Diffusion Matrix Research Articles

Trend to equilibrium of renormalized solutions to reaction–cross-diffusion systems

The convergence to equilibrium of renormalized solutions to reaction–cross-diffusion systems in a bounded domain under no-flux boundary conditions is studied. The reactions model complex balanced chemical reaction networks coming from mass-action kinetics and thus do not obey any growth condition, while the diffusion matrix is of cross-diffusion type and hence nondiagonal and neither symmetric nor positive semi-definite, but the system admits a formal gradient-flow or entropy structure. The diffusion term generalizes the population model of Shigesada, Kawasaki and Teramoto to an arbitrary number of species. By showing that any renormalized solution satisfies the conservation of masses and a weak entropy–entropyproduction inequality, it can be proved under the assumption of no boundary equilibria that all renormalized solutions converge exponentially to the complex balanced equilibrium with a rate which is explicit up to a finite dimensional inequality.

On Algebraic condition for null controllability of some coupled degenerate systems

In this paper we will generalize the Kalman rank condition for the null controllability to $n$-coupled linear degenerate parabolic systems with constant coefficients, diagonalizable diffusion matrix, and $m$-controls. For that we prove a global Carleman estimate for the solutions of a scalar $2n$-order parabolic equation then we infer from it an observability inequality for the corresponding adjoint system, and thus the null controllability.

Low-rank diffusion matrix estimation for high-dimensional time-changed Lévy processes

Nous étudions le problème de l’estimation de la matrice de diffusion $\Sigma$ d’un processus de Lévy en grande dimension, qui peut être changé de temps, en se basant sur des observations discrètes du processus à une distance fixée. Nous imposons une condition de faible rang sur $\Sigma$. À l’aide d’une méthode spectrale, nous construisons un estimateur pondéré des moindres carrés avec une pénalisation par une norme nucléaire. Nous prouvons des inégalités oracle et obtenons des vitesses de convergence pour l’estimateur de la matrice de diffusion. Nous constatons que ces vitesses dépendent du rang de $\Sigma$ d’une façon surprenante, et qu’elles sont optimales au sens minimax pour une dimension fixée. Ces résultats théoriques sont illustrés par une étude de simulations.

Global existence of solutions for an [formula omitted]-component cross-diffusion system with a 3-component case study

In this paper, we examine a general m-component reaction–diffusion matrix with a full diffusion matrix and polynomially growing reaction terms through its diagonalization. We establish the invariant regions of the system and derive the necessary conditions for the existence of solutions. The 3 × 3 case is taken as a case study, where we determine the exact conditions for the positivity of the eigenvalues, which is necessary for the diagonalization process. Numerical examples are used to illustrate and confirm the findings of this paper.

Stationary distribution of a stochastic SIQR epidemic model with saturated incidence and degenerate diffusion

The objective of this paper is to study stationary distribution of a stochastic SIQR epidemic model with saturated incidence and degenerate diffusion. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory will be used to obtain the existence of a unique stable stationary distribution. A threshold dynamic determined by the basic reproduction number R0s is established: the disease can be eradicated almost surely if R0s<1, whereas if R0s>1, it has an endemic stationary distribution which leads to the stochastic persistence of the disease.

Correction: Premature aortic smooth muscle cell differentiation contributes to matrix dysregulation in Marfan Syndrome

[This corrects the article DOI: 10.1371/journal.pone.0186603.].

On an Optimal Filtration Problem for One-Dimensional Diffusion Processes

We find a method that reduces the solution of a problem of nonlinear filtration of one-dimensional diffusion processes to the solution of a linear parabolic equation with constant diffusion coefficients whose remaining coefficients are random and depend on the trajectory of the observable process. The method consists in reducing the initial filtration problem to a simpler problem with identity diffusion matrix and subsequently reducing the solution of the parabolic Ito equation for the filtered density to solving the above-mentioned parabolic equation. In addition, the filtered densities of both problems are connected by a sufficiently simple formula.

Stochastic invariance of closed sets with non-Lipschitz coefficients

This paper provides a new characterization of the stochastic invariance of a closed subset of Rd with respect to a diffusion. We extend the well-known inward pointing Stratonovich drift condition to the case where the diffusion matrix can fail to be differentiable: we only assume that the covariance matrix is. In particular, our result can be applied to construct affine and polynomial diffusions on any arbitrary closed set.

