{
  "abstract": "We consider the collisional evolution towards equilibrium of a spatially homogeneous and isotropic phonon gas of a three-dimensional superfluid with a concave acoustic excitation branch, at a non-zero but arbitrarily low temperature $T$. Three-phonon collisions $1\\phi\\leftrightarrow 2\\phi$ are forbidden by conservation of energy-momentum. Four-phonon collisions $2\\phi\\to 2\\phi$ of Landau and Khalatnikov lead, after a time $\\propto T^{-7}$, only to a partial thermal equilibrium, a Bose law of non-zero chemical potential for the phonons, because they conserve the total number of phonons. Relaxation towards complete thermochemical equilibrium is therefore ensured by the much slower five-phonon collisions $2\\phi\\leftrightarrow 3\\phi$ of Khalatnikov, in a time $\\propto T^{-9}$. Using kinetic equations on the occupation numbers of the phonon modes and explicitly calculating the $2\\phi\\to 3\\phi$ collisional amplitude with quantum hydrodynamics at low temperature, we determine the corresponding evolution of the fugacity $z_\\phi$ of the phonon gas from the non-degenerate regime $z_\\phi=0^+$ to complete equilibrium $z_\\phi=1^-$. Using the conservation of total energy, we find that the fugacity varies with a non-integer power law $\\propto t^{4/5}$ at short times and an exponential law at long times; the speed of change of entropy, always positive, is asymptotically proportional to the square of the speed of change of fugacity, $(\\mathrm{d}/\\mathrm{d}t)S_\\phi\\propto[(\\mathrm{d}/\\mathrm{d}t)z_\\phi]^2$, as Landau predicted for an arbitrarily slow adiabatic transformation. Our results bring to a close the study initiated by Khalatnikov in 1950 and could be experimentally verified in a gas of cold spin-1/2 fermionic atoms in the sector of the BEC-BCS crossover where the acoustic branch has a concave start, or in superfluid liquid helium-4 at sufficiently high pressure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2604.18298",
  "author_total": 2,
  "authors": [
    {
      "name": "Yvan Castin"
    },
    {
      "name": "Mariia Tsimokha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bfdx8gx2mctc1.bib",
  "cited_by": 0,
  "doi": "10.5802/crphys.292",
  "fulltext_url": null,
  "id": "bfdx8gx2mctc1",
  "issns": [
    "1631-0705",
    "1878-1535"
  ],
  "landing_url": "https://hal.science/hal-05595595v2",
  "language": "fr",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.phys"
    ],
    [
      "hal",
      "1.phys.cond"
    ],
    [
      "hal",
      "2.phys.cond.gas"
    ],
    [
      "openalex",
      "3107|Quantum, superfluid, helium dynamics"
    ]
  ],
  "openalex": "W7155582171",
  "pdf_url": "https://hal.science/hal-05595595/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": "Académie des sciences (Paris)",
  "ris": "https://research.draconitic.com/paper/bfdx8gx2mctc1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:57Z",
      "native_id": "hal-05595595",
      "source": "hal",
      "url": "https://hal.science/hal-05595595v2"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "hal:1.phys.cond",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Phonon number relaxation in a 3D superfluid with a concave acoustic branch.",
  "title_text": "Phonon number relaxation in a 3D superfluid with a concave acoustic branch.",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:17:57Z",
  "url": "https://research.draconitic.com/paper/bfdx8gx2mctc1",
  "venue": "Comptes Rendus. Physique",
  "versions": []
}