Transport in ultradilute solutions of 3^3He in superfluid 4^4He

We calculate the effect of a heat current on transporting $^3$He dissolved in superfluid $^4$He at ultralow concentration, as will be utilized in a proposed experimental search for the electric dipole moment of the neutron (nEDM). In this experiment, a phonon wind will generated to drive (partly depolarized) $^3$He down a long pipe. In the regime of $^3$He concentrations $\tilde < 10^{-9}$ and temperatures $\sim 0.5$ K, the phonons comprising the heat current are kept in a flowing local equilibrium by small angle phonon-phonon scattering, while they transfer momentum to the walls via the $^4$He first viscosity. On the other hand, the phonon wind drives the $^3$He out of local equilibrium via phonon-$^3$He scattering. For temperatures below $0.5$ K, both the phonon and $^3$He mean free paths can reach the centimeter scale, and we calculate the effects on the transport coefficients. We derive the relevant transport coefficients, the phonon thermal conductivity and the $^3$He diffusion constants from the Boltzmann equation. We calculate the effect of scattering from the walls of the pipe and show that it may be characterized by the average distance from points inside the pipe to the walls. The temporal evolution of the spatial distribution of the $^3$He atoms is determined by the time dependent $^3$He diffusion equation, which describes the competition between advection by the phonon wind and $^3$He diffusion. As a consequence of the thermal diffusivity being small compared with the $^3$He diffusivity, the scale height of the final $^3$He distribution is much smaller than that of the temperature gradient. We present exact solutions of the time dependent temperature and $^3$He distributions in terms of a complete set of normal modes.Comment: NORDITA PREPRINT 2015-37, 9 pages, 6 figure

Transport in very dilute solutions of 3^3He in superfluid 4^4 He

Motivated by a proposed experimental search for the electric dipole moment of the neutron (nEDM) utilizing neutron-$^3$He capture in a dilute solution of $^3$He in superfluid $^4$He, we derive the transport properties of dilute solutions in the regime where the $^3$He are classically distributed and rapid $^3$He-$^3$He scatterings keep the $^3$He in equilibrium. Our microscopic framework takes into account phonon-phonon, phonon-$^3$He, and $^3$He-$^3$He scatterings. We then apply these calculations to measurements by Rosenbaum et al. [J.Low Temp.Phys. {\bf 16}, 131 (1974)] and by Lamoreaux et al. [Europhys.Lett. {\bf 58}, 718 (2002)] of dilute solutions in the presence of a heat flow. We find satisfactory agreement of theory with the data, serving to confirm our understanding of the microscopics of the helium in the future nEDM experiment.Comment: 10 pages, 5 figures, v

Low Temperature Transport Properties of Very Dilute Classical Solutions of 3^3He in Superfluid 4^4He

We report microscopic calculations of the thermal conductivity, diffusion constant and thermal diffusion constant for classical solutions of $^3$He in superfluid $^4$He at temperatures T \la 0.6~K, where phonons are the dominant excitations of the $^4$He. We focus on solutions with $^3$He concentrations \la \,10^{-3}, for which the main scattering mechanisms are phonon-phonon scattering via 3-phonon Landau and Beliaev processes, which maintain the phonons in a drifting equilibrium distribution, and the slower process of $^3$He-phonon scattering, which is crucial for determining the $^3$He distribution function in transport. We use the fact that the relative changes in the energy and momentum of a $^3$He atom in a collision with a phonon are small to derive a Fokker-Planck equation for the $^3$He distribution function, which we show has an analytical solution in terms of Sonine polynomials. We also calculate the corrections to the Fokker-Planck results for the transport coefficients.Comment: 29 pages, 2 figure

Measuring Extinction Curves of Lensing Galaxies

We critique the method of constructing extinction curves of lensing galaxies using multiply imaged QSOs. If one of the two QSO images is lightly reddened or if the dust along both sightlines has the same properties then the method works well and produces an extinction curve for the lensing galaxy. These cases are likely rare and hard to confirm. However, if the dust along each sightline has different properties then the resulting curve is no longer a measurement of extinction. Instead, it is a measurement of the difference between two extinction curves. This "lens difference curve'' does contain information about the dust properties, but extracting a meaningful extinction curve is not possible without additional, currently unknown information. As a quantitative example, we show that the combination of two Cardelli, Clayton, & Mathis (CCM) type extinction curves having different values of R(V) will produce a CCM extinction curve with a value of R(V) which is dependent on the individual R(V) values and the ratio of V band extinctions. The resulting lens difference curve is not an average of the dust along the two sightlines. We find that lens difference curves with any value of R(V), even negative values, can be produced by a combination of two reddened sightlines with different CCM extinction curves with R(V) values consistent with Milky Way dust (2.1 < R(V) < 5.6). This may explain extreme values of R(V) inferred by this method in previous studies. But lens difference curves with more normal values of R(V) are just as likely to be composed of two dust extinction curves with R(V) values different than that of the lens difference curve. While it is not possible to determine the individual extinction curves making up a lens difference curve, there is information about a galaxy's dust contained in the lens difference curves.Comment: 15 pages, 4 figues, ApJ in pres

Money and happiness : rank of income, not income, affects life satisfaction

Does money buy happiness, or does happiness come indirectly from the higher rank in society that money brings? Here we test a rank hypothesis, according to which people gain utility from the ranked position of their income within a comparison group. The rank hypothesis contrasts with traditional reference income hypotheses, which suggest utility from income depends on comparison to a social group reference norm. We find that the ranked position of an individual’s income predicts general life satisfaction, while absolute income and reference income have no effect. Furthermore, individuals weight upward comparisons more than downward comparisons. According to the rank hypothesis, income and utility are not directly linked: Increasing an individual’s income will only increase their utility if ranked position also increases and will necessarily reduce the utility of others who will lose rank

Landau critical velocity in weakly interacting Bose gases

The flow of a uniform Bose gas at speeds greater than the Landau critical velocity, v_c, does not necessarily destroy superfluidity, but rather need only lead to a decrease of the superfluid mass density, {\rho}_s. Analyzing a weakly interacting Bose gas with a finite range interparticle interaction that leads to a Landau critical velocity at non-zero quasiparticle momentum, we explicitly construct the (non-uniform) condensate for fluid flow faster than v_c and calculate the accompanying decrease in {\rho}_s. We briefly comment on the relation of the physics to other problems in superfluids, e.g., solitons, and vortices in Bose-Einstein condensates, and critical currents in superconductors.Comment: 5 pages, 1 figur