The Mathematics of Diffusion.

Type: Article
Publication Date: 1957-03-01
Citations: 4106
DOI: https://doi.org/10.1021/ja01562a070

Abstract

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTThe Mathematics of Diffusion.J. W. Westwater and H. G. DrickamerCite this: J. Am. Chem. Soc. 1957, 79, 5, 1267Publication Date (Print):March 1, 1957Publication History Published online1 May 2002Published inissue 1 March 1957https://pubs.acs.org/doi/10.1021/ja01562a070https://doi.org/10.1021/ja01562a070research-articleACS PublicationsRequest reuse permissionsArticle Views349Altmetric-Citations25LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access optionsGet e-Alertsclose Get e-Alerts

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The concept of diffusion describes regularities in the temporal propagation of people, objects, practices, or information in geographical space. The Swedish geographer Torsten Hägerstrand attached his name as early as … The concept of diffusion describes regularities in the temporal propagation of people, objects, practices, or information in geographical space. The Swedish geographer Torsten Hägerstrand attached his name as early as the 1950s to the first formulation of the theory of the spatial diffusion of innovations. The spatial constraint on movement is expressed both in terms of proximity and in terms of connectivity. The theory establishes a generic link between behaviors observed at the microscale (those of individuals) and the resulting configurations at the meso‐ or macroscale (regional, continental, and global). Indeed, spatial diffusion theory, with associated concepts and simulation models, has found applications for analyzing and predicting a large variety of spatiotemporal processes. The main ones concern migrations, innovation diffusion, and the spread of epidemics. Recent research focuses on the transmission of epidemics, information, and opinions, particularly via digital networks.
It has already been demonstrated that concentration C depends upon space co-ordinates and upon time. An example of such a relationship is given on page 21, equation (2.1). If M … It has already been demonstrated that concentration C depends upon space co-ordinates and upon time. An example of such a relationship is given on page 21, equation (2.1). If M and D are both taken to be unity, the relationship becomes (3.1) <math display='block'> <mrow> <mi>C</mi><mo>=</mo><mfrac> <mn>1</mn> <mrow> <mn>2</mn><mo>&#x221A;</mo><mrow><mo>(</mo> <mrow> <mi>&#x03C0;</mi><mi>t</mi> </mrow> <mo>)</mo></mrow> </mrow> </mfrac> <mi>exp</mi><mrow><mo>(</mo> <mrow> <mo>&#x2212;</mo><msup> <mi>x</mi> <mn>2</mn> </msup> <mo>/</mo><mn>4</mn><mi>t</mi> </mrow> <mo>)</mo></mrow> </mrow> </math> $$C = \frac{1}{{2\surd \left( {\pi t} \right)}}\exp \left( { - {x^2}/4t} \right)$$
This book presents a full account of the differential equations applicable to many diffusion problems. Analytical discussions are given of a wide variety of problems, and the solutions are completed … This book presents a full account of the differential equations applicable to many diffusion problems. Analytical discussions are given of a wide variety of problems, and the solutions are completed with well-drawn graphical representations.
The famous diffusion equation, also known as the heat equation, reads $$\frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}},$$ where $$u(x,t)$$ is the unknown function to be solved for, x is a coordinate in … The famous diffusion equation, also known as the heat equation, reads $$\frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}},$$ where $$u(x,t)$$ is the unknown function to be solved for, x is a coordinate in space, and t is time. The coefficient α is the diffusion coefficient and determines how fast u changes in time. A quick short form for the diffusion equation is $$u_{t}=\alpha u_{xx}$$ . Compared to the wave equation, $$u_{tt}=c^{2}u_{xx}$$ , which looks very similar, the diffusion equation features solutions that are very different from those of the wave equation. Also, the diffusion equation makes quite different demands to the numerical methods.
Abstract This chapter addresses diffusion, random walks and congestion in multilayer networks. Here it is revealed that diffusion on a multilayer network can be significantly speed up with respect to … Abstract This chapter addresses diffusion, random walks and congestion in multilayer networks. Here it is revealed that diffusion on a multilayer network can be significantly speed up with respect to diffusion taking place on its single layers taken in isolation, and that sometimes it is possible also to observe super-diffusion. Diffusion is here characterized on multilayer network structures by studying the spectral properties of the supra-Laplacian and the dependence on the diffusion constant among different layers. Random walks and its variations including the Lévy Walk are shown to reflect the improved navigability of multilayer networks with more layers. These results are here compared with the results of traffic on multilayer networks that, on the contrary, point out that increasing the number of layers could be detrimental and could lead to congestion.
