Space-dependent Diffusion Coefficient Research Articles

Large deviations at level 2.5 and for trajectories observables of diffusion processes: the missing parts with respect to their random-walks counterparts

Behind the nice unification provided by the notion of the level 2.5 in the field of large deviations for time-averages over a long Markov trajectory, there are nevertheless very important qualitative differences between the meaning of the level 2.5 for diffusion processes on one hand, and the meaning of the level 2.5 for Markov chains either in discrete-time or in continuous-time on the other hand. In order to analyze these differences in detail, it is thus useful to consider two types of random walks converging towards a given diffusion process in dimension d involving arbitrary space-dependent forces and diffusion coefficients, namely (i) continuous-time random walks on the regular lattice of spacing b; (ii) discrete-time random walks in continuous space with a small time-step τ. One can then analyze how the large deviations at level 2.5 for these two types of random walks behave in the limits b → 0 and τ → 0 respectively, in order to describe how the fluctuations of some empirical observables of the random walks are suppressed in the limit of diffusion processes. One can then also study the limits b → 0 and τ → 0 for any trajectory observable of the random walks that can be decomposed on its empirical density and its empirical flows in order to see how it is projected on the appropriate trajectory observable of the diffusion process involving its empirical density and its empirical current.

Random-search efficiency in a bounded interval with spatially heterogeneous diffusion coefficient

We consider random walkers searching for a target in a bounded 1D heterogeneous environment, in the interval , where diffusion is described by a space-dependent diffusion coefficient D(x). Boundary conditions are absorbing at the position of the target (set at x = 0) and reflecting at the border x = L. We calculate and compare the estimates of efficiency and . In the Stratonovich framework of the multiplicative random process, both measures are analytically calculated for arbitrary D(x). For other interpretations of the stochastic integrals (e.g. Itô and anti-Itô), we obtain general results for ɛ 2, while ɛ 1 is obtained for particular forms of D(x). The impact of the diffusivity profile on these measures of efficiency is discussed. Symmetries and peculiar properties arise when the search starts at the border (). In particular, heterogeneity spoils the efficiency of the search within the Stratonovich framework, while for other interpretations the searcher can perform better for certain heterogeneous diffusivity profiles.

On regularization results for a two-dimensional nonlinear time-fractional inverse diffusion problem

Ill-posed inverse problems arise in a variety of practically important applications including image restoration, medicine, ecology as well as many other branches of pure and applied sciences. In this paper, we consider an ill-posed inverse problem for a two-dimensional nonlinear time-fractional inverse diffusion model involving the Caputo time-fractional derivative as follows:{Dtβu(x,y,t)=−a(x)(ux(x,y,t)+uy(x,y,t))+ℓ(x,y,t,u(x,y,t)),x>0,y>0,t>0,u(x,y,0)=0,x>0,y>0,limx→∞⁡u(x,y,t)=0,y>0,t>0,u(1,y,t)=u1(y,t),y>0,t>0, where β∈(0,1) is fixed, a is a space-dependent diffusion coefficient, ℓ is a nonlinear function satisfying a Lipschitz condition, and the data u1 is given approximately. We want to determine u(x,y,t) for 0≤x<1. There exists a vast literature on regularization results related to the considered problem. We are concerned with regularization results for the nonlinear case. The first regularization results in such a case were studied by Tuan, Hoan and Tatar in [17]. While these results apply to a constant diffusivity coefficient in the one-dimensional setting, the results of Vo and co-worker in [18], more generally, apply to the case of the space-dependent diffusion coefficient in the two-dimensional setting. In all these papers relatively strong a priori assumptions on the regularity of the exact solution are made. In this work, we weaken such a priori assumptions via two regularization strategies based on the integral equation method. We obtain several explicit error estimates including an error estimate of the Hölder-Logarithmic type for all x∈[0,1), which can be seen as the improvement and the generalization of error estimates presented by Zheng and Wei in [20,21]. Eventually, several numerical examples are presented for the purpose of illustrating the theoretical results.

Efficiency of random search with space-dependent diffusivity.

We address the problem of random search for a target in an environment with a space-dependent diffusion coefficient D(x). Considering a general form of the diffusion differential operator that includes Itô, Stratonovich, and Hänggi-Klimontovich interpretations of the associated stochastic process, we obtain and analyze the first-passage-time distribution and use it to compute the search efficiency E=〈1/t〉. For the paradigmatic power-law diffusion coefficient D(x)=D_{0}|x|^{α}, where x is the distance from the target and α<2, we show the impact of the different interpretations. For the Stratonovich framework, we obtain a closed-form expression for E, valid for arbitrary diffusion coefficient D(x). This result depends only on the distribution of diffusivity values and not on its spatial organization. Furthermore, the analytical expression predicts that a heterogeneous diffusivity profile leads to a lower efficiency than the homogeneous one with the same average level within the space between the target and the searcher initial position, but this efficiency can be exceeded for other interpretations.

Analysis of a scheme which preserves the dissipation and positivity of Gibbs' energy for a nonlinear parabolic equation with variable diffusion

In this work, we design and analyze a discrete model to approximate the solutions of a parabolic partial differential equation in multiple dimensions. The mathematical model considers a nonlinear reaction term and a space-dependent diffusion coefficient. The system has a Gibbs' free energy, we establish rigorously that it is non-negative under suitable conditions, and that it is dissipated with respect to time. The discrete model proposed in this work has also a discrete form of the Gibbs' free energy. Using a fixed-point theorem, we prove the existence of solutions for the numerical model under suitable assumptions on the regularity of the component functions. We prove that the scheme preserves the positivity and the dissipation of the discrete Gibbs' free energy. We establish theoretically that the discrete model is a second-order consistent scheme. We prove the stability of the method along with its quadratic convergence. Some simulations illustrating the capability of the scheme to preserve the dissipation of Gibb's energy are presented.

Molecular Communication in Inhomogeneous Diffusion Channels

We consider an inhomogeneous diffusion channel (IDC) for molecular communication, where the diffusivities vary in space. We first study the statistical properties of molecular noise. These properties can be fully characterized by two IDC parameters related to a space-dependent diffusion coefficient and a constant drift velocity. We then characterize a noise power for the IDC by introducing the concept of geometric noise power since the molecular noise can be a heavy-tailed distribution. Finally, we derive the bit error rate of molecular communication with timing modulation in terms of two IDC parameters.

New Analytical Results and Comparison of 14 Numerical Schemes for the Diffusion Equation with Space-Dependent Diffusion Coefficient

We examine the one-dimensional transient diffusion equation with a space-dependent diffusion coefficient. Such equations can be derived from the Fokker–Planck equation and are essential for understanding the diffusion mechanisms, e.g., in carbon nanotubes. First, we construct new, nontrivial analytical solutions with the classical self-similar Ansatz in one space dimension. Then we apply 14 different explicit numerical time integration methods, most of which are recently introduced unconditionally stable schemes, to reproduce the analytical solution. The test results show that the best algorithms, especially the leapfrog-hopscotch, are very efficient and severely outperform the conventional Runge–Kutta methods. Our results may attract attention in the community who develops multi-physics engineering software.

Statistical physics of inhomogeneous transport: Unification of diffusion laws and inference from first-passage statistics.

Characterization of composite materials, whose properties vary in space over microscopic scales, has become a problem of broad interdisciplinary interest. In particular, estimation of the inhomogeneous transport coefficients, e.g., the diffusion coefficient or the heat conductivity, which shape important processes in biology and engineering, is a challenging task. The analysis of such systems is further complicated because two alternative formulations of the inhomogeneous transport equationsexist in the literature-the Smoluchowski and Fokker-Planck equations, which are also related to the so-called Ito-Stratonovich dilemma. Using the theory of statistical physics, we show that the two formulations, usually regarded as distinct models, are physically equivalent. From this result we develop efficient estimates for the transverse space-dependent diffusion coefficient in fluids near a phase boundary. Our method requires only measurements of escape probabilities and mean exit times of molecules leaving a narrow spatial region. We test our estimates in three case studies: (i) a Langevin model of a Büttikker-Landauer ratchet; atomistic molecular-dynamics simulations of liquid-water molecules in contact with (ii) vapor, and (iii) soap (surfactant) film which has promising applications in physical chemistry. Our analysis reveals that near the surfactant monolayer the mobility of water molecules is slowed down almost twice with respect to the bulk liquid. Moreover, the diffusion coefficient of water correlates with the transition from hydrophilic to hydrophobic parts of the film.

Attaining ergodicity in a Langevin equation with heterogeneous media

Generalizations of the mean-squared displacement and time-averaged mean-squared displacement are proposed to attaining the ergodicity of a Langevin equation with a broad class of the space-dependent diffusion coefficient.

Physics-informed Karhunen-Loéve and neural network approximations for solving inverse differential equation problems

We present the PI-CKL-NN method for parameter estimation in differential equation (DE) models given sparse measurements of the parameters and states. In the proposed approach, the space- or time-dependent parameters are approximated by Karhunen-Loéve (KL) expansions that are conditioned on the parameters' measurements, and the states are approximated by deep neural networks (DNNs). The unknown weights in the KL expansions and DNNs are found by minimizing the cost function that enforces the measurements of the states and the DE constraint. Regularization is achieved by adding the l2 norm of the conditional KL coefficients into the loss function. Our approach assumes that the parameter fields are correlated in space or time and enforces the statistical knowledge (the mean and the covariance function) in addition to the DE constraints and measurements as opposed to the physics-informed neural network (PINN) and other similar physics-informed machine learning methods where only DE constraints and data are used for parameter estimation. We use the PI-CKL-NN method for parameter estimation in an ordinary differential equation with an unknown time-dependent parameter and the one- and two-dimensional partial differential diffusion equations with unknown space-dependent diffusion coefficients. We also demonstrate that PI-CKL-NN is more accurate than the PINN method, especially when the observations of the parameters are very sparse.

NON-FICKIAN DIFFUSION IN TIME–SPACE FLUCTUATING DIFFUSIVITY LANDSCAPES: FROM SUPERFAST TO ULTRASLOW

In this study, the local time and space structural derivative diffusion model is introduced to describe non-Fickian diffusion in heterogeneous media from ultraslow to superfast. Its equivalent heterogeneous diffusion process (HDP) model with time and space-dependent diffusion coefficient is also given. The connection and difference between the proposed model and the existing HDP models are compared. Different types of diffusion processes can be captured by selecting proper structural functions in the structural derivative model, which is determined based on the concentration of tracers, mean squared displacement (MSD) and diffusion coefficient of the heterogeneous media. The dynamics of the free cooling granular gases with constant restitution coefficient and velocity-dependent restitution coefficient is investigated to test the proposed model. The results show that the diffusion law of the granular particles is not uniform for the short and long-time scales, but one uniform diffusion law derived from the proposed model can well fit the data in different time scales. The patterns of the time-dependent diffusion coefficient are also simulated to deeply understand the complicated non-Fickian diffusion in complex system.

A class of integro-differential Fokker-Planck equations with space-dependent coefficient

Exact solution for the probability distribution function (PDF) in Laplace space is obtained for a class of integro-differential Fokker-Planck equations with different kernels and generic space-dependent diffusion coefficient. Besides, exact solutions for the PDF and n-moment are also obtained for different kernels and a power-law diffusion coefficient; they are analyzed and compared for different kernels.

Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes.

Heterogeneous diffusion processes (HDPs) with space-dependent diffusion coefficients D(x) are found in a number of real-world systems, such as for diffusion of macromolecules or submicron tracers in biological cells. Here, we examine HDPs in quenched-disorder systems with Gaussian colored noise (GCN) characterized by a diffusion coefficient with a power-law dependence on the particle position and with a spatially random scaling exponent. Typically, D(x) is considered to be centerd at the origin and the entire x axis is characterized by a single scaling exponent α. In this work we consider a spatially random scenario: in periodic intervals ("layers") in space D(x) is centerd to the midpoint of each interval. In each interval the scaling exponent α is randomly chosen from a Gaussian distribution. The effects of the variation of the scaling exponents, the periodicity of the domains ("layer thickness") of the diffusion coefficient in this stratified system, and the correlation time of the GCN are analyzed numerically in detail. We discuss the regimes of superdiffusion, subdiffusion, and normal diffusion realisable in this system. We observe and quantify the domains where nonergodic and non-Gaussian behaviors emerge in this system. Our results provide new insights into the understanding of weak ergodicity breaking for HDPs driven by colored noise, with potential applications in quenched layered systems, typical model systems for diffusion in biological cells and tissues, as well as for diffusion in geophysical systems.

Critical patch size reduction by heterogeneous diffusion.

Population survival depends on a large set of factors and on how they are distributed in space. Due to landscape heterogeneity, species can occupy particular regions that provide the ideal scenario for development, working as a refuge from harmful environmental conditions. Survival occurs if population growth overcomes the losses caused by adventurous individuals that cross the patch edge. In this work, we consider a single species dynamics in a patch with a space-dependent diffusion coefficient. We show analytically, within the Stratonovich framework, that heterogeneous diffusion reduces the minimal patch size for population survival when contrasted with the homogeneous case with the same average diffusivity. Furthermore, this result is robust regardless of the particular choice of the diffusion coefficient profile. We also discuss how this picture changes beyond the Stratonovich framework. Particularly, the Itô case, which is nonanticipative, can promote the opposite effect, while Hänggi-Klimontovich interpretation reinforces the reduction effect.

Small-time expansion of the Fokker–Planck kernel for space and time dependent diffusion and drift coefficients

We study the general solution of the Fokker–Planck equation in d dimensions with arbitrary space and time dependent diffusion matrix and drift terms. We show how to construct the solution for arbitrary initial distributions as an asymptotic expansion for small time. This generalizes the well-known asymptotic expansion of the heat kernel for the Laplace operator on a general Riemannian manifold. We explicitly work out the general solution to leading and next-to-leading order in this small-time expansion, as well as to next-to-next-to-leading order for vanishing drift. We illustrate our results on a several examples.

Location of sources in reaction-diffusion equations using support vector machines.

The reaction-diffusion equation serves to model systems in the diffusion regime with sources. Specific applications include diffusion processes in chemical reactions, as well as the propagation of species, diseases, and populations in general. In some of these applications the location of an outbreak, for instance, the source point of a disease or the nest of a vector spreading a virus is important. Also important are the environmental parameters of the domain where the process diffuses, namely the space-dependent diffusion coefficient and the proliferation parameter of the process. Determining both, the location of a source and the environmental parameters, define an inverse problem that in turn, involves a partial differential equation. In this paper we classify the values of these parameters using Support Vector Machines (SVM) trained with numerical solutions of the reaction-diffusion problem. Our set up has accuracy of classifying the outbreak location above 90% and 77% of classifying both, the location and the environmental parameters. The approach presented in our analysis can be directly implemented by measuring the population under study at specific locations in the spatial domain as function of time.

Stability of a general HIV-1 reaction–diffusion model with multiple delays and immune response

The aim of this paper is to study a general HIV-1 reaction–diffusion model with space-dependent diffusion coefficients and antibody immune response. The model contains four discrete time delays and three categories of infected cells: Latently infected cells, short-lived productively infected cells and long-lived productively infected cells. The effect of highly active antiretroviral therapies (HAART) used to treat HIV-1 is incorporated into the model. The existence, non-negativity and boundedness of solutions are studied in order to verify the well-posedness of the model. All possible equilibria of the model are evaluated and the threshold parameters needed for their existence and stability are investigated. The global asymptotic stability of the corresponding equilibria are proved by building suitable Lyapunov functionals. The numerical simulations are conducted in order to confirm the theoretical results and show the behavior of solutions in space and time.

Backstepping design for a class of coupled parabolic PDEs with spatially varying coefficient

AbstractIn this paper, we concentrate on the state feedback stabilization of a class of coupled parabolic partial differential equations with space dependent diffusion coefficients. To stabilize the system, we design a state feedback controller using the backstepping technique. Namely, we convert the original system into a stable desired system called the target system to obtain a backstepping feedback controller. We prove that the feedback controller causes the main system to be exponential stable. Also, the numerical simulations show that validity and efficiency of theoretical results.

A Simultaneous Inversion Problem for the Variable-Order Time Fractional Differential Equation with Variable Coefficient

This paper deals with an inverse problem of simultaneously determining the space-dependent diffusion coefficient and the fractional order in the variable-order time fractional diffusion equation by the measurements at one interior point. Numerical solution to the forward problem is given by the finite difference scheme, and the homotopy regularization algorithm is applied to solve the inverse problem utilizing Legendre polynomials as the basis functions of the approximate space. The inversion solutions with noisy data which give good approximations to the exact solution demonstrate effectiveness of the inversion algorithm for the simultaneous inversion problem.

Determination of initial data for a reaction-diffusion system with variable coefficients

In this paper, we study a final value problem for a reaction-diffusion system with time and space dependent diffusion coefficients. In general, the inverse problem of identifying the initial data is not well-posed, and herein the Hadamard-instability occurs. Applying a new version of a modified quasi-reversibility method, we propose a stable approximate (regularized) problem. The existence, uniqueness and stability of the corresponding regularized problem are obtained. Furthermore, we also investigate the error estimate and show that the approximate solution converges to the exact solution in \begin{document}$L_2$\end{document} and \begin{document}$\stackrel{0}{H_1}$\end{document} norms. Our method can be applied to some concrete models that arise in biology, chemical engineering, etc.