On Diffusion in Narrow Random Channels
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On Diffusion in Narrow Random Channels Mark Freidlin · Wenqing Hu
Received: 23 October 2012 / Accepted: 3 May 2013 / Published online: 10 May 2013 © Springer Science+Business Media New York 2013
Abstract We consider in this paper a solvable model for the motion of molecular motors. Based on the averaging principle, we reduce the problem to a diffusion process on a graph. We then calculate the effective speed of transportation of these motors. Keywords Brownian motors/ratchets · Averaging principle · Diffusion processes on graphs · Random environment
1 Introduction One of the possible ways to model Brownian motors/ratchets is to describe them as particles (which model the protein molecules) traveling along a designated track (see [7]). At a microscopic scale such a motion is conveniently described as a diffusion process with a deterministic drift. On the other hand, the designated track along which the molecule is traveling can be viewed as a tubular domain of some random shape. In particular, such a domain can have many random “wings” added to it. (See Fig. 1. The shaded areas represent the “wings”.) In this paper we are going to introduce a mathematically solvable model of the Brownian motor and discuss some interesting relevant questions around this problem. Our model is based on ideas similar to that of [5] and [2, Chap. 7]. + The model is as follows. Let h± 0 (x) be a pair of piecewise smooth functions with h0 (x) − − − + h0 (x) = l0 (x) > 0. Let D0 = {(x, z) : x ∈ R, h0 (x) ≤ z ≤ h0 (x)} be a tubular 2-d domain of infinite length, i.e. it goes along the whole x-axis. At the discontinuities of h± 0 (x), we connect the pieces of the boundary via straight vertical lines. The domain D0 models the “main” channel in which the motor is traveling. Let a sequence of “wings” Dj (j ≥ 1) be attached to D0 . These wings are attached to D0 at the discontinuities of the functions h± 0 (x). M. Freidlin · W. Hu () Department of Mathematics, University of Maryland at College Park, College Park, MD, USA e-mail: [email protected] M. Freidlin e-mail: [email protected]
On Diffusion in Narrow Random Channels
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Fig. 1 A model of the molecular motor
Consider the union D = D0 ∪ ( ∞ j =1 Dj ). An example of such a domain D is shown in Fig. 1, in which one can see four “wings” D1 , D2 , D3 , D4 . We assume that, after adding the “wings”, for the domain D, the boundary ∂D has two smooth pieces: the upper boundary and the lower boundary. Let n(x, z) = (n1 (x, z), n2 (x, z)) be the inward unit normal vector to ∂D. We make some assumptions on the domain D. Assumption 1 The set of points x ∈ R for which there are points (x, z) ∈ ∂D at which the unit normal vector n(x, z) is parallel to the x-axis: n2 (x, z) = 0 has no limit points in R. Each such point x corresponds to only one point (x, z) ∈ ∂D for which n2 (x, z) = 0. Assumption 2 For every x the cross-section of the region D at level x, i.e., the set of all points belonging to D with the first coordinate equal to x, consists of either one or two intervals that are its connec
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