Diffusion in a crowded environment

We analyze a pair of diffusion equations which are derived in the infinite system--size limit from a microscopic, individual--based, stochastic model. Deviations from the conventional Fickian picture are found which ultimately relate to the depletion of resources on which the particles rely. The macroscopic equations are studied both analytically and numerically, and are shown to yield anomalous diffusion which does not follow a power law with time, as is frequently assumed when fitting data for such phenomena. These anomalies are here understood within a consistent dynamical picture which applies to a wide range of physical and biological systems, underlining the need for clearly defined mechanisms which are systematically analyzed to give definite predictions.Comment: 4 pages, 3 figures, minor change

Reaction-Diffusion Process Driven by a Localized Source: First Passage Properties

We study a reaction-diffusion process that involves two species of atoms, immobile and diffusing. We assume that initially only immobile atoms, uniformly distributed throughout the entire space, are present. Diffusing atoms are injected at the origin by a source which is turned on at time t=0. When a diffusing atom collides with an immobile atom, the two atoms form an immobile stable molecule. The region occupied by molecules is asymptotically spherical with radius growing as t^{1/d} in d>=2 dimensions. We investigate the survival probability that a diffusing atom has not become a part of a molecule during the time interval t after its injection and the probability density of such a particle. We show that asymptotically the survival probability (i) saturates in one dimension, (ii) vanishes algebraically with time in two dimensions (with exponent being a function of the dimensionless flux and determined as a zero of a confluent hypergeometric function), and (iii) exhibits a stretched exponential decay in three dimensions.Comment: 7 pages; version 2: section IV is re-written, references added, 8 pages (final version

Role of proton irradiation and relative air humidity on iron corrosion

This paper presents a study of the effects of proton irradiation on iron corrosion. Since it is known that in humid atmospheres, iron corrosion is enhanced by the double influence of air and humidity, we studied the iron corrosion under irradiation with a 45% relative humidity. Three proton beam intensities (5, 10 and 20 nA) were used. To characterise the corrosion layer, we used ion beam methods (Rutherford Backscattering Spectrometry (RBS), Elastic Recoil Detection Analysis (ERDA)) and X-ray Diffraction (XRD) analysis. The corrosion kinetics are plotted for each proton flux. A diffusion model of the oxidant species is proposed, taking into account the fact that the flux through the surface is dependent on the kinetic factor K. This model provides evidence for the dependence of the diffusion coefficient, D, and the kinetic factor, K, on the proton beam intensity. Comparison of the values for D with the diffusion coefficients for thermal oxygen diffusion in iron at 300 K suggests an enhancement due to irradiation of 6 orders of magnitude

2D pattern evolution constrained by complex network dynamics

Complex networks have established themselves along the last years as being particularly suitable and flexible for representing and modeling several complex natural and human-made systems. At the same time in which the structural intricacies of such networks are being revealed and understood, efforts have also been directed at investigating how such connectivity properties define and constrain the dynamics of systems unfolding on such structures. However, lesser attention has been focused on hybrid systems, \textit{i.e.} involving more than one type of network and/or dynamics. Because several real systems present such an organization (\textit{e.g.} the dynamics of a disease coexisting with the dynamics of the immune system), it becomes important to address such hybrid systems. The current paper investigates a specific system involving a diffusive (linear and non-linear) dynamics taking place in a regular network while interacting with a complex network of defensive agents following Erd\"os-R\'enyi and Barab\'asi-Albert graph models, whose nodes can be displaced spatially. More specifically, the complex network is expected to control, and if possible to extinguish, the diffusion of some given unwanted process (\textit{e.g.} fire, oil spilling, pest dissemination, and virus or bacteria reproduction during an infection). Two types of pattern evolution are considered: Fick and Gray-Scott. The nodes of the defensive network then interact with the diffusing patterns and communicate between themselves in order to control the spreading. The main findings include the identification of higher efficiency for the Barab\'asi-Albert control networks.Comment: 18 pages, 32 figures. A working manuscript, comments are welcome

