Brownian motion, boundary layers, and quantized vortex production in thinHe4films

Abstract

Using the law of friction proposed for two-dimensional (2-D) vortices by Vinen, Ambegaokar, Halperin, Nelson, and Siggia, we show that the two-body Fokker-Planck equation describing the Brownian motion of a pair of interacting vortices in phase space is equivalent to the Onsager configuration-space Brownian motion for an interacting pair, and that the two-body equation reduces to the one-body Smoluchowski equation for a 2-D Coulomb charge in a logarithmic and external potential for the case of vortices of opposite circulation. Our earlier calculation of the dissociation rate for 2-D Coulomb charges is then used to predict the flow-induced dissociation rate for vortices in the limit of small (uniform) flow rates, and the dissociation rate for arbitrary field strengths is formulated by using a certain scaling law. We discuss the relation of our work to that of Myerson, Huberman, Myerson, and Doniach and also Ambegaokar, Halperin, Nelson, and Siggia, and we analyze various saddle-point approximations within the framework of our general theory. We have discovered that the theory of dissociation of 2-D vortices cannot be developed by an analytic perturbation theory, and develop the theory by using a singular perturbation method. The singular perturbation method allows us to draw an analogy between the concept of a boundary layer in a classical viscous fluid and the production of quantized vortex pairs of opposite circulation in thin ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ films. Finally, we point out that an experimental test of the velocity dependence of the dissociation rate is a test of the short-range interaction between vortices, with the velocity dependence $R\sim {U_s}^{\lambda }$ reflecting the (classical) assumption that two vortices attract logarithmically when their separation is only slightly greater than the vortex-core size.