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arXiv:1412.0706

Superfluid Mutual-friction Coefficients from Vortex Dynamics in the Two-dimensional Galerkin-truncated Gross-Pitaevskii Equation

Vishwanath Shukla, Marc Brächet, Rahul Anil Pandit

Published Nov 23, 2014• 0 citations
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Abstract

We present algorithms for the ab-initio determination of the temperature ($T$) dependence of the mutual-friction coefficients $α$ and $α'$ and the normal-fluid density $ρ_{\rm n}$ in the two-dimensional (2D) Galerkin-truncated Gross-Pitaevskii system. Our algorithms enable us to determine $α(T)$, even though fluctuations in 2D are considerably larger than they are in 3D. We also examine the implications of our measurements of $α'(T)$ for the Iordanskii force, whose existence is often questioned.

DOI
10.48550/arxiv.1412.0706

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The microscopic mechanism of thermal dissipation in quantum turbulence is numerically studied by solving the coupled system involving the Gross-Pitaevskii equation and the Bogoliubov-de Gennes equation. At low temperatures, the obtained dissipation does not work at scales greater than the vortex core size. However, as the temperature increases, dissipation works at large scales and it affects the vortex dynamics. We successfully obtain the mutual friction coefficients of the vortex in dilute Bose-Einstein condensates dynamics as functions of temperature.

