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Partially Reflected Brownian Motion: a Stochastic Approach to Transport Partially Reflected Brownian Motion: A Stochastic Approach to Transport Phenomena ∗ Denis S. Grebenkov† Laboratoire de Physique de la Mati`ere Condens´ee, C.N.R.S. – Ecole Polytechnique 91128 Palaiseau Cedex, France Received: August 2004; Published: September 2006 Abstract Transport phenomena are ubiquitous in nature and known to be impor- tant for various scientific domains. Examples can be found in physics, elec- trochemistry, heterogeneous catalysis, physiology, etc. To obtain new infor- mation about diffusive or Laplacian transport towards a semi-permeable or resistive interface, one can study the random trajectories of diffusing parti- cles modeled, in a first approximation, by the partially reflected Brownian motion. This stochastic process turns out to be a convenient mathematical foundation for discrete, semi-continuous and continuous theoretical descrip- tions of diffusive transport. This paper presents an overview of these topics with a special empha- sis on the close relation between stochastic processes with partial reflections and Laplacian transport phenomena. We give selected examples of these phenomena followed by a brief introduction to the partially reflected Brow- nian motion and related probabilistic topics (e.g., local time process and spread harmonic measure). A particular attention is paid to the use of the Dirichlet-to-Neumann operator. Some practical consequences and further arXiv:math/0610080v1 [math.PR] 2 Oct 2006 perspectives are discussed. Keywords: Diffusion with Reflections; Mixed Boundary Value Problems; Lapla- cian Transport Phenomena. ∗This article partially reproduces the chapter which has been written by the author for the volume “Focus on Probability Theory”, and it should be referenced as D. S. Grebenkov, in Focus on Probability Theory, Ed. L. R. Velle, pp. 135-169 (Nova Science Publishers, 2006). The bibligraphic reference (ISBN) is 1-59454-474-3. Further information about this volume can be found on https://www.novapublishers.com/catalog/ †E-mail address: [email protected] 1 Partially Reflected Brownian Motion... 2 Introduction An erratic motion of pollens of Clarkia (primrose family) discovered by Robert Brown in 1827 and quantitatively described by Albert Einstein in 1905 gave a substantial impact for developing mathematical theory of stochastic processes, an important branch of modern mathematics. Supported by rigorous mathematical foundations, the Brownian motion and related stochastic processes found numer- ous applications in different scientific domains, from theoretical physics to biol- ogy and economics. To study the transport of species diffusing from a remote source towards and across semi-permeable or resistive interface (e.g., cellular membrane), one can employ either an averaged description in terms of an ap- propriate boundary value problem for the concentration of species, or stochastic analysis of their random trajectories. In the first case, a finite permeability (re- activity, resistivity, etc.) of the interface leads to the mixed or Fourier boundary condition, while in the second case it can be modeled as partial reflections on the boundary. Physical or chemical processes governed by the Laplace equation (sta- tionary diffusion) with mixed boundary condition are generally called Laplacian transport phenomena. Their examples are found in physiology (oxygen diffusion towards and across alveolar membranes), in electrochemistry (electric transport in electrolytic cells), in heterogeneous catalysis (diffusion of reactive molecules towards catalytic surfaces), in nuclear magnetic resonance (diffusion of spins in confining porous media). Studying random trajectories of diffusing species, one can extract a subtle information about the system in question. However, the theo- retical or numerical analysis of these phenomena is in general complicated by an irregular geometry of the interface (e.g., microroughness of metallic electrodes, see Section 1). In this paper, we focus on a particular stochastic process, called partially re- flected Brownian motion (PRBM), and its application to study Laplacian transport phenomena. Our main purpose is to capture the attention to this interesting pro- cess itself, and its use for understanding the influence of a geometrical irregularity of the interface on transport properties. Bearing in mind the particular role of the geometry, we would like to “bridge” theoretical, numerical and experimental stud- ies of Laplacian transport phenomena on the one side, and powerful mathematical methods of stochastic analysis on the other side. Since the extensive literature existing on both topics is generally difficult to get through for non-specialists, we prefer to use a descriptive style of writing in order to give the whole vision of the problem, without specifying particular details which can be found anywhere else (e.g., see references at the end of this paper). In the first section, we present three examples of Laplacian transport phenom- ena in different scientific fields. Their mathematical description by the mixed boundary value problem opens encouraging possibilities to apply powerful tools Partially Reflected Brownian Motion... 