BROWNIAN-MOTION, BOUNDARY-LAYERS, AND QUANTIZED VORTEX PRODUCTION IN THIN HE-4 FILMS

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 He4 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∼Usλ reflecting the (classical) assumption that two vortices attract logarithmically when their separation is only slightly greater than the vortex-core size. © 1979 The American Physical Society.