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arXiv:2604.18298

Phonon number relaxation in a 3D superfluid with a concave acoustic branch

Yvan Castin, Mariia Tsimokha

Published Apr 20, 2026• 0 citations
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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 $1ϕ\leftrightarrow 2ϕ$ are forbidden by conservation of energy-momentum. Four-phonon collisions $2ϕ\to 2ϕ$ 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 $2ϕ\leftrightarrow 3ϕ$ of Khalatnikov, in a time $\propto T^{-9}$. Using kinetic equations on the occupation numbers of the phonon modes and explicitly calculating the $2ϕ\to 3ϕ$ collisional amplitude with quantum hydrodynamics at low temperature, we determine the corresponding evolution of the fugacity $z_ϕ$ of the phonon gas from the non-degenerate regime $z_ϕ=0^+$ to complete equilibrium $z_ϕ=1^-$. 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, $(\mathrm{d}/\mathrm{d}t)S_ϕ\propto[(\mathrm{d}/\mathrm{d}t)z_ϕ]^2$, 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.

DOI
10.48550/arxiv.2604.18298

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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 TTT. Three-phonon collisions 1ϕ↔2ϕ1ϕ\leftrightarrow 2ϕ1ϕ↔2ϕ are forbidden by conservation of energy-momentum. Four-phonon collisions 2ϕ→2ϕ2ϕ\to 2ϕ2ϕ→2ϕ of Landau and Khalatnikov lead, after a time ∝T−7\propto T^{-7}∝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 2ϕ↔3ϕ2ϕ\leftrightarrow 3ϕ2ϕ↔3ϕ of Khalatnikov, in a time ∝T−9\propto T^{-9}∝T−9. Using kinetic equations on the occupation numbers of the phonon modes and explicitly calculating the 2ϕ→3ϕ2ϕ\to 3ϕ2ϕ→3ϕ collisional amplitude with quantum hydrodynamics at low temperature, we determine the corresponding evolution of the fugacity zϕz_ϕzϕ​ of the phonon gas from the non-degenerate regime zϕ=0+z_ϕ=0^+zϕ​=0+ to complete equilibrium zϕ=1−z_ϕ=1^-zϕ​=1−. Using the conservation of total energy, we find that the fugacity varies with a non-integer power law ∝t4/5\propto t^{4/5}∝t4/5 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, (d/dt)Sϕ∝[(d/dt)zϕ]2(\mathrm{d}/\mathrm{d}t)S_ϕ\propto[(\mathrm{d}/\mathrm{d}t)z_ϕ]^2(d/dt)Sϕ​∝[(d/dt)zϕ​]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.
Nous considérons l’évolution collisionnelle vers l’équilibre du gaz de phonons spatialement homogène et isotrope d’un superfluide tridimensionnel à branche d’excitation acoustique concave, à une température T non nulle mais arbitrairement basse. Les collisions à trois phonons 1 ϕ ↔ 2 ϕ sont interdites par la conservation de l’énergie-impulsion. Les collisions à quatre phonons 2 ϕ → 2 ϕ de Landau et Khalatnikov ne conduisent, au bout d’un temps ∝ T –7 , qu’à un équilibre thermique partiel, une loi de Bose de potentiel chimique non nul pour les phonons, car elles en conservent le nombre total. La relaxation vers l’équilibre thermochimique complet est donc assurée par les collisions beaucoup plus lentes à cinq phonons 2 ϕ ↔ 3 ϕ de Khalatnikov, en un temps ∝ T –9 . Au moyen d’équations cinétiques sur les nombres d’occupation des modes de phonons et après calcul explicite par l’hydrodynamique quantique de l’amplitude de collision 2 ϕ → 3 ϕ à basse température, nous déterminons l’évolution correspondante de la fugacité z ϕ du gaz de phonons depuis le régime non dégénéré z ϕ = 0 + jusqu’à l’équilibre complet z ϕ = 1 – . En utilisant la conservation de l’énergie totale, nous trouvons que la fugacité varie avec une loi de puissance non entière ∝ t 4/5 aux temps courts et une loi exponentielle aux temps longs ; la vitesse de variation de l’entropie, toujours positive, est asymptotiquement proportionnelle au carré de la vitesse de variation de la fugacité, (d/d t ) S ϕ ∝ [(d/d t ) z ϕ ] 2 , comme l’avait prédit Landau pour une transformation adiabatique arbitrairement lente. Nos résultats portent à son terme l’étude commencée par Khalatnikov en 1950 et pourraient être vérifiés expérimentalement dans un gaz d’atomes froids fermioniques de spin 1/2 dans le secteur du raccordement CBE-BCS où la branche acoustique est de départ concave, ou dans l’hélium 4 liquide superfluide à suffisamment haute pression.
It is generally accepted that the phonon gas of a superfluid always enters a weak coupling regime at sufficiently low temperatures, whatever the strength of the interactions between the underlying particles (constitutive of the superfluid). Thus, in this limit, we should always be able to calculate the damping rate of thermal phonons by applying Fermi's golden rule to the Hamiltonian H3H_3H3​ of cubic phonon-phonon coupling taken from quantum hydrodynamics, at least in the case of a convex acoustic branch and in the collisionless regime (where the eigenfrequency of the considered phonons remains much greater than the gas thermalization rate). Using the many-body Green's function method, we predict that, unexpectedly, this is not true in two dimensions, contrary to the three-dimensional case. We confirm this prediction with classical phonon-field simulations and a non-perturbative theory in H3H_3H3​, where the fourth order is regularized by hand, giving a complex energy to the virtual phonons of the four-phonon collisional processes. For a weakly interacting fluid and a phonon mode in the long-wavelength limit, we predict a damping rate about three times lower than that of the golden rule.