Solving two-phase freezing Stefan problems: Stability and monotonicity

[EN] The two-phase Stefan problems with phase formation and depletion are special cases ofmoving boundary problemswith interest in science and industry. In this work, we study a solidification problem, introducing a front-fixing transformation. The resulting non-linear partial differential system involves singularities, both at the beginning of the freezing process and when the depletion is complete, that are treated with special attention in the numerical modelling. The problem is decomposed in three stages, in which implicit and explicit finite difference schemes are used. Numerical analysis reveals qualitative properties of the numerical solution spatial monotonicity of both solid and liquid temperatures and the evolution of the solidification front. Numerical experiments illustrate the behaviour of the temperatures profiles with time, as well as the dynamics of the solidification front.Ministerio de Ciencia, Innovacion y Universidades, Grant/Award Number: MTM2017-89664-P.Piqueras, MA.; Company Rossi, R.; Jódar Sánchez, LA. (2020). Solving two-phase freezing Stefan problems: Stability and monotonicity. Mathematical Methods in the Applied Sciences. 43(14):7948-7960. https://doi.org/10.1002/mma.5787S794879604314Schmidt, A. (1996). Computation of Three Dimensional Dendrites with Finite Elements. Journal of Computational Physics, 125(2), 293-312. doi:10.1006/jcph.1996.0095Singh, S., & Bhargava, R. (2014). Simulation of Phase Transition During Cryosurgical Treatment of a Tumor Tissue Loaded With Nanoparticles Using Meshfree Approach. Journal of Heat Transfer, 136(12). doi:10.1115/1.4028730Company, R., Egorova, V. N., & Jódar, L. (2014). Solving American Option Pricing Models by the Front Fixing Method: Numerical Analysis and Computing. Abstract and Applied Analysis, 2014, 1-9. doi:10.1155/2014/146745Griewank, P. J., & Notz, D. (2013). Insights into brine dynamics and sea ice desalination from a 1-D model study of gravity drainage. Journal of Geophysical Research: Oceans, 118(7), 3370-3386. doi:10.1002/jgrc.20247Javierre, E., Vuik, C., Vermolen, F. J., & van der Zwaag, S. (2006). A comparison of numerical models for one-dimensional Stefan problems. Journal of Computational and Applied Mathematics, 192(2), 445-459. doi:10.1016/j.cam.2005.04.062Briozzo, A. C., Natale, M. F., & Tarzia, D. A. (2007). Explicit solutions for a two-phase unidimensional Lamé–Clapeyron–Stefan problem with source terms in both phases. Journal of Mathematical Analysis and Applications, 329(1), 145-162. doi:10.1016/j.jmaa.2006.05.083Caldwell, J., & Chan, C.-C. (2000). Spherical solidification by the enthalpy method and the heat balance integral method. Applied Mathematical Modelling, 24(1), 45-53. doi:10.1016/s0307-904x(99)00031-1Chantasiriwan, S., Johansson, B. T., & Lesnic, D. (2009). The method of fundamental solutions for free surface Stefan problems. Engineering Analysis with Boundary Elements, 33(4), 529-538. doi:10.1016/j.enganabound.2008.08.010Hon, Y. C., & Li, M. (2008). A computational method for inverse free boundary determination problem. International Journal for Numerical Methods in Engineering, 73(9), 1291-1309. doi:10.1002/nme.2122RIZWAN-UDDIN. (1999). A Nodal Method for Phase Change Moving Boundary Problems. International Journal of Computational Fluid Dynamics, 11(3-4), 211-221. doi:10.1080/10618569908940875Caldwell, J., & Kwan, Y. Y. (2003). On the perturbation method for the Stefan problem with time-dependent boundary conditions. International Journal of Heat and Mass Transfer, 46(8), 1497-1501. doi:10.1016/s0017-9310(02)00415-5Stephan, K., & Holzknecht, B. (1976). Die asymptotischen lösungen für vorgänge des erstarrens. International Journal of Heat and Mass Transfer, 19(6), 597-602. doi:10.1016/0017-9310(76)90042-9Savović, S., & Caldwell, J. (2003). Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions. International Journal of Heat and Mass Transfer, 46(15), 2911-2916. doi:10.1016/s0017-9310(03)00050-4Kutluay, S., Bahadir, A. R., & Özdeş, A. (1997). The numerical solution of one-phase classical Stefan problem. Journal of Computational and Applied Mathematics, 81(1), 135-144. doi:10.1016/s0377-0427(97)00034-4Asaithambi, N. S. (1997). A variable time step Galerkin method for a one-dimensional Stefan problem. Applied Mathematics and Computation, 81(2-3), 189-200. doi:10.1016/0096-3003(95)00329-0Landau, H. G. (1950). Heat conduction in a melting solid. Quarterly of Applied Mathematics, 8(1), 81-94. doi:10.1090/qam/33441Churchill, S. W., & Gupta, J. P. (1977). Approximations for conduction with freezing or melting. International Journal of Heat and Mass Transfer, 20(11), 1251-1253. doi:10.1016/0017-9310(77)90134-xKutluay, S., & Esen, A. (2004). An isotherm migration formulation for one-phase Stefan problem with a time dependent Neumann condition. Applied Mathematics and Computation, 150(1), 59-67. doi:10.1016/s0096-3003(03)00197-8Esen, A., & Kutluay, S. (2004). A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method. Applied Mathematics and Computation, 148(2), 321-329. doi:10.1016/s0096-3003(02)00846-9Mitchell, S. L., & Vynnycky, M. (2016). On the accurate numerical solution of a two-phase Stefan problem with phase formation and depletion. Journal of Computational and Applied Mathematics, 300, 259-274. doi:10.1016/j.cam.2015.12.021Meek, P. C., & Norbury, J. (1984). Nonlinear Moving Boundary Problems and a Keller Box Scheme. SIAM Journal on Numerical Analysis, 21(5), 883-893. doi:10.1137/0721057Tarzia, D. (2017). Relationship between Neumann solutions for two-phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions. Thermal Science, 21(1 Part A), 187-197. doi:10.2298/tsci140607003tPlemmons, R. J. (1977). M-matrix characterizations.I—nonsingular M-matrices. Linear Algebra and its Applications, 18(2), 175-188. doi:10.1016/0024-3795(77)90073-8Axelsson, O. (1994). Iterative Solution Methods. doi:10.1017/cbo978051162410

