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arXiv:2602.07701

Damping of phonons in Bose gas at low temperatures

Jan Dereziński, Lorenzo Pettinari

Published Feb 7, 2026• 0 citations
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Abstract

We consider a homogeneous Bose gas in a large cubic box with periodic boundary conditions interacting with a small potential with a positive Fourier transform. We compute the imaginary part of the phononic excitation spectrum in the lowest order of perturbation theory in the thermodynamic limit at low temperatures and low momentum. Our analysis is based on perturbation theory of the standard Liouvillean. We use two approaches: the first, motivated by the standard representation of operator algebras, examines resonances near zero; the second analyzes the 2-point correlation function in the energy-momentum space.

DOI
10.48550/arxiv.2602.07701

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We consider second-quantized homogeneous Bose gas in a large cubic box with periodic boundary conditions at zero temperature. We discuss the energy-momentum spectrum of the Bose gas and its physical significance. We review various rigorous and heuristic results as well as open conjectures about its properties. Our main aim is to convince the readers, including those with mainly mathematical background, that this subject has many interesting problems for rigorous research. In particular, we investigate the upper bound on the infimum of the energy for a fixed total momentum k given by the expectation value of one-particle excitations over a squeezed states. This bound can be viewed as a rigorous version of the famous Bogoliubov method. We show that this approach seems to lead to a (nonphysical) energy gap. The variational problem involving squeezed states can serve as the preparatory step in a perturbative approach that should be useful in computing excitation spectrum. This version of a perturbative approach to the Bose gas seems (at least in principle) superior to the commonly used approach based on the c-number substitution.
We develop a finite-temperature perturbation theory (beyond the mean field) for a Bose-condensed gas and calculate temperature-dependent damping rates and energy shifts for Bogolyubov excitations of any energy. The theory is generalized for the case of excitations in a spatially inhomogeneous (trapped) Bose-condensed gas, where we emphasize the principal importance of inhomogeneity of the condensate density profile and develop the method of calculating the self-energy functions. The use of the theory is demonstrated by calculating the damping rates and energy shifts of low-energy quasiclassical excitations, i.e., the quasiclassical excitations with energies much smaller than the mean-field interaction between particles. In this case the boundary region of the condensate plays a crucial role and the result for the damping rates and energy shifts is completely different from that in spatially homogeneous gases. We also analyze the frequency shifts and damping of sound waves in cylindrical Bose condensates and discuss the role of damping in the recent MIT experiment on the sound propagation.
The low-temperature internal energy and specific heat of a hard sphere Bose gas confined to a general cuboidal enclosure are calculated under periodic boundary conditions. The special cases of a film and a channel geometry are considered in some detail. It is shown that, in the appropriate limits, one recovers the recent results of Pajkowski and Pathria, and of Zasada and Pathria, for a system of phonons confined to different geometries. Systematic applications of the Poisson summation formula enable us to obtain strongly convergent expansions, in both λ/L≪1 and λ/L≫1 regimes, and to see how the magnitude of the finite size corrections in the asymptotic limit (L/λ≫1) is directly linked with the dimensionality of the system. Special attention is paid to examining the combined influence of the dispersion in the phonon spectrum of the system and the finiteness of the enclosure.
We study the low-energy collective oscillations of a dilute Bose gas at finite temperature in the collisionless regime. By using a time-dependent mean-field scheme we derive for the dynamics of the condensate and noncondensate components a set of coupled equations, which we solve perturbatively to second order in the interaction coupling constant. This approach is equivalent to the finite-temperature extension of the Beliaev approximation and includes corrections to the Gross-Pitaevskii theory due to both quantum and thermal fluctuations. For a homogeneous system we explicitly calculate the temperature dependence of the velocity of propagation and damping rate of zero sound. In the case of harmonically trapped systems in the thermodynamic limit, we calculate, as a function of temperature, the frequency shift of the low-energy compressional and surface modes.