Diffusion of Conserved Charges in Relativistic Heavy Ion Collisions.

We demonstrate that the diffusion currents do not depend only on gradients of their corresponding charge density, but that the different diffusion charge currents are coupled. This happens in such a way that it is possible for density gradients of a given charge to generate dissipative currents of another charge. Within this scheme, the charge diffusion coefficient is best viewed as a matrix, in which the diagonal terms correspond to the usual charge diffusion coefficients, while the off-diagonal terms describe the coupling between the different currents. In this Letter, we calculate for the first time the complete diffusion matrix for hot and dense nuclear matter, including baryon, electric, and strangeness charges. We find that the baryon diffusion current is strongly affected by baryon charge gradients but also by its coupling to gradients in strangeness. The electric charge diffusion current is found to be strongly affected by electric and strangeness gradients, whereas strangeness currents depend mostly on strange and baryon gradients.

Stochastic response determination and optimization of a class of nonlinear electromechanical energy harvesters: A Wiener path integral approach

A methodology based on the Wiener path integral technique (WPI) is developed for stochastic response determination and optimization of a class of nonlinear electromechanical energy harvesters. To this aim, first, the WPI technique is extended to address the particular form of the coupled electromechanical governing equations, which possess a singular diffusion matrix. Specifically, a constrained variational problem is formulated and solved for determining the joint response probability density function (PDF) of the nonlinear energy harvesters. This is done either by resorting to a Lagrange multipliers approach, or by utilizing the nullspace of the constraint equation. Next, the herein extended WPI technique is coupled with an appropriate optimization algorithm for determining optimal energy harvester parameters. It is shown that due to the relatively high accuracy exhibited in determining the joint response PDF, the WPI technique is particularly well-suited for constrained optimization problems, where the constraint refers to low probability events (e.g. probabilities of failure). In this regard, the WPI technique outperforms significantly an alternative statistical linearization solution treatment commonly utilized in the literature, which fails to capture even basic features of the response PDF. This inadequacy of statistical linearization becomes even more prevalent in cases of nonlinear harvesters with asymmetric potentials, where the response PDF deviates significantly from the Gaussian. Several numerical examples are included, while comparisons with pertinent Monte Carlo simulation data demonstrate the robustness and reliability of the methodology.

Long-time behavior of a stochastic logistic equation with distributed delay and nonlinear perturbation

In this paper, we study a stochastic logistic model with distributed delay and nonlinear perturbation. We first use the linear chain technique to transfer the one-dimensional stochastic model with strong kernel or weak kernel into an equivalent system with degenerate diffusion. Then we establish sufficient and necessary criteria for extinction of the population with probability one. Moreover, since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We show that the densities of the distributions of the solutions can converge in L1 to an invariant density. The results show that the larger white noise can lead to the extinction of the population while the smaller white noise can guarantee the existence of a stable stationary distribution which leads to the stochastic persistence of the population. Numerical simulations are provided to demonstrate the theoretical results.

An Euclidean Distance Based on Tensor Product Graph Diffusion Related Attribute Value Embedding for Nominal Data Clustering

Not like numerical data clustering, nominal data clustering is a very difficult problem because there exists no natural relative ordering between nominal attribute values. This paper mainly aims to make the Euclidean distance measure appropriate to nominal data clustering, and the core idea is the attribute value embedding, namely, transforming each nominal attribute value into a numerical vector. This embedding method consists of four steps. In the first step, the weights, which can quantify the amount of information in attribute values, is calculated for each value in each nominal attribute based on each object and its k nearest neighbors. In the second step, an intra-attribute value similarity matrix is created for each nominal attribute by using the attribute value's weights. In the third step, for each nominal attribute, we find another attribute with the maximal dependence on it, and build an inter-attribute value similarity matrix on the basis of the attribute value's weights related to these two attributes. In the last step, a diffusion matrix of each nominal attribute is constructed by the tensor product graph diffusion process, and this step can cause the acquired value embedding to contain simultaneously the intra- and inter-attribute value similarities information. To evaluate the effectiveness of our proposed method, experiments are done on 10 data sets. Experimental results demonstrate that our method not only enables the Euclidean distance to be used for nominal data clustering, but also can acquire the better clustering performance than several existing state-of-the-art approaches.