Dissolution in a field

We study the dissolution of a solid by continuous injection of reactive ``acid'' particles at a single point, with the reactive particles undergoing biased diffusion in the dissolved region. When acid encounters the substrate material, both an acid particle and a unit of the material disappear. We find that the lengths of the dissolved cavity parallel and perpendicular to the bias grow as t^{2/(d+1)} and t^{1/(d+1)}, respectively, in d-dimensions, while the number of reactive particles within the cavity grows as t^{2/(d+1)}. We also obtain the exact density profile of the reactive particles and the relation between this profile and the motion of the dissolution boundary. The extension to variable acid strength is also discussed.Comment: 6 pages, 6 figures, 2-column format, for submission to PR

From the solutions of diffusion equation to the solutions of subdiffusive one

Starting with the Green's functions found for normal diffusion, we construct exact time-dependent Green's functions for subdiffusive equation (with fractional time derivatives), with the boundary conditions involving a linear combination of fluxes and concentrations. The method is particularly useful to calculate the concentration profiles in a multi-part system where different kind of transport occurs in each part of it. As an example, we find the solutions of subdiffusive equation for the system composed from two parts with normal diffusion and subdiffusion, respectively.Comment: 11 pages, 2 figure

Realistic boundary conditions for stochastic simulations of reaction-diffusion processes

Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). Such reaction-diffusion processes can be mathematically modelled using either deterministic partial-differential equations or stochastic simulation algorithms. The latter provide a more detailed and precise picture, and several stochastic simulation algorithms have been proposed in recent years. Such models typically give the same description of the reaction-diffusion processes far from the boundary of the simulated domain, but the behaviour close to a reactive boundary (e.g. a membrane with receptors) is unfortunately model-dependent. In this paper, we study four different approaches to stochastic modelling of reaction-diffusion problems and show the correct choice of the boundary condition for each model. The reactive boundary is treated as partially reflective, which means that some molecules hitting the boundary are adsorbed (e.g. bound to the receptor) and some molecules are reflected. The probability that the molecule is adsorbed rather than reflected depends on the reactivity of the boundary (e.g. on the rate constant of the adsorbing chemical reaction and on the number of available receptors), and on the stochastic model used. This dependence is derived for each model.Comment: 24 pages, submitted to Physical Biolog

Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation