Abstract Based on the theory of the thermodynamic equilibrium in a system of quantum vortices in superfluid liquids and in a Bose – Einstein condensate in the presence of a counterflow of normal and superfluid components, we investigate the structure of a chaotic tangle of quantum vortices in turbulent superfluid liquids. Using the characteristic functional method, the properties of hydrodynamic vortex filaments are examined. It is shown that the average curvature of the vortex lines is on the order of the intervortex distance, with the proportionality coefficient being independent of the counterflow velocity. It is found that the degree of anisotropy of the vortex loops does not depend on the applied counterflow velocity. The obtained results explain the origin of the anisotropy as well as the relationship between the curvature of the lines and the intervortex space and their dependence on the parameters of the problem.
Abstract Based on the theory of the thermodynamic equilibrium in a system of quantum vortices in superfluid liquids and in a Bose – Einstein condensate in the presence of a counterflow of normal and superfluid components, we investigate the structure of a chaotic tangle of quantum vortices in turbulent superfluid liquids. Using the characteristic functional method, the properties of hydrodynamic vortex filaments are examined. It is shown that the average curvature of the vortex lines is on the order of the intervortex distance, with the proportionality coefficient being independent of the counterflow velocity. It is found that the degree of anisotropy of the vortex loops does not depend on the applied counterflow velocity. The obtained results explain the origin of the anisotropy as well as the relationship between the curvature of the lines and the intervortex space and their dependence on the parameters of the problem.
A statement of the quantum turbulence problem in both a Bose – Einstein condensate (BEC) and superfluid helium is formulated. In superfluid helium use is made of a so-called vortex filament method, in which quantum vortices are represented by string-like objects, i.e. vortex lines. The dynamics of the vortex lines is determined by deterministic equations of motion, supplemented by random reconnections. Unlike He II, the laws of the dynamics of quantum vortices in BEC are based on the nonlinear Schrödinger equation. This makes it possible to obtain a microscopic description of the collision of vortices, the structure of a vortex filament, etc. A comparative analysis of these complementary approaches is carried out. It is shown that there are some features that do not automatically transfer the results obtained for BEC to vortices in He II and vice versa.
The paper describes several features of vortex filaments in a weakly non-ideal Bose-Einstein condensate (BEC) with a counterflow. The study focuses on a process of reconnection and equations of motion. A primary motivation of this study is a fact that quantum vortex dynamics for superfluid helium was obtained previously at a phenomenological level. But some critical problems can be solved through a study of a corresponding problem in BEC. By considering a vortex filament as a crossing line for surfaces where the real and imaginary parts of a macroscopic wave function ψ ( r , t ) turn into zero, we derived the equations of motion for a vortex filament. It is remarkable that the equation structure coincides completely with a structure of equation for vortex filament dynamics in superfluid helium that has been obtained from other (phenomenological) foundations.
The aim of this paper is to derive the equations of motion of quantum vortices in a weakly non-ideal Bose–Einstein condensate (BEC) at finite temperatures and in the presence of macroscopic superfluid and normal velocities. The motivation for this work is that the dynamics of quantum vortices in superfluid helium has been studied at the phenomenological level, and a number of issues require clarification, which can be addressed by studying the corresponding problem in BEC. In BEC, quantum vortices are located at points where the macroscopic wave function (MWF) ψ(r,T), which describes the BEC dynamics, vanishes ψ(r,T)=0. The intersection of the surfaces Reψ(r,T)=0 and Imψ(r,T)=0 forms a one-dimensional line, which is just the quantum vortex line. During the evolution of the MWF, the position of the zero surfaces changes, that is, the vortex lines move. Conversely, the motion of these vortex filaments determines the evolution of the MWF. Thus, there is a mutual correspondence, and the quantity ψ(r,T) can be considered as a functional of the entire vortex configuration, i.e., {s(ξ,t)}. This viewpoint allows us to derive the equations of motion for vortices from the Gross–Pitaevskii equation (GPE), which describes the dynamics of MWF. The key finding is the confirmation that the equation derived from the fundamental quantum description (the GPE) for a BEC has the same mathematical structure as the equation derived from macroscopic, phenomenological reasoning for superfluid helium.
Describing superfluid turbulence at intermediate scales between the intervortex distance and the macroscale requires an acceptable equation of motion for the density of quantized vortex lines L. The closure of such an equation for superfluid inhomogeneous flows requires additional inputs besides L and the normal and superfluid velocity fields. In this paper, we offer a minimal closure using one additional anisotropy parameter Il0. Using the example of counterflow superfluid turbulence, we derive two coupled closure equations for the vortex line density and the anisotropy parameter Il0 with an input of the normal and superfluid velocity fields. The various closure assumptions and the predictions of the resulting theory are tested against numerical simulations.
Quantum vortices are pivotal for understanding of phenomena in quantum hydrodynamics. Vortices were observed in different physical systems like trapped dilute Bose-Einstein condensates, liquid helium, exciton-polariton condensates and other types of systems. Foreign particles attached to the vortices often serve for a visualization of the vortex shape and kinematics in superfluid experiments. Fascinating discoveries were made in the field of cold quantum mixtures, where vortices created in one component may interact with the other component. This works raise the fundamental question of the interaction between quantum vortices and matter. The generalized nonlinear Schrodinger equation based formalism is applied here to model three different processes involving the interaction of quantum vortices with foreign particles: propagation of a fast classical particle in a superfluid under the influence of sound waves, scattering of a single fermion by a quantized vortex line and dynamics of vortex pairs doped with heavy bosonic matter. The obtained results allow to to clarify the details of recent experiments and acquire a better understanding of the multicomponent quantum turbulence.
We study the stochastic behavior of a set of chaotic vortex loops appeared in imperfect Bose gas. Dynamics of Bose-gas is supposed to obey Gross-Pitaevskii equation with additional noise satisfying fluctuation-dissipation relation. The corresponding Fokker-Planck equation for probability functional has solution P({ψ(r)})=Nexp(−H{ψ(r)}/T), where $H\{ψ({\bf r})\} $ is the Ginzburg-Landau free energy. Considering vortex filaments as topological defects of field ψ(r) we derive a Langevin-type equation of motion of the line with the correspondingly transformed stirring force. The respective Fokker-Planck equation for probability functional P({s(ξ)}) in vortex loop configuration space is shown to have a solution in the form of P({s(ξ)})=Nexp(−H{s}/T), where N is the normalizing factor and $H\{{\bf s}\} $ is energy of vortex line configurations. Analyzing this result we discuss possible reasons for destruction of the thermodynamic equilibrium and follow the mechanisms of transition to the turbulent state