investigates more particularly diffusivity effects of interphase areas and fibre packings. For this purpose, a “ n -phase” Generalized Self-Consistent Scheme (GSCS) coupled with a Morphologically Representative Pattern (MRP) approach has been developed. This first part is focused on the modelling strategy and the concept of “transfer matrices” has been used. This approach is applied in Part II (Joannès and Hervé-Luanco, 2016) to study the influence of fibre packings on the effective behaviour of composite materials made of insulated fibres embedded in a diffusive matrix.

A new Lp-antieigenvalue condition for Ornstein–Uhlenbeck operators

In this paper we study perturbed Ornstein–Uhlenbeck operators[L∞v](x)=A△v(x)+〈Sx,∇v(x)〉−Bv(x),x∈Rd,d⩾2, for simultaneously diagonalizable matrices A,B∈CN,N. The unbounded drift term is defined by a skew-symmetric matrix S∈Rd,d. Differential operators of this form appear investigating rotating waves in time-dependent reaction diffusion systems. As shown in a companion paper, one key assumption to prove resolvent estimates of L∞ in Lp(Rd,CN), 1<p<∞, is the following Lp-dissipativity condition|z|2Re〈w,Aw〉+(p−2)Re〈w,z〉Re〈z,Aw〉⩾γA|z|2|w|2∀z,w∈CN for some γA>0. We prove that the Lp-dissipativity condition is equivalent to a new Lp-antieigenvalue conditionA invertibleandμ1(A)>|p−2|p,1<p<∞,μ1(A) first antieigenvalue of A, which is a lower p-dependent bound of the first antieigenvalue of the diffusion matrix A. This relation provides a complete algebraic characterization and a geometric meaning of Lp-dissipativity for complex-valued Ornstein–Uhlenbeck operators in terms of the antieigenvalues of A. The proof is based on the method of Lagrange multipliers. We also discuss several special cases in which the first antieigenvalue can be given explicitly.

WEAK DIFFUSION LIMITS OF DYNAMIC CONDITIONAL CORRELATION MODELS

The properties of dynamic conditional correlation (DCC) models, introduced more than a decade ago, are still not entirely known. This paper fills one of the gaps by deriving weak diffusion limits of a modified version of the classical DCC model. The limiting system of stochastic differential equations is characterized by a diffusion matrix of reduced rank. The degeneracy is due to perfect collinearity between the innovations of the volatility and correlation dynamics. For the special case of constant conditional correlations, a nondegenerate diffusion limit can be obtained. Alternative sets of conditions are considered for the rate of convergence of the parameters, obtaining time-varying but deterministic variances and/or correlations. A Monte Carlo experiment confirms that the often used quasi-approximate maximum likelihood (QAML) method to estimate the diffusion parameters is inconsistent for any fixed frequency, but that it may provide reasonable approximations for sufficiently large frequencies and sample sizes.

Operator estimates in homogenization theory

This paper gives a systematic treatment of two methods for obtaining operator estimates: the shift method and the spectral method. Though substantially different in mathematical technique and physical motivation, these methods produce basically the same results. Besides the classical formulation of the homogenization problem, other formulations of the problem are also considered: homogenization in perforated domains, the case of an unbounded diffusion matrix, non-self- adjoint evolution equations, and higher-order elliptic operators. Bibliography: 62 titles.

Analytical solutions of solute transport in a fracture–matrix system with different reaction rates for fracture and matrix

Summary This study deals with the problem of reactive solute transport in a fracture–matrix system using both analytical and numerical modeling methods. The groundwater flow velocity in the fracture is assumed to be high enough (no less than 0.1 m/day) to ensure the advection-dominant transport in the fracture. The problem includes advection along the fracture, transverse diffusion in the matrix, with linear sorption as well as first-order reactions operative in both the fracture and the matrix. A constant-concentration boundary condition and a decay source boundary condition in the fracture are considered. With a constant-concentration source, we obtain closed-form analytical solutions that account for the transport without reaction as well as steady-state solutions with different first-order reactions in the two media. With a decay source, a semi-analytical solution is obtained. The analytical and semi-analytical solutions are in excellent agreement with the numerical simulation results obtained using COMSOL Multiphysics. Sensitivity analysis is conducted to assess the relative importance of matrix diffusion coefficient, fracture aperture, and matrix porosity. We conclude that the first-order reaction as well as the matrix diffusion in the fractured rock would decrease the solute peak concentration and shorten the penetration distance into the fracture. The solutions can be applied to assess the spatial–temporal distribution of concentrations in the fracture and the matrix as well as to assess the contaminant mass stored in the rock matrix. All of these are useful for designing remediation plans for contaminated fractured rocks or for risk assessment of contaminated fracture–matrix systems.

