Resolving the puzzle of sound propagation in liquid helium at low temperatures

Abstract

Experimental data suggests that, at temperatures below 1 K, the pressure in liquid helium has a cubic dependence on density. Thus the speed of sound scales as a cubic root of pressure. Near a critical pressure point, this speed approaches zero whereby the critical pressure is negative, thus indicating a cavitation instability regime. We demonstrate that to explain this dependence, one has to view liquid helium as a mixture of three quantum Bose liquids: dilute (Gross-Pitaevskii-type) Bose-Einstein condensate, Ginzburg-Sobyanin-type fluid, and logarithmic superfluid. Therefore, the dynamics of such a mixture is described by a quantum wave equation, which contains not only the polynomial (Gross-Pitaevskii and Ginzburg-Sobyanin) nonlinearities with respect to a condensate wavefunction, but also a non-polynomial logarithmic nonlinearity. We derive an equation of state and speed of sound in our model, and show their agreement with experiment.

An Alternative to Dark Matter and Dark Energy: Scale-Dependent Gravity in Superfluid Vacuum Theory

“…One example of such a material is helium II, the superfluid phase of helium-4. For the latter, the logarithmic superfluid model is known to have been well verified by experimental data [10,43]. Among other things, the logarithmic superfluid model does reproduce the sought-after Landau-type spectrum of excitations, discussed in the previous section; detailed derivations can be found in [10].…”

“…Such equations find fruitful applications in the theory of strongly-interacting quantum fluids, and have been successfully applied to laboratory superfluids [10,43,49]. We note that, in principle, one is not precluded from adding other types of nonlinearity, such as polynomial ones, into the condensate wave equations, but the role of logarithmic nonlinearity is crucial.…”

“…[42]; but non-logarithmic terms can also come into play: one example is to be found in Ref. [43]. The nonlinear coupling function b = b(r, t) can also be made more detailed, to account for the thermodynamic environment in a more realistic way.…”

Solution of the 3D logarithmic Schrödinger equation with a central potential

“…The log SE appears in many fields of physics, including quantum optics [7], nuclear physics [8], transport and diffusion phenomena [9], dispersion studies [10], stochastic quantum mechanics [11], Bose-Einstein condensation [5,12], quantum gravity [12][13][14] and superfluids [15,16]. For example, it was recently shown that the logarithmic term y S r ln ; | ( )|  of equation (1) dominates the standard Gross-Pitaevskii terms for Helium-4 at low temperatures in certain regimes [16]. A superfluid vacuum theory using a log SE appears in one formulation for the Higgs potential [17,18].…”

Regularised Finite Difference Methods for the Logarithmic Klein-Gordon Equation

“…, x d ) T ∈ d , d = 1, 2, 3 is the spatial coordinate, t time, u := u(x, t) a realvalued scalar field, and λ shows the force of the non-linear interaction. Such non-linearities appear in relativistic wave equations, which describe dilatonic quantum gravity [43], superfluid [44], spinless particles [16,17] and non-relativistic spinning particles moving in an external electromagnetic field. Besides, such non-linearity effects often arise in various areas of physics such as inflation cosmology [14,25], supersymmetric field theories, geophysics [32,37], optics [20], and nuclear physics [34].…”