arXiv:cond-mat/0607434•Physical Review Letters•Oct 2, 2006•48 citations
We present algorithms for the ab-initio determination of the temperature (TTT) dependence of the mutual-friction coefficients ααα and α′α'α′ and the normal-fluid density ρnρ_{\rm n}ρn​ in the two-dimensional (2D) Galerkin-truncated Gross-Pitaevskii system. Our algorithms enable us to determine α(T)α(T)α(T), even though fluctuations in 2D are considerably larger than they are in 3D. We also examine the implications of our measurements of α′(T)α'(T)α′(T) for the Iordanskii force, whose existence is often questioned.
Abstract The physical origin of mutual friction in quantum fluids is deeply connected to the fundamental nature of superfluidity. It stems from the interaction between the superfluid and normal components, mediated by the dynamics of quantized vortices that induce the exchange of momentum and energy. Despite the complexity of these interactions, their essential features can be effectively described by the dissipative point vortex model—an extension of classical vortex dynamics that incorporates finite-temperature dissipation. Mutual friction is parametrized by the longitudinal (dissipative) coefficient α\alphaα α and the transverse (reactive) coefficient α′\alpha 'α′ α ′ . Accurate measurement of these parameters provides critical insights into the microscopic mechanisms governing vortex motion and dissipation in quantum fluids, serving as a key benchmark for theoretical models. In this work, we employ the dissipative point vortex model to study how fluctuations in the initial conditions influence the inference of α\alphaα α and α′\alpha 'α′ α ′ from the time evolution of the vortex trajectories. Using experimentally realistic parameters, we show that fluctuations can introduce significant biases in the extracted values of the mutual friction coefficients. We compare our findings with recent experimental measurements in strongly interacting atomic superfluids. Applying this analysis to our recent experimental results allowed us to account for fluctuations in the correct determination of α\alphaα α and α′\alpha 'α′ α ′ .
The mutual friction alpha on a quantized vortex in a two-dimensional Bose field is commonly taken to scale with the normal fraction rho_n/rho. In an earlier paper (doi:10.5281/zenodo.23225293) we measured alpha(T) in the projected Gross-Pitaevskii equation at one cutoff and found alpha ~ 0.24 rho_n/rho; we read the coefficient as a mode-averaged transport cross-section of the vortex (1.4 healing lengths) and registered its cutoff dependence as the open question, with the Born phonon-scattering law (coefficient proportional to the cutoff) as the rival. Here the same energy-conserving instrument, with its T = 0 known answer reproduced to 0.3 % at each cutoff, is run at two further cutoffs (k_c xi = 2 pi/3 and 2 pi). The registered prediction of a cutoff-independent coefficient fails and so does its rival: the coefficient is 0.31 +- 0.03, 0.232 +- 0.014, 0.102 +- 0.011 at k_c xi = 2.1, 3.1, 6.3. What does not move is alpha itself: alpha/T = 0.054 +- 0.003 over six arms and three cutoffs (chi2 = 1.3 for 5 degrees of freedom; 0.060 +- 0.002 with the regression estimator, because the energy estimator, the primary one, is biased low by diffusion -- a flaw of the registered synthetic gate exposed by a second, Rust implementation of the estimators). The proportionality to rho_n was a degeneracy of a single cutoff, where rho_n is proportional to T; the 'geometric core size' reading is withdrawn. The temperature law was found after the data were read; it is registered as a hypothesis, with its prediction for the next arm fixed before that arm was read (reported in the paper). Pre-registrations, controls, scripts, analysis file and Lean companion (FrictionKinetic.lean) are in the software record.
We report numerical simulations of the dissipative Gross-Pitaevskii equation for a bulk region of thermal-counterflow turbulence. Quasistationary states are obtained over a range of forcing, damping, and healing-length parameters. The mutual-friction acceleration exhibits cubic scaling with the mean relative velocity between the superfluid and normal-fluid components, and the coefficient of this scaling is linked to the phenomenological damping parameter. The intervortex spacing follows the expected dimensional scaling in the weak-forcing regime. Comparison with a straight-vortex-line model suggests that the vortex-line orientations are nearly isotropic.
We investigate the two-dimensional motion of a single vortex orbiting a pinned anti-vortex in a unitary Fermi superfluid. By analyzing its trajectory, we measure the yet-unknown longitudinal and transverse mutual friction coefficients, which quantify the vortex-mediated coupling between the normal and superfluid components. Both coefficients increase while approaching the superfluid transition. They provide access to the vortex Hall angle, which is linked to the relaxation time of the localized quasiparticles occupying Andreev bound states within the vortex core, and to the vortex Reynolds number Reα associated with the transition from laminar to quantum turbulent flows. We compare our results with numerical simulations and an analytic model originally formulated for superfluid 3He, finding good agreement. Our work suggests that vortex dynamics in unitary Fermi superfluids is essentially affected by the interplay between delocalized thermal excitations and vortex-bound quasiparticles. Further, it provides a novel testbed for studying vortex dynamics at finite temperatures. Vortex dynamics and mutual friction in quantum fluids are intimately connected to the fundamental properties of superfluids. Here, the authors reveal previously unexplored mechanisms underlying the mutual friction coefficients in ultracold Fermi superfluids in the unitary limit, suggesting bound quasiparticles within the vortex core play a significant role.
A quantized vortex in a thermal two-dimensional Bose gas is dragged by the normal fluid (friction α), pushed sideways by it (transverse coefficient α′) and kicked by it (diffusion η). We measure the three coefficients in the projected Gross-Pitaevskii equation — microcanonical, no damping term, no noise put in by hand — with sub-grid vortex tracking and estimators that are identities of the dissipative point-vortex model, machine-checked in Lean 4. Every hypothesis, gate and decision rule was committed before the corresponding run; the instrument failed its T = 0 control twice before it passed, an earlier gate verdict built on the flawed instrument is withdrawn, and both are reported. Results. At T/TBKT = 0.14, 0.27, 0.43: α = 0.0062(4), 0.014(2), 0.02(1), size- and estimator-independent, α ≈ 0.24 ρn/ρ. α′ is zero within 1% once the T = 0 compressible pair speed is subtracted; the Iordanskii value ρn/ρ is excluded at 18σ and the registered Iordanskii-like prediction is refuted. The Einstein relation is inconclusive by the registered rule (one admitted temperature at ηK/α = 2.1 ± 0.7; informally 1.4-3x, not 100x). Single pairs stop shrinking while the phonon band takes the impulse they shed (a self-generated counterflow with a published precedent in the truncated GPE); the decisive box-size control is not run. The single non-equilibrium pair that corrupts the superfluid-stiffness estimator in finite boxes decays by 4-27x given Lz-scaled time (six of six L = 192 trajectories admitted, one unfinished at t = 14 500). The pre-registered mode-by-mode dielectric relation fails in the thermal pair gas (coherence 0.06-0.28), while the vortex-charge structure factor equals the Kosterlitz-Thouless polarisability Σd2 from the measured pair sizes to 5-21%. The Josephson offset ηK − 1 = +0.3 survives at L = 192 in equilibrium, independent of fit window and torus geometry. Formal companions (Lean 4 / Mathlib, standard axioms): energy dissipation of any skew-plus-gradient vortex flow, the d2 law and lifetime bound, the centre-velocity identity, the wind stall, the zero-flux/Einstein equivalence, the pair-polarisation form and response ceiling, the charge-density/matching bound, the Kosterlitz-flow invariant, the bounded increments of quasi-periodic signals. Pre-registrations, amendments, gate records, data and code are in the software repository (concept DOI 10.5281/zenodo.22855581). No physics is claimed as new; the numbers, the refutations and the method are.
The microscopic mechanism of thermal dissipation in quantum turbulence is numerically studied by solving the coupled system involving the Gross-Pitaevskii equation and the Bogoliubov-de Gennes equation. At low temperatures, the obtained dissipation does not work at scales greater than the vortex core size. However, as the temperature increases, dissipation works at large scales and it affects the vortex dynamics. We successfully obtain the mutual friction coefficients of the vortex in dilute Bose-Einstein condensates dynamics as functions of temperature.