On analytic characteristic functions and processes governed by SDEs

We study the analyticity of a characteristic function of a process defined by means of SDEs. Namely, starting with the simple case of a scalar Ito SDE we show that the corresponding characteristic function is entire. The proof is based on Grönwall's inequality technique and the classic analyticity criterion in terms of moments. Further, we extend this criterion and derive a handy sufficient condition of analyticity in the multidimensional case, which is used to prove the corresponding general result. We assume that the drift vector obeys the linear growth condition and the diffusion matrix is time-only-dependent but possibly degenerate. The approach used in this article can be extended to more general types of SDEs.

On maximum likelihood estimation of the drift matrix of a degenerated O–U process

In this work, we consider a 2n-dimension Ornstein–Uhlenbeck (O–U) process with a singular diffusion matrix. This process represents a currently used model for mechanical systems subject to random vibrations. We study the problem of estimating the drift parameters of the stochastic differential equation that governs the O–U process. The maximum likelihood estimator proposed and explored in Koncz (J Anal Math 13(1):75–91, 1987) is revisited and applied to our model. We prove the local asymptotic normality property and the convergence of moments of the estimator. Simulation studies based on representative examples taken from the literature illustrate the obtained theoretical results.

Mass and heat diffusion in ternary polymer solutions: A classical irreversible thermodynamics approach

Governing equations for evolution of concentration and temperature in three-component systems were derived in the framework of classical irreversible thermodynamics using Onsager’s variational principle and were presented for solvent/solvent/polymer and solvent/polymer/polymer systems. The derivation was developed from the Gibbs equation of equilibrium thermodynamics using the local equilibrium hypothesis, Onsager reciprocal relations and Prigogine’s theorem for systems in mechanical equilibrium. It was shown that the details of mass and heat diffusion phenomena in a ternary system are completely expressed by a 3 × 3 matrix whose entries are mass diffusion coefficients (4 entries), thermal diffusion coefficients (2 entries) and three entries that describe the evolution of heat in the system. The entries of the diffusion matrix are related to the elements of Onsager matrix that are bounded by some constraints to satisfy the positive definiteness of entropy production in the system. All the elements of diffusion matrix were expressed in terms of derivatives of exchange chemical potentials of the components with respect to concentration and temperature. The spinodal curves of ternary polymer solutions were derived from the governing equations and their correctness was checked by the Hessian of free energy density. Moreover, it was proved that setting cross-diffusion coefficients to zero results in a contradiction, and the governing equations without cross-diffusion coefficients do not express the actual phase behavior of the system.

Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also prove weak existence, hence obtain unique strong solutions by the Yamada–Watanabe theorem. The proofs are based in part on a recent maximal regularity result in infinite dimensions, the theory of quasi-regular Dirichlet forms and an infinite dimensional version of a Zvonkin-type transformation. As a main application, we show pathwise uniqueness for stochastic reaction diffusion equations perturbed by a Borel measurable bounded drift. Hence, such SDE have a unique strong solution.

A mixed derivative terms removing method in multi-asset option pricing problems

The challenge of removing the mixed derivative terms of a second order multidimensional partial differential equation is addressed in this paper. The proposed method, which is based on proper algebraic factorization of the so-called diffusion matrix, depends on the semidefinite or indefinite character of this matrix. Computational cost of the transformed equation is considerably reduced and well-known numerical drawbacks are avoided.

The effect of flake orientational order on the permeability of barrier membranes: numerical simulations and predictive models

This work presents results of numerical simulations along with the development of simple analytical forms aimed at predicting diffusivity in barrier membranes with randomly dispersed flakes. Simulations are performed using Boundary Element models of representative volume elements that account for the barrier microstructures with high level of detail. Microstructural features such as flake aspect ratio (α) and volume fraction (ϕ) are varied in the range of practical interest (0.1≤αϕ≤5). Numerical simulations also address the effects of the flake orientational order. Simulation results are used to develop a new model that predicts the elements of the diffusivity matrix as a function of flake arrangement. The basic idea behind the proposed model is to assimilate the parameter proposed by Bharadwaj to describe flake orientational order into the diffusivity model by Lape, which was originally developed for uniformly oriented flakes. The model predictions are shown to be consistent with theoretical limiting behaviors and with those of other models in the literature. The proposed model is among the few ones that accounts for the disorder in the flake orientation, which is found to have a noticeable impact on diffusivity in the direction parallel to the flake orientation.

