A Brownian Needle in Dire Straits

Brownian motion of anisotropic objects such as an ellipsoid or a needle in two-dimensions is anisotropic and governed by longitudinal, transversal, and rotational diffusion constants. [1–5]. In a planar strip, when the needle is only slightly shorter than the width of the strip its turning around becomes a rare event, because there is not much room in configuration space for the vertical position of the needle in the strip. Thus the computation of the mean time to turn around becomes a narrow escape problem. It does not fall, however, under any one of the previously studied narrow escape time (NET) in planar or solid geometries [6–16] and many more, because the passage of the needle through a vertical position in a narrow strip is equivalent to the diffusion of anisotropic Brownian motion through a narrow neck cut from a boundary cusp. The problem of reaching an absorbing arc near a boundary cusp was reduced in [12] by a conformal map to the standard NET problem in a half plane and solved by extracting the singularity of the planar Neumann function. This method, as well as the boundary layer method of [6], fail in the problem at hand and the solution requires different mapping, as used in [17], and altogether different boundary layer analysis. More specifically, we consider the motion of a Brownian needle of length l in a narrow strip of width l0 > l such