arXiv:2610.27810

Phonon Dispersion Relation in Weakly Coupled Bose Gas Beyond Bogoliubov Approximation

Jan Dereziński, Lorenzo Pettinari

Published Oct 20, 2026• 0 citations

Abstract

We study the quasiparticle dispersion relation for a homogeneous Bose gas in a large cubic box with periodic boundary conditions at zero temperature. Inspired by the Bogoliubov method, we rearrange the usual Bose gas Hamiltonian in a form adapted to perturbation theory in the weak-coupling regime, both in the canonical and grand-canonical setting. We compute the first non-trivial corrections to the dispersion relation in both settings. We show that for small momenta they are linear in the absolute value of the momentum. Thus, in this approximation phonons have a well-defined velocity, which we compute.

DOI: 10.48550/arxiv.2610.27810

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arXiv:2610.27810

Phonon Dispersion Relation in Weakly Coupled Bose Gas Beyond Bogoliubov Approximation

Jan Dereziński, Lorenzo Pettinari

Published Oct 20, 2026• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2610.27810

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{n}_{0}

(T)

{V}_{0}

{]}^{1/2}

, where

{n}_{0}

(T) is the temperature-dependent condensate of a free Bose gas. In this paper the phonon velocity is worked out with use of the one-loop diagrams for the regular quantities. The result is [

{n}_{S}

(T)

{V}_{0}

{]}^{1/2}

[1-f(T)], where

{n}_{S}

(T) is the Landau superfluid density defined in terms of the Bogoliubov excitations and f(T) is a relatively small correction. Numerical calculations are given for spin-polarized ${\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}$ with density n=5\ifmmode\times\else\texttimes\fi{}

{10}^{18}

{\mathrm{cm}}^{3}

. .AE

We study condensate statistics and thermodynamics of weakly interacting Bose gas with a fixed total number

N

of particles in a cubic box. We find the exact recursion relation for the canonical ensemble partition function. Using this relation, we calculate the distribution function of condensate particles for

N=200

. We also calculate the distribution function based on multinomial expansion of the characteristic function. Similar to the ideal gas, both approaches give exact statistical moments for all temperatures in the framework of Bogoliubov model. We compare them with the results of unconstraint canonical ensemble quasiparticle formalism and the hybrid master equation approach. The present recursion relation can be used for any external potential and boundary conditions. We investigate the temperature dependence of the first few statistical moments of condensate fluctuations as well as thermodynamic potentials and heat capacity analytically and numerically in the whole temperature range.