3 of potential theory, variational analysis and probability theory. The second section is devoted to remind some basic definitions of stochastic process theory: stop- ping times, reflected Brownian motion, local time process, harmonic measure, etc. In the third section, we introduce the partially reflected Brownian motion and show its properties for a planar surface. An important relation to the Dirichlet- to-Neumann operator is revealed and then illustrated by several examples. The last section presents different stochastic descriptions of Laplacian transport phe- nomena: a recently developed continuous approach and two other methods. In the conclusion, we summarize the essential issues of the paper. 1 Laplacian Transport Phenomena The transport of species between two distinct “regions” separated by an interface occurs in various biological systems: water and minerals are pumped by roots from the earth, ions and biological species penetrate through cellular membranes, oxygen molecules diffuse towards and pass through alveolar ducts, and so on. Transport processes are relevant for many other scientific domains, for example, heterogeneous catalysis and electrochemistry. In this section, we shall give three1 important examples of the particular transport process, called Laplacian or diffu- sive transport. 1.1 Stationary Diffusion across Semi-permeable Membranes Let us begin by considering the respiration process of mammals. Inbreathing a fresh air, one makes it flow from the mouth to the dichotomic bronchial tree of the lungs (Fig. 1). For humans, first fifteen generations of this tree serve for convec- tional transport of the air towards pulmonary acini, terminal gas exchange units [1]. A gradual increase of the total cross-section of bronchiae leads to a decrease of air velocity. At the entrance of the acinus, it becomes lower than the character- istic diffusion velocity [2]. As a consequence, one can describe the gas exchange inside the acinus as stationary diffusion of oxygen molecules in air from the en- trance (“source” with constant concentration C0 during one cycle of respiration) to the alveolar membranes [3]. In the bulk, the flux density is proportional to the gradient of concentration (Fick’s law), J = D C, where D is the diffusion coefficient. The mass conservation law, written− locally∇ as div J = 0, leads to the Laplace equation ∆C = 0 in the bulk. The flux density towards the interface is simply Jn = D ∂C/∂n, where the normal derivative ∂/∂n is directed to the bulk. 1Diffusive NMR phenomena present another important example when the transport properties are considerably affected by irregular geometry. In this paper, we do not discuss this case since it has been recently reviewed in a separate paper [4]. Partially Reflected Brownian Motion... 4 Arrived to the alveolar membrane, oxygen molecules can penetrate across the boundary for further absorption in blood, or to be “bounced” on it and to continue the motion. The “proportion” of absorbed and reflected molecules can be charac- terized by permeability W varying from 0 (perfectly reflecting boundary) to in- finity (perfectly absorbing boundary). In this description, the flux density across the alveolar membrane is proportional to the concentration, Jn = WC. Equat- ing these two densities on the alveolar membrane, one gets a mixed boundary condition, D(∂C/∂n) = WC, called also Fourier or Robin boundary condition. Resuming these relations, one provides the following mathematical description for the diffusion regime of human or, in general, mammalian respiration: ∆C =0 inthebulk (1) C = C0 onthesource (2) ∂ I Λ C =0 on the alveolar membrane (3) − ∂n where the underlying physics and physiology are characterized by a single pa- rameter Λ = D/W , which is homogeneous to a length (I stands for the identity operator). Note also that the dependence on constant C0 is irrelevant. In what follows, we address to this “classical” boundary value problem. The essential complication resides in a very irregular geometry of the pulmonary acinus, which presents a branched structure of eight generations (for humans), “sticked” by alve- olar ducts (Fig. 1). For small Λ, only a minor part of the boundary is involved to the transport process (so-called Dirichlet active zone), whereas the flux across the rest of the boundary is almost zero (this effect is called diffusional screening [3, 5,
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