Image encryption scheme based on block‐based confusion and multiple levels of diffusion

This study proposes a chaos‐based image encryption scheme using Henon map and Lorenz equation with multiple levels of diffusion. The Henon map is used for confusion and the Lorenz equation for diffusion. Apart from the Lorenz equation, another matrix with the same size as the original image is generated which is a complex function of the original image. This matrix which is configured as a diffusion matrix permits two stages of diffusion. Due to this step, there is a strong sensitivity to input image. This encryption algorithm has high key space, entropy very close to eight (for grey images) and very less correlation among adjacent pixels. The highlight of this method is the ideal number of pixels change rate and unified average changing intensity it offers. These ideal values indicate that the encrypted images produced by this proposed scheme are random‐like. Further, a cryptanalysis study has been carried out to prove that the proposed algorithm is resistant to known attacks.

A comparison of Fick and Maxwell–Stefan diffusion formulations in PEMFC gas diffusion layers

This paper explores the mathematical formulations of Fick and Maxwell–Stefan diffusion in the context of polymer electrolyte membrane fuel cell cathode gas diffusion layers. The simple Fick law with a diagonal diffusion matrix is an approximation of Maxwell–Stefan. Formulations of diffusion combined with mass-averaged Darcy flow are considered for three component gases. For this application, the formulations can be compared computationally in a simple, one dimensional setting. Despite the models’ seemingly different structure, it is observed that the predictions of the formulations are very similar on the cathode when air is used as oxidant. The two formulations give quite different results when the Nitrogen in the air oxidant is replaced by helium (this is often done as a diagnostic for fuel cells designs). The two formulations also give quite different results for the anode with a dilute Hydrogen stream. These results give direction to when Maxwell–Stefan diffusion, which is more complicated to implement computationally in many codes, should be used in fuel cell simulations.

Grassmann phase space methods for fermions. I. Mode theory

In both quantum optics and cold atom physics, the behaviour of bosonic photons and atoms is often treated using phase space methods, where mode annihilation and creation operators are represented by c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggest the possibility of using anti-commuting Grassmann variables to represent these operators. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of Grassmann phase space methods in quantum–atom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics.The theory of Grassmann phase space methods for fermions based on separate modes is developed, showing how the distribution function is defined and used to determine quantum correlation functions, Fock state populations and coherences via Grassmann phase space integrals, how the Fokker–Planck equations are obtained and then converted into equivalent Ito equations for stochastic Grassmann variables. The fermion distribution function is an even Grassmann function, and is unique. The number of c-number Wiener increments involved is 2n2, if there are n modes. The situation is somewhat different to the bosonic c-number case where only 2n Wiener increments are involved, the sign of the drift term in the Ito equation is reversed and the diffusion matrix in the Fokker–Planck equation is anti-symmetric rather than symmetric. The un-normalised B distribution is of particular importance for determining Fock state populations and coherences, and as pointed out by Plimak, Collett and Olsen, the drift vector in its Fokker–Planck equation only depends linearly on the Grassmann variables. Using this key feature we show how the Ito stochastic equations can be solved numerically for finite times in terms of c-number stochastic quantities. Averages of products of Grassmann stochastic variables at the initial time are also involved, but these are determined from the initial conditions for the quantum state. The detailed approach to the numerics is outlined, showing that (apart from standard issues in such numerics) numerical calculations for Grassmann phase space theories of fermion systems could be carried out without needing to represent Grassmann phase space variables on the computer, and only involving processes using c-numbers. We compare our approach to that of Plimak, Collett and Olsen and show that the two approaches differ.As a simple test case we apply the B distribution theory and solve the Ito stochastic equations to demonstrate coupling between degenerate Cooper pairs in a four mode fermionic system involving spin conserving interactions between the spin 1/2 fermions, where modes with momenta −k,+k—each associated with spin up, spin down states, are involved.

Time-Asymptotic Evolution of Spatially Uniform Gaussian Vlasov Fluctuation Fields

We consider Vlasov fluctuations from a spatially uniform, infinitely extended plasma equilibrium and derive the appropriate Braun–Hepp evolution equations (with regularized potential). Assuming a Gaussian initial fluctuation field (of the form associated with the relevant central limit theorem), we show that at long times the fluctuating force field converges in law to a stationary Gaussian process. We also illustrate how the time-asymptotic fluctuating force field is formally associated with the diffusion matrix and drift vector for the Balescu–Guernsey–Lenard kinetic equation.