Damping of phonons in Bose gas at low temperatures

TL;DR

The paper analyzes damping of phonons in a dilute, homogeneous Bose gas at low temperature and small momentum by perturbing around a Bogoliubov-described quasiparticle framework with a c-number condensate. It develops two complementary formalisms—the standard Liouvillean (operator-algebra) approach and a Green-function (two-point) method—and derives the imaginary part of the phonon dispersion to leading order in the weak coupling $\kappa$, decomposed into Beliaev and Landau damping contributions. The main results are explicit integral expressions for $\gamma_{\mathrm{B}}$ and $\gamma_{\mathrm{L}}$ and their asymptotic behavior in various regimes of $|\mathbf{k}|$, $\beta$, and $\nu$, including high-temperature corrections and the thermodynamic limit. These findings connect microscopic three-body interaction processes to observable linewidths, aligning with and extending Beliaev's and Hohenberg–Martin's classic results and providing a controlled framework for finite-temperature damping in Bose gases.

Abstract

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 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.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Remark 2.2
• proof
• Proposition 2.3
• proof
• Example 3.1
• Lemma 3.2
• proof