Application of the theory of continuous media to the description of thermal excitations in superfluid helium

Abstract

We propose a model for the quasiparticles of superfluid 4 He which describes both phonons and rotons in a unified way. The theory is based on the fact that the thermal de Broglie wavelengths of the atoms overlap each other. This allows us to treat superfluid 4 He as a continuous medium at all length scales. Then the parameters of the continouous medium ͑density, pressure, and velocity͒ can be given a probabilistic value at each point in space. The quasiparticles of superfluid 4 He are small fluctuations in these parameters; the frequency and wave vector of a fluctuation correspond to the energy and momentum of the quasiparticle, respectively. Using the Lagrange formalism we derive equations for the potential associated with these fluctuations, and this leads to a generalized wave equation. From the Hamiltonian formalism we derive a system of equations for the variables of a continuous medium, and show that in the general case there is a non-local dependence between pressure and density. Applying the methods of the mechanics of continuous media, we calculate the creation probabilities for both phonons and rotons by phonons in a solid, in a unified way. This theory explains why R Ϫ rotons are not created by a heater. The theory is compared with those of others, and the results with experiments.

Short-wave excitations in non-local Gross-Pitaevskii model

“…This non-local relationship between pressure and density was used in the phenomenological approach of [35] (neglecting the quantum term) to describe the rotons in the superfluid helium. From our consideration, it follows that the non-locality kernel, which relates the pressure and the density in a phenomenological approach, is determined by the inter-particle interaction potential.…”

Normal transmission of phonons with anomalous dispersion through the interface of two continuous media

“…We describe it with the help of the nonlocal hydrodynamics developed in [15]. According to it, in the infinite space the quantum fluid obeys the ordinary linearized equations of a continuous medium:…”

“…In this form it does not contain a convolution product in the sense of the Fourier transform because of the finite lower limit. There is no convolution product in the sense of either the Laplace or one-sided Fourier transform because the function h x ( ) is even [15] and cannot become zero at x < 0. In order to solve equation (7) by the Fourier transform we have first to define the unknown function in the region x < 0 and to widen the limits of integration to infinity.…”

“…We describe a quantum fluid with nonlinear dispersion relation within the framework of the theory suggested in [15], in which it is considered as a continuous medium with correlations. This theory is based on the fact that the thermal de Broglie wavelength of a particle of a quantum fluid exceeds the average interatomic separation.…”

“…It turned out that the theory introduced in [15], which describes a quantum fluid in the infinite space, cannot be trivially generalized to the half-space. In the brief work [17] a way of such generalization was offered, and the equation describing a quantum fluid in half-space was obtained.…”