A condensate of pairs in an isolated, homogeneous, unpolarised, finite-size spin 1/2 Fermi gas at low nonzero temperature T, undergoes with time a phase change with a random component, due to coupling to the gas thermal phonons. With the quantum second Josephson relation relating the derivative of the condensate phase operator to the phonon occupation numbers, and linearised kinetic equations giving the evolution of occupation number fluctuations, we access the behaviour of the phase change variance at times much longer than the phonon collision time. The case of a convex phonon branch is similar to the Bose gas case: the leading collisional processes are the Beliaev-Landau 3-phonons processes, and the variance is the sum of a ballistic term and of a delayed diffusive term, whose analytical expressions are given in the thermodynamic limit. The concave case is more exotic. It is analysed at time scales T−9T^{-9}T−9, allowing one to restrict to 2 phonons →\rightarrow→ 2 phonons small-angle Landau-Khalatnikov processes. The total number of phonons is conserved and the phonon mean occupation numbers at equilibrium can exhibit a negative chemical potential, assumed isotropic. The phase change variance is then the sum of a ballistic term, of a diffusive term, of exotic subsubleading terms and of a constant term. The analytic expression of some of the corresponding coefficients is obtained, as well as the diverging leading behavior of the other ones when the phonon chemical potential tends to 0. When this chemical potential is 0, the variance sub-ballistic part becomes superdiffusive, with an exponent 5/3 and an exactly-known coefficient. For a nonzero infinitesimal phonon chemical potential, a law is found, interpolating between superdiffusive and diffusive phase spreading. Also new results are obtained on the phonon Landau-Khalatnikov damping rate, in particular at negative phonon chemical potential.
Abstract We study the interactions among phonons and the phonon lifetime in a pair‐condensed Fermi gas in the BEC‐BCS crossover in the collisionless regime. To compute the phonon‐phonon coupling amplitudes we use a microscopic model based on a generalized BCS Ansatz including moving pairs, which allows for a systematic expansion around the mean field BCS approximation of the ground state. We show that the quantum hydrodynamic expression of the amplitudes obtained by Landau and Khalatnikov apply only on the energy shell, that is for resonant processes that conserve energy. The microscopic model yields the same excitation spectrum as the Random Phase Approximation, with a linear (phononic) start and a concavity at low wave number that changes from upwards to downwards in the BEC‐BCS crossover. When the concavity of the dispersion relation is upwards at low wave number, the leading damping mechanism at low temperature is the Beliaev‐Landau process 2 phonons ↔ 1 phonon while, when the concavity is downwards, it is the Landau‐Khalatnikov process 2 phonons ↔ 2 phonons. In both cases, by rescaling the wave vectors to absorb the dependence on the interaction strength, we obtain a universal formula for the damping rate. This universal formula corrects and extends the original analytic results of Landau and Khalatnikov [ZhETF 19, 637 (1949)] for the 2↔2 processes in the downward concavity case. In the upward concavity case, for the Beliaev 1↔ 2 process for the unitary gas at zero temperature, we calculate the damping rate of an excitation with wave number q including the first correction proportional to q7 to the q5 hydrodynamic prediction, which was never done before in a systematic way.
Liquid helium and spin-1/2 cold-atom Fermi gases both exhibit in their superfluid phase two distinct types of excitations, gapless phonons and gapped rotons or fermionic pair-breaking excitations. In the long wavelength limit, revising and extending Landau and Khalatnikov's theory initially developed for helium [ZhETF 19, 637 (1949)], we obtain universal expressions for three- and four-body couplings among these two types of excitations. We calculate the corresponding phonon damping rates at low temperature and compare them to those of a pure phononic origin in high-pressure liquid helium and in strongly interacting Fermi gases, paving the way to experimental observations.
We consider a 3D homogeneous superfluid at low temperature TTT with 2 types of excitations, gapless phonons with a linear dispersion relation at low wavenumber, and gapped quasiparticles with a quadratic dispersion relation around extrema. We calculate the scattering amplitude of a phonon on a quasiparticle to leading order in TTT for all subsonic quasiparticle velocities, with a SSS-matrix formalism between exact asymptotic states dressed by virtual phonons. We then characterize the erratic motion of the quasiparticle in the superfluid due to its unceasing collisions with thermal phonons through mean force F(k)F(k)F(k), longitudinal and transverse kkk-dependent momentum diffusion coefficients, and spatial diffusion coefficient. At the minimum location k0k_0k0​ of the dispersion relation, where the velocity vanishes, F(k)F(k)F(k) varies linearly with velocity with an isotropic friction coefficient; if k0=0k_0=0k0​=0, the momentum diffusion is also isotropic and F(k0)=0F(k_0)=0F(k0​)=0; if k0>0k_0>0k0​>0, it is not, and F(k0)F(k_0)F(k0​) is nonzero but subleading with respect to friction by one order in TTT. The velocity time correlation function, whose integral is the spatial diffusion coefficient, decays with the mean velocity damping rate if k0=0k_0=0k0​=0; if k0>0k_0>0k0​>0, it has a second exponential component, with an amplitude and a damping rate lower by a factor ∝T\propto T∝T (it is the velocity direction thermalization rate). We also characterize force and momentum diffusion close to the stability domain sonic edge. Our general expressions are expected to be exact to leading order in TTT. We illustrate them in the BCS approximation, for a fermionic quasiparticle (an unpaired fermion) in a superfluid of spin 1/2 fermions, realisable with cold atoms in flat bottom traps. We also refute the statement of Lerch, Bartosch and Kopietz (2008), that there would be no fermionic quasiparticle in such a superfluid.