This paper investigates the temporal anomalous diffusion and drift of particles in a comb backbone. Fractional Cattaneo–Christov flux is introduced and the formulated governing equation has a Dirac delta function … This paper investigates the temporal anomalous diffusion and drift of particles in a comb backbone. Fractional Cattaneo–Christov flux is introduced and the formulated governing equation has a Dirac delta function and mixed partial derivative which displays evolutional characteristics from parabolic (α → 0) to hyperbolic (α = 1). A solution is obtained numerically with the L1-scheme and the shifted Grunwald formula. Two completely opposite distributions are found: one is the monotonically decreasing convex distribution for the particle numbers and the other is the monotonically increasing concave distribution for the fractional order moments. The change rates of both the distributions decrease with the increase of time. Moreover, the influences of related parameters on the temporal evolution characteristics are discussed and analyzed in detail.
Abstract In this paper, we consider a diffusive predator-prey system with spatial memory and predator-taxis. Since in this system, the memory delay appears in the diffusion term, and the diffusion … Abstract In this paper, we consider a diffusive predator-prey system with spatial memory and predator-taxis. Since in this system, the memory delay appears in the diffusion term, and the diffusion term is nonlinear, the classical normal form of Hopf bifurcation in the reaction-diffusion system with delay can't be applied to this system. Thus, in this paper, we first derive an algorithm for calculating the normal form of Hopf bifurcation in this system. Then in order to illustrate the effectiveness of our newly developed algorithm, we consider the diffusive Holling-Tanner model with spatial memory and predator-taxis. The stability and Hopf bifurcation analysis of this model are investigated, and the direction and stability of Hopf bifurcation periodic solution are also researched by using our newly developed algorithm for calculating the normal form of Hopf bifurcation. At last, we carry out some numerical simulations, two stable spatially inhomogeneous periodic solutions corresponding to the mode-1 and mode-2 Hopf bifurcations are found, which verifies our theoretical analysis results.
Abstract This chapter addresses the problem of the construction of solutions of the GPME by viewing it as an abstract evolution equation, or more precisely as an ordinary differential equation … Abstract This chapter addresses the problem of the construction of solutions of the GPME by viewing it as an abstract evolution equation, or more precisely as an ordinary differential equation with values in a Hilbert or Banach space. The outline of the chapter is as follows. Section 10.1 deals with the theory of maximal monotone operators in Hilbert spaces. Section 10.2 introduces time discretizations and the concepts of mild solutions and the accretive operators in Banach spaces. Section 10.3 applies the theory of accretive operators to the filtration equation. The chapter ends with some new ideas of mass transportation and gradient flows and a review of different extensions to more general equations where new concepts of solution are needed.
We present a simple model to describe the evolution of the structure factor during irreversible diffusion limited cluster-cluster aggregation. For growing compact clusters, the scattered intensity is predicted to scale … We present a simple model to describe the evolution of the structure factor during irreversible diffusion limited cluster-cluster aggregation. For growing compact clusters, the scattered intensity is predicted to scale as in spinodal decomposition, i.e., as ${q}_{m}^{d}I(\frac{q}{{q}_{m}})$. For fractal clusters, the scattered intensity is predicted to apparently scale in ${q}_{m}^{{d}_{f}}I(\frac{q}{{q}_{m}})$ only in late stages. We find an excellent agreement with the experimental results of Carpineti and Giglio [Phys. Rev. Lett. 68, 3327 (1992)] and with novel data from a dynamic simulation.
Abstract This chapter reviews the main applications of the porous medium equation (PME). Section 2.1 describes the standard model of gas flow through a porous medium (Darcy-Leibenzon-Muskat). Section 2.2 describes … Abstract This chapter reviews the main applications of the porous medium equation (PME). Section 2.1 describes the standard model of gas flow through a porous medium (Darcy-Leibenzon-Muskat). Section 2.2 describes the model of nonlinear heat transfer (Zel'dovich-Raizer). Section 2.3 describes Boussinesq's model of groundwater flow, and Section 2.4 presents a model of population dynamics (Gurtin-McCamy).
Summary The effects of geometry upon diffusion from instantaneous sources is considered. Although surface area is important initially, it is the square of the volume of the source which determines … Summary The effects of geometry upon diffusion from instantaneous sources is considered. Although surface area is important initially, it is the square of the volume of the source which determines flux from the source at large times. This asymptotic shape independence is demonstrated for spherical and cuboidal sources.
Tikhonov’s regularization procedure is applied to the operation of differentiation, resulting in a procedure for numerical differentiation for which the effects of errors in the values of the function being … Tikhonov’s regularization procedure is applied to the operation of differentiation, resulting in a procedure for numerical differentiation for which the effects of errors in the values of the function being differentiated on the values for the derivative obtained in the procedure can be studied. The theoretical discussion is complemented by the results of numerical experiments.