Kinetics of dissolution of sapphire in melts in the CaO–Al2O3–SiO2 system

The dissolution rate of sapphire in melts in the CAS system of varying silica activity, viscosity and degree of alumina saturation has been determined at 1600 °C and 1.5 GPa. After an initiation period of up to 1800 s, dissolution is controlled by diffusion of cations through the boundary layer adjacent to the dissolving sapphire. The dissolution rate decreases with increasing silica activity, viscosity and molar Al2O3/CaO. The calculated diffusion matrix for each solvent melt shows that CAS 1 and 9 which have molar Al2O3/CaO of 0.33 and 0.6 and dissolution rate constants of 0.65 × 10−6 and 0.59 × 10−6 m/s0.5 have similar directions and magnitudes of diffusive coupling: DCaO-Al2O3 and DAl2O3-CaO are both negative are approximately equal. The solvent with the fastest dissolution rate: CAS 4, which has a rate constant of 1.5 × 10−6 m/s0.5 and Al2O3/CaO of 0.31 has positive DCaO-Al2O3 and negative DAl2O3-CaO and the absolute values vary by a factor of 4. Although many studies show that aluminium is added to the melts via the reaction:Si4+=Al3++0.5Ca2+the compositional profiles show that this reaction is not the only one involved in accommodating the aluminium added during sapphire dissolution. Rather, aluminium is incorporated as both tetrahedrally coordinated Al charge balanced by Ca and as aluminium not charge balanced by Ca (termed Alxs). This reaction:AlIV–Ca=Alxs+CaNBOwhere CaNBO is a non-bridging oxygen associated with calcium, may involve the formation of aluminium triclusters. The shape of the compositional profiles and oxide-oxide composition paths is controlled by the aluminium addition reaction. When Alxs exceeds 2%, CaO diffusion becomes increasingly anomalous and since the bond strength of Alxs correlates with CaO/CaO + Al2O3, the presence of more than 2% Alxs leads to significantly slower dissolution than when Alxs is absent or at low concentration. Thus, dissolution is controlled by diffusion of cations through the boundary layer, but this diffusion is itself controlled by the structural modifications required by the addition of new components to the melt. Comparison of quartz dissolution rates in similar melts shows that dissolution is much faster for quartz than for sapphire and that dissolution rates show the same correlation with silica activity and viscosity. We suggest that diffusive fluxes are related to changes in melt structure and the nature of the reaction that incorporates the added component. For the slow eigendirection, SiO2 addition occurs by a single reaction whereas Al2O3 addition requires a more complex two part reaction in which Al is accommodated by charge balance with Ca until Al is in excess of that which can be charge balanced. The Alxs incorporation reaction, is slower than the Si incorporation reaction which inhibits sapphire dissolution relative to quartz in melts of the same composition.

On models for predicting thermodynamic regimes in high-pressure turbulent mixing and combustion of multispecies mixtures

The thermodynamic regime of a complex mixture depends on the composition, the pressure and the temperature; the spinodal locus separates the regime of thermodynamic instability from the remainder of the phase space. Since diffusion is one of the phenomena affecting the local chemical composition, the first focus is here on evaluating diffusion models in the context of high-pressure (high-$p$) multispecies mixing and combustion. It is shown that the diffusion model equations previously used to create two high-$p$ direct numerical simulation (DNS) databases can reproduce classical experimental observations of uphill diffusion in an accurate spatiotemporal manner, whereas the popular model which has a diagonal diffusion matrix and uses a velocity correction lacks spatiotemporal accuracy. Further, a mathematical formalism is used to compute the spinodal locus for mixtures for which either experimental data or previous computations from the literature are available, and it is shown that the agreement of the present calculations with that previously existing information is excellent. Using the spinodal-calculation mathematical formalism, the aforementioned DNS databases are then examined to determine the thermodynamic regime of the mixture at important stages of the simulations. In the first subset of the DNS databases that portrays mixing of five species under high-$p$ conditions, this stage is that of the transitional state representing the individual time station at which each simulation, having been initiated in a laminar state, transitions to a state having turbulent characteristics. In the second subset of the DNS databases that portrays high-$p$ turbulent combustion, this stage represents the individual time station at the peak $p$ achieved during the calculations. In both databases, the influence of the initial Reynolds number, the free-stream composition and the free-stream $p$ is studied. The results show that in all cases the mixture is in the single-phase regime. The present DNS databases have only five species, but it is shown that the methodology for computing the spinodal locus can be applied to very complex mixtures, with examples given for a twelve-species mixture and surrogate diesel fuels, thereby boding well for determining the thermodynamic regime of practical mixtures in high-$p$ turbulent flow simulations for engineering applications. According to these calculations, diesel-fuel surrogates are always in the single-phase regime at injection-conditions $p$ and temperatures existing in diesel-engine combustion chambers.

