We revisit an inequality due to Onsager, which states that the (quantum) liquid structure factor has an upper bound of the form (const.) x |k|, for not too large modulus of the wave vector k. This inequality implies the validity of the Landau criterion in the theory of superfluidity with a definite, nonzero critical velocity. We prove an auxiliary proposition for general Bose systems, together with which we arrive at a rigorous proof of the inequality for one of the very few soluble examples of an interacting Bose fluid, Girardeau's model. The latter proof demonstrates the importance of the thermodynamic limit of the structure factor, which must be taken initially at k different from 0. It also substantiates very well the heuristic density functional arguments, which are also shown to hold exactly in the limit of large wave-lengths. We also briefly discuss which features of the proof may be present in higher dimensions, as well as some open problems related to superfluidity of trapped gases.
We revisit an inequality due to Onsager, which states that the (quantum) liquid structure factor has an upper bound of the form (const.) x |k|, for not too large modulus of the wave vector k. This inequality implies the validity of the Landau criterion in the theory of superfluidity with a definite, nonzero critical velocity. We prove an auxiliary proposition for general Bose systems, together with which we arrive at a rigorous proof of the inequality for one of the very few soluble examples of an interacting Bose fluid, Girardeau's model. The latter proof demonstrates the importance of the thermodynamic limit of the structure factor, which must be taken initially at k different from 0. It also substantiates very well the heuristic density functional arguments, which are also shown to hold exactly in the limit of large wave-lengths. We also briefly discuss which features of the proof may be present in higher dimensions, as well as some open problems related to superfluidity of trapped gases.
We revisit an inequality due to Onsager, which states that the (quantum) liquid structure factor has an upper bound of the form (const) × |k|, for not too large modulus of the wave vector k. This inequality implies the validity of the Landau criterion in the theory of superfluidity with a definite, nonzero critical velocity. We prove an auxiliary proposition for general Bose systems, together with which we arrive at a rigorous proof of the inequality for one of the very few solvable examples of an interacting Bose fluid, Girardeau’s model. The latter proof demonstrates the importance of the thermodynamic limit of the structure factor, which must be taken initially at k = 0. It also substantiates very well the heuristic density functional arguments, which are also shown to hold exactly in the limit of large wavelengths. We also briefly discuss which features of the proof may be present in higher dimensions, as well as some open problems related to superfluidity of trapped gases. PACS numbers: 67.40.−w, 67.40.Db 1.
The two-fluid and Landau criteria for superfluidity are compared for trapped Bose gases. While the two-fluid criterion predicts translational superfluidity, it is suggested, on the basis of the homogeneous Gross–Pitaevski limit, that a necessary part of Landau's criterion, adequate for non-translationally invariant systems, does not hold for trapped Bose gases in the GP limit. As a consequence, if the compressibility is detected to be very large (infinite by experimental standards), the two-fluid criterion is seen to be the relevant one in case the system is a translational superfluid, while the Landau criterion is the relevant one if translational superfluidity is absent.
This lecture describes certain properties of interacting Bose gases and superconductors which have recently been considered at Harvard. It is a brief resumé of work by the lecturer, P.C. Hohenberg, A. Fetter, R. Lange, and C. De Dominicis, which will be reported in mathematical detail elsewhere. It concerns attempts to derive from fundamental principles, several aspects of the macroscopic two-fluid model of London, Landau, and Tisza, the Landau phonon spectrum for bosons, and the structured condensate envisaged by Onsager and Feynman. The lecture describes new results which have been obtained concerning: (1) methods for calculating properties of large, but not necessarily infinite, condensed systems at finite temperatures; (2) methods for performing calculations in boson systems consistent with current conservation and sum rules; (3) results obtained by these methods for the phonon spectrum; (4) more accurate results for the less physical single-particle spectrum at finite temperatures; (5) results obtained for the macroscopically structured condensate in rotating systems which agree with the macroscopically inferred results of Feynman and Onsager; (6) results which treat depletion of the condensate as a result of interaction consistently and verify that nonetheless, the superfluid density, as operationally determined, is not correspondingly depleted; (7) methods in which the entropy plays a central role, for casting the macroscopic theory in a form which lays stress on the renormalized excitations of renormalized interactions; (8) the almost identical features of the mathematics for condensed Fermi and Bose systems; (9) a restatement, in this microscopic transcription, of the Feynman-Onsager argument for flux and vorticity quantization.
The flow of a uniform Bose gas at speeds greater than the Landau critical velocity vc does not necessarily destroy superfluidity, but rather needs only to lead to a decrease of the superfluid mass density ${\ensuremath{\rho}}_{s}$. Analyzing a weakly interacting Bose gas with a finite-range interparticle interaction that leads to a Landau critical velocity at nonzero quasiparticle momentum, we explicitly construct the (nonuniform) condensate for fluid flow faster than vc and calculate the accompanying decrease in ${\ensuremath{\rho}}_{s}$. 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.
The probability distribution of the total momentum P is studied in N-particle interacting homogeneous quantum systems at positive temperatures. Using Galilean invariance we prove that in one dimension, the asymptotic distribution of P/N is normal at all temperatures and densities, and in two dimensions, the tail distribution of P/N is normal. We introduce the notion of the density matrix reduced to the center of mass and show that its eigenvalues are N times the probabilities of the different eigenvalues of P. A series of results is presented for the limit of sequences of positive definite atomic probability measures, relevant for the probability distribution of both the single-particle and the total momentum. The P = 0 ensemble is shown to be equivalent to the canonical ensemble. Through some conjectures we associate the properties of the asymptotic distribution of the total momentum with the characteristics of fluid, solid, and superfluid phases. Our main suggestion is that in interacting quantum systems above one dimension, in infinite space, the total momentum is finite with a nonzero probability at all temperatures and densities. In solids, this probability is 1, and in a crystal, it is distributed on a lattice. Since it is less than 1 in two dimensions, we conclude that a 2D system is always in a fluid phase, so if the hexatic phase existed classically, it would be destroyed by quantum fluctuations. For a superfluid, we conjecture that the total momentum is zero with a nonzero probability and otherwise its distribution is continuous. We define a macroscopic wave function based on the density matrix reduced to the center of mass. We discuss how dissipation can give rise to a critical velocity, predict the temperature dependence of the latter, and argue that Landau’s criterion cannot explain superfluidity and its breakdown at positive temperatures. We also comment on the relation between superfluidity and Bose-Einstein condensation.
We report a measurement of the excitation spectrum omega(k) and the static structure factor S(k) of a Bose-Einstein condensate. The excitation spectrum displays a linear phonon regime, as well as a parabolic single-particle regime. The linear regime provides an upper limit for the superfluid critical velocity, by the Landau criterion. The excitation spectrum agrees well with the Bogoliubov spectrum in the local density approximation, even close to the long-wavelength limit of the region of applicability. Feynman's relation between omega(k) and S(k) is verified, within an overall constant.