Virtually every ecosystem has been invaded by exotic organisms with potentially drastic consequences for the native fauna or flora. Studying the forms and rates of invading species has been an … Virtually every ecosystem has been invaded by exotic organisms with potentially drastic consequences for the native fauna or flora. Studying the forms and rates of invading species has been an important topic in spatial ecology. We investigate two two-species competition models with Allee effects in the forms of reaction-diffusion equations and integro-difference equations. We discuss the spatial transitions from a mono-culture equilibrium to a coexistence equilibrium or a different mono-culture equilibrium in these models. We provide formulas for the spreading speeds based on the linear determinacy and show the results on the existence of traveling waves. We also study a two-sex stage-structured model. We carry out initial analysis for the spreading speed and conduct numerical simulations on the traveling waves and spreading speeds in the two-sex model.
We consider the problem of drug diffusion in the dermal layer of human body. Two existing mathematical models of the drug diffusion problem are recalled. We obtain that the existing … We consider the problem of drug diffusion in the dermal layer of human body. Two existing mathematical models of the drug diffusion problem are recalled. We obtain that the existing models lead to inconsistent equations for the steady state condition. We also obtain that solutions to the existing models are unrealistic for some cases of the unsteady state condition, because negative drug concentrations occur due to the inappropriate assumption of the model. Therefore, in this paper, we propose a modified mathematical model, so that the model is consistent, and the solution is nonnegative for both steady and unsteady state conditions of the drug diffusion problem in the dermal layer of human body. For the steady state condition, the exact solution to the proposed model is given. For unsteady state condition, we use a finite difference method for solving the models numerically, where the discretisation is centred in space and forward in time. Simulation results confirm that our proposed model and method preserve the non-negativity of the solution to the problem, so the solution is more realistic than that of the old model.
A simple mathematical model of tumor growth by diffusion is constructed in order to examine and evaluate different hypotheses concerning the evolution of a solid carcinoma. A primary objective is … A simple mathematical model of tumor growth by diffusion is constructed in order to examine and evaluate different hypotheses concerning the evolution of a solid carcinoma. A primary objective is to infer the chemical source of growth inhibition from the most easily obtained data, namely, the outer radius of the nodule as a function of time and a histological examination of the final dormant state. In section 6 some of the conclusions of this study relating to a prototype experiment and described with as little mathematics as possible.
We study the hopping diffusion on a linear chain with different hopping rates on the left or right of the origin. The relevance of our model with respect to experiments … We study the hopping diffusion on a linear chain with different hopping rates on the left or right of the origin. The relevance of our model with respect to experiments on interdiffusion between different solid materials is pointed out.
The evolution of a one-dimensional array of reactive lamellae with distributed striation thickness is studied by means of simulations, scaling analysis, and space-averaged kinetics. An infinitely fast, diffusion-controlled reaction A+B\ensuremath{\rightarrow}2P … The evolution of a one-dimensional array of reactive lamellae with distributed striation thickness is studied by means of simulations, scaling analysis, and space-averaged kinetics. An infinitely fast, diffusion-controlled reaction A+B\ensuremath{\rightarrow}2P occurs at the interfaces between striations. As time increases, thin striations are eaten by thicker neighbors resulting in a modification of the striation thickness distribution (STD). Scaling analysis suggests that the STD evolves into a universal form and that the behavior of the system at short and long times is characterized by two different kinetic regimes. These predictions are confirmed by means of a novel numerical algorithm.
The concept of differentiation and integration to non-integer order has its origins in the nineteen century. However, only in the second-half of the twenty century appeared the first applications related … The concept of differentiation and integration to non-integer order has its origins in the nineteen century. However, only in the second-half of the twenty century appeared the first applications related to the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control methodologies are investigated and compared, such as the adoption of a Smith predictor structure for the better control of systems with time delays. Several simulations are presented assessing the performance of the proposed fractional-order algorithms.
Abstract Three general methods are developed for solving moving‐boundary problems which are governed by diffusional processes such as heat and mass transfer. Examples of such problems include melting, evaporation, and … Abstract Three general methods are developed for solving moving‐boundary problems which are governed by diffusional processes such as heat and mass transfer. Examples of such problems include melting, evaporation, and ablation. A method based upon a Riemann‐Volterra integration of the diffusion equation leads to nonlinear integrodifferential equations for the boundary motion that are in terms of definite integrals involving Green's functions. An analytical method, which is more convenient for problems involving phase motion, is based on the method of intermediate integrals. A numerical method based on finite difference approximations is implemented on the differential analyzer (analogue computer).
Abstract A method is described for determining the diffusion coefficient in dilute solid solutions by measurements of the electrical resistivity. The diffusion equation is solved numerically for the selected experimental … Abstract A method is described for determining the diffusion coefficient in dilute solid solutions by measurements of the electrical resistivity. The diffusion equation is solved numerically for the selected experimental conditions and this provides curves suitable for direct comparison with experimental data. Results obtained for the diffusion parameters of Zn in pure Al indicate the accuracy of the method.