Self-similar solutions for multi-species plasma mixing by gradient driven transport

Multi-species transport of plasma ions across an initial interface between DT and CH is shown to exhibit self-similar species density profiles under 1D isobaric conditions. Results using transport theory from recent studies and using a Maxwell–Stephan multi-species approximation are found to be in good agreement for the self-similar mix profiles of the four ions under isothermal and isobaric conditions. The individual ion species mass flux and molar flux profile results through the mixing layer are examined using transport theory. The sum over species mass flux is confirmed to be zero as required, and the sum over species molar flux is related to a local velocity divergence needed to maintain pressure equilibrium during the transport process. The light ion species mass fluxes are dominated by the diagonal coefficients of the diffusion transport matrix, while for the heaviest ion species (C in this case), the ion flux with only the diagonal term is reduced by about a factor two from that using the full diffusion matrix, implying the heavy species moves more by frictional collisions with the lighter species than by its own gradient force. Temperature gradient forces were examined by comparing profile results with and without imposing constant temperature gradients chosen to be of realistic magnitude for ICF experimental conditions at a fuel-capsule interface (10 μm scale length or greater). The temperature gradients clearly modify the relative concentrations of the ions, for example near the fuel center, however the mixing across the fuel-capsule interface appears to be minimally influenced by the temperature gradient forces within the expected compression and burn time. Discussion considers the application of the self-similar profiles to specific conditions in ICF.

Assessment of Remote Myocardium Heterogeneity in Patients with Ventricular Tachycardia Using Texture Analysis of Late Iodine Enhancement (LIE) Cardiac Computed Tomography (cCT) Images

PurposeDiffuse remodeling of myocardial extra-cellular matrix is largely responsible for left ventricle (LV) dysfunction and arrhythmias. Our hypothesis is that the texture analysis of late iodine enhancement (LIE) cardiac computed tomography (cCT) images may improve characterization of the diffuse extra-cellular matrix changes. Our aim was to extract volumetric extracellular volume (ECV) and LIE texture features of non-scarred (remote) myocardium from cCT of patients with recurrent ventricular tachycardia (rVT), and to compare these radiomic features with LV-function, LV-remodeling, and underlying cardiac disease.ProceduresForty-eight patients suffering from rVT were prospectively enrolled: 5/48 with idiopathic VT (IVT), 23/48 with post-ischemic dilated cardiomyopathy (ICM), 9/48 with idiopathic dilated cardiomyopathy (IDCM), and 11/48 with scars from a previous healed myocarditis (MYO). All patients underwent echocardiography to assess LV systolic and diastolic function and cCT with pre-contrast, angiographic, and LIE scan to obtain end-diastolic volume (EDV), ECV, and first-order texture parameters of Hounsfield Unit (HU) of remote myocardium in LIE [energy, entropy, HU-mean, HU-median, standard deviation (SD), and mean absolute deviation (MAD)].ResultsEnergy, HU mean, and HU median by cCT texture analysis correlated with ECV (rho = 0.5650, rho = 0.5741, rho = 0.5068; p < 0.0005). cCT-derived ECV, HU-mean, HU-median, SD, and MAD correlated directly to EDV by cCT and inversely to ejection fraction by echocardiography (p < 0.05). SD and MAD correlated with diastolic function by echocardiography (rho = 0.3837, p = 0.0071; rho = 0.3330, p = 0.0208). MYO and IVT patients were characterized by significantly lower values of SD and MAD when compared with ICM and IDCM patients, independently of LV-volume systolic and diastolic function.ConclusionsTexture analysis of LIE may expand cCT capability of myocardial characterization. Myocardial heterogeneity (SD and MAD) was associated with LV dilatation, systolic and diastolic function, and is able to potentially identify the different patterns of structural remodeling characterizing patients with rVT of different etiology.

