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Journal of Statistical Mechanics Theory and Experiment

Gauge-field model of superfluid turbulence in the zero-temperature limit

Mohammad Mehrafarin

Published Feb 1, 2018• 0 citations
Publisher

Abstract

Abstract We present a gauge-field extension of the Bose condensate model that describes T ≈ 0 superfluid turbulence generated by the macroscopic motion of the superfluid. We first establish that the condensate model is dual to the short-range interacting loop gas model, wherein the loops represent quantum vortex lines. Vortex lines form, interact and proliferate as a result of the superfluid motion. Our extension is based on incorporating the Biot–Savart interaction between vortex lines, which is lacking in the loop gas model. We show that the extended loop gas is dual to a Ginzburg–Landau model, wherein the gauge coupling is between the macroscopic velocity field of the superfluid and the condensate. Applying the model to cylindrical and pipe flows, we describe how turbulence transitions with and without intermediate vortex flow, respectively.

DOI
10.1088/1742-5468/aaa8f2

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Abstract We present a gauge-field extension of the Bose condensate model that describes T ≈ 0 superfluid turbulence generated by the macroscopic motion of the superfluid. We first establish that the condensate model is dual to the short-range interacting loop gas model, wherein the loops represent quantum vortex lines. Vortex lines form, interact and proliferate as a result of the superfluid motion. Our extension is based on incorporating the Biot–Savart interaction between vortex lines, which is lacking in the loop gas model. We show that the extended loop gas is dual to a Ginzburg–Landau model, wherein the gauge coupling is between the macroscopic velocity field of the superfluid and the condensate. Applying the model to cylindrical and pipe flows, we describe how turbulence transitions with and without intermediate vortex flow, respectively.
Due to strong interactions between superfluid particles, many particles are not in the condensate near absolute zero. In nonequilibrium situations produced under external stirring, they contribute to the flow via their induced average macroscopic motion. We present a modification of the Bose condensate model that considers this induced internal motion. Specifically, we show that the velocity field of the internal motion acts as a gauge field with which the condensate interacts, thereby coupling the applied vorticity to the superflow. The resulting Ginzburg-Landau-like model predicts the flow phenomena observed near absolute zero, in particular, transition to turbulence.
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\mathcal{L}L. The closure of such an equation for superfluid inhomogeneous flows requires additional inputs besides L\mathcal{L}L and the normal and superfluid velocity fields. In this paper, we offer a minimal closure using one additional anisotropy parameter Il0{I}_{l0}Il0​. Using the example of counterflow superfluid turbulence, we derive two coupled closure equations for the vortex line density and the anisotropy parameter Il0{I}_{l0}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.
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),{\cal P}(\{ψ({\bf r})\})={\cal N}\exp (-H\{ψ({\bf r)}\} /T),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)ψ({\bf r})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)}){\cal P}(\{{\bf s}(ξ)\})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),{\cal P}(\{{\bf s}(ξ)\})={\cal N}\exp (-H\{{\bf s}\} /T),P({s(ξ)})=Nexp(−H{s}/T), where N{\cal N}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
The detailed understanding of the intricate dynamics of quantum fluids, in particular in the rapidly growing subfield of quantum turbulence which elucidates the evolution of a vortex tangle in a superfluid, requires an in-depth understanding of the role of finite temperature in such systems. The Landau two-fluid model is the most successful hydrodynamical theory of superfluid helium, but by the nature of the scale separations it cannot give an adequate description of the processes involving vortex dynamics and interactions. In our contribution we introduce a framework based on a nonlinear classical-field equation that is mathematically identical to the Landau model and provides a mechanism for severing and coalescence of vortex lines, so that the questions related to the behavior of quantized vortices can be addressed self-consistently. The correct equation of state as well as nonlocality of interactions that leads to the existence of the roton minimum can also be introduced in such description. We review and apply the ideas developed for finite-temperature description of weakly interacting Bose gases as possible extensions and numerical refinements of the proposed method. We apply this method to elucidate the behavior of the vortices during expansion and contraction following the change in applied pressure. We show that at low temperatures, during the contraction of the vortex core as the negative pressure grows back to positive values, the vortex line density grows through a mechanism of vortex multiplication. This mechanism is suppressed at high temperatures.
Simulating superfluid turbulence using the localized induction approximation allows neighboring parallel vortices to proliferate. In many circumstances a turbulent tangle becomes unsustainable, degenerating into a series of parallel, noninteracting vortex lines. Calculating with the fully nonlocal Biot-Savart law prevents this difficulty but also increases computation time. Here we use a truncated Biot-Savart integral to investigate the effects of nonlocality on homogeneous turbulence. We find that including the nonlocal interaction up to roughly the spacing between nearest-neighbor vortex segments prevents the parallel alignment from developing, yielding an accurate model of homogeneous superfluid turbulence with less computation time.
This work rectifies the hydrodynamic equations commonly used to describe the superfluid velocity field in such a way that vortex dynamics are also taken into account. In the field of quantum turbulence, it is of fundamental importance to know the correct form of the equations which play similar roles to the Navier-Stokes equation in classical turbulence. Here, such equations are obtained by carefully taking into account the frequently overlooked multivalued nature of the U(1)U(1)U(1) phase field. Such an approach provides exact analytical explanations to some numerically observed features involving the dynamics of quantum vortices in Bose-Einstein condensates, such as the universal t1/2{t}^{1/2}t1/2 behavior of reconnecting vortex lines. It also expands these results beyond the Gross-Pitaevskii theory so that some features can be generalized to other systems such as superfluid 4He^{4}\mathrm{He}4He, dipolar condensates, and mixtures of different superfluid systems.