Multicomponent diffusion in basaltic melts at 1350 °C

Nine successful diffusion couple experiments were conducted in an 8-component SiO2–TiO2–Al2O3–FeO–MgO–CaO–Na2O–K2O system at ∼1350 °C and at 1 GPa, to study multicomponent diffusion in basaltic melts. At least 3 traverses were measured to obtain diffusion profiles for each experiment. Multicomponent diffusion matrix at 1350 °C was obtained by simultaneously fitting diffusion profiles of diffusion couple experiments. Furthermore, in order to better constrain the diffusion matrix and reconcile mineral dissolution data, mineral dissolution experiments in the literature and diffusion couple experiments from this study, were fit together. All features of diffusion profiles in both diffusion couple and mineral dissolution experiments were well reproduced by the diffusion matrix. Diffusion mechanism is inferred from eigenvectors of the diffusion matrix, and it shows that the diffusive exchange between network-formers SiO2 and Al2O3 is the slowest, the exchange of SiO2 with other oxide components is the second slowest with an eigenvalue that is only ∼10% larger, then the exchange between divalent oxide components and all the other oxide components is the third slowest with an eigenvalue that is twice the smallest eigenvalue, then the exchange of FeO + K2O with all the other oxide components is the fourth slowest with an eigenvalue that is 5 times the smallest eigenvalue, then the exchange of MgO with FeO + CaO is the third fastest with an eigenvalue that is 6.3 times the smallest eigenvalue, then the exchange of CaO + K2O with all the other oxide components is the second fastest with an eigenvalue that is 7.5 times the smallest eigenvalue, and the exchange of Na2O with all other oxide components is the fastest with an eigenvalue that is 31 times the smallest eigenvalue. The slowest and fastest eigenvectors are consistent with those for simpler systems in most literature. The obtained diffusion matrix was successfully applied to predict diffusion profiles during mineral dissolution in basaltic melts.

Variational method for multiple parameter identification in elliptic PDEs

In the present paper we investigate the inverse problem of identifying simultaneously the diffusion matrix, source term and boundary condition in the Neumann boundary value problem for an elliptic partial differential equation (PDE) from a measurement data, which is weaker than required of the exact state. A variational method based on energy functions with Tikhonov regularization is here proposed to treat the identification problem. We discretize the PDE with the finite element method and prove the convergence as well as analyze error bounds of this approach.

Analysis of a degenerate parabolic cross-diffusion system for ion transport

A cross-diffusion system describing ion transport through biological membranes or nanopores in a bounded domain with mixed Dirichlet–Neumann boundary conditions is analyzed. The ion concentrations solve strongly coupled diffusion equations with a drift term involving the electric potential which is coupled to the concentrations through a Poisson equation. The global-in-time existence of bounded weak solutions and the uniqueness of weak solutions under moderate regularity assumptions are shown. The main difficulties of the analysis are the cross-diffusion terms and the degeneracy of the diffusion matrix, preventing the use of standard tools. The proofs are based on the boundedness-by-entropy method, extended to nonhomogeneous boundary conditions, and the uniqueness technique of Gajewski. A finite-volume discretization in one space dimension illustrates the large-time behavior of the numerical solutions and shows that the equilibration rates may be very small.

Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces

We study the large deviation behaviour of the trajectories of empirical distributions of independent copies of time-homogeneous Feller processes on locally compact metric spaces. Under the condition that we can find a suitable core for the generator of the Feller process, we are able to define a notion of absolutely continuous trajectories of measures in terms of some topology on this core. Also, we define a Hamiltonian in terms of the linear generator and a Lagrangian as its Legendre transform. We prove the large deviation principle and show that the rate function can be decomposed as a rate function for the initial time and an integral over the Lagrangian, finite only for absolutely continuous trajectories of measures. We apply this result for diffusion and Lévy processes on $\mathbb{R}^{d}$, for pure jump processes with bounded jump kernel on arbitrary locally compact spaces and for discrete interacting particle systems. For diffusion processes, the theorem partly extends the Dawson and Gärtner theorem for noninteracting copies in the sense that it only holds for time-homogeneous processes, but on the other hand it holds for processes with degenerate diffusion matrix.