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

This paper theoretically determines the relaxation dynamics of phonon number in a 3D superfluid with a concave acoustic branch, demonstrating that while four-phonon collisions only achieve partial equilibrium, the slower five-phonon collisions drive the system to complete thermochemical equilibrium with a specific time-dependent evolution of fugacity and entropy, thereby concluding a study initiated by Khalatnikov in 1950.

The Gist

The Big Picture: A Dance Floor of Sound Waves

Imagine a superfluid (like liquid helium or a cloud of ultra-cold atoms) not as a liquid, but as a giant, perfectly smooth dance floor. On this floor, the "dancers" are phonons—tiny packets of sound energy.

Usually, when you have a crowd of dancers, they bump into each other, swap partners, and eventually settle into a perfect rhythm where everyone is happy and the energy is evenly distributed. This is called thermal equilibrium.

However, this paper studies a very specific, tricky scenario:

1. The Dance Floor is Weird: The physics of this superfluid is shaped like a "concave" bowl (curving inward).
2. The Rules are Strict: The dancers must follow strict rules of conservation (you can't create or destroy energy or momentum out of thin air).
3. The Problem: Because of the weird shape of the floor, the usual way dancers bump into each other (3 dancers meeting) is forbidden. It's like trying to do a specific dance move that the laws of physics simply won't allow on this specific floor.

The Three-Act Story of Relaxation

The paper explains how these sound waves eventually calm down and reach a perfect state of rest. It happens in three distinct stages, like a slow-motion movie.

Act 1: The Stalemate (The 4-Phonon Collision)

Since the usual "3-dancer" bump is forbidden, the phonons try a different move: 4 phonons colliding.

• The Analogy: Imagine four people trying to swap dance partners. They can do this, and they can shuffle their energy around so everyone feels the same temperature.
• The Catch: While they can shuffle energy, they cannot change the total number of dancers. If you start with 100 phonons, you still have 100.
• The Result: The system reaches a "partial equilibrium." It feels warm and balanced, but it has a "chemical potential" (a pressure to gain or lose particles) that isn't zero. It's stuck in a holding pattern. This happens relatively quickly (in the world of quantum physics), but it's not the final answer.

Act 2: The Long Wait (The 5-Phonon Collision)

To reach the true final equilibrium (where the chemical potential is zero and the system is perfectly relaxed), the phonons need to change their total number. They need to do a 5-phonon collision (2 phonons turning into 3, or vice versa).

• The Analogy: This is like trying to get five people to coordinate a complex, synchronized dance move at the exact same time. It's incredibly rare.
• The Result: This process is extremely slow. The paper calculates that the time it takes is proportional to $T^{-9}$ (where $T$ is temperature). As the temperature drops, this process becomes agonizingly slow. It's the "bottleneck" that determines how long the system takes to fully relax.

Act 3: The Final Stretch (The Fugacity Journey)

The authors calculated exactly how the system moves from a state of "no particles" (non-degenerate) to "full equilibrium."

• The Fugacity ($z_\phi$): Think of this as a "crowd meter."
  • At the start, the meter is at 0 (empty).
  • At the end, the meter is at 1 (perfectly full/balanced).
• The Speed:
  • Early on: The crowd builds up quickly, following a strange, non-integer power law (like $t^{0.8}$). It's a rapid initial rush.
  • Later on: As it gets closer to the finish line, the speed slows down exponentially, like a car gently braking to a stop. It takes a long time to get the last few percent right.

Why Does This Matter?

1. It Solves a 70-Year-Old Mystery
The famous physicist Lev Landau and his student Khalatnikov started studying this in the 1950s. They knew the 4-phonon collisions happened, but they couldn't fully calculate the slow, final step involving 5 phonons. This paper finally does the math to close the loop on their work.

2. It's Not Just Theory
The authors suggest you can actually test this in a lab!

• Cold Atoms: You can trap fermionic atoms (like Lithium or Potassium) in a magnetic field and tune them to the "BCS side" of a superfluid transition. In this state, the physics creates that "concave" shape needed for the experiment.
• Liquid Helium: If you squeeze liquid helium-4 hard enough (high pressure), it also behaves this way.

The "Entropy" Surprise

The paper also looks at Entropy (disorder).

• Landau predicted that if you change a system very slowly, the rate at which entropy increases is proportional to the square of the rate at which the system changes.
• The authors confirmed this. Even though the system is far from equilibrium, the "speed of disorder" is perfectly linked to the "speed of the crowd meter" (fugacity). It's a beautiful confirmation of a fundamental thermodynamic rule.

Summary in One Sentence

This paper explains how sound waves in a weirdly curved superfluid get stuck in a "traffic jam" because they can't bump into each other normally, forcing them to wait for a rare, five-way collision to finally reach a state of perfect peace, a process that takes a very long time and follows a specific mathematical rhythm.

Technical Summary

1. Problem Statement

The paper investigates the collisional evolution of a spatially homogeneous and isotropic phonon gas in a three-dimensional superfluid at low but non-zero temperature ($T$). The specific focus is on superfluids where the acoustic excitation branch is concave (curvature parameter $\gamma < 0$) near zero momentum.

• The Constraint: In such systems, standard three-phonon processes ($1\phi \leftrightarrow 2\phi$) are forbidden by energy-momentum conservation because the dispersion relation does not allow three phonons to satisfy both conservation laws simultaneously.
• The Hierarchy of Relaxation:
  • Four-phonon collisions ($2\phi \leftrightarrow 2\phi$): These are the dominant processes, occurring on a timescale $\propto T^{-7}$. They drive the system to thermal equilibrium (a Bose distribution with a specific temperature $T$) but conserve the total number of phonons ($N_\phi$). Consequently, the phonon chemical potential $\mu_\phi$ remains non-zero.
  • Five-phonon collisions ($2\phi \leftrightarrow 3\phi$): These are the sub-dominant processes required to change the total phonon number. They drive the system toward complete thermochemical equilibrium (where $\mu_\phi \to 0$). The paper aims to quantify the kinetics of this slower relaxation process, which occurs on a timescale $\propto T^{-9}$.

2. Methodology

The authors employ a rigorous theoretical framework combining kinetic theory and quantum hydrodynamics:

• Kinetic Equations: They derive the Boltzmann-like kinetic equations for the occupation numbers $n_q$ of phonon modes, specifically focusing on the contribution of $2\phi \leftrightarrow 3\phi$ collisions.
• Collision Amplitude Calculation: A central contribution is the explicit calculation of the transition amplitude $A_{2\phi \to 3\phi}$ using Landau and Khalatnikov's cubic quantum hydrodynamics.
  • The calculation is performed in the low-temperature limit where phonon wavevectors and scattering angles are small ($O(T)$).
  • The authors utilize the generalized Fermi's Golden Rule (third-order perturbation theory) to handle the sub-dominant nature of these processes.
  • Key Technical Insight: They prove that for a concave branch, the collision amplitude calculated in a vacuum is identical to that in a thermal bath (despite the presence of intermediate phonon occupation numbers), provided the system is on the energy shell ($E_i = E_f$). This simplifies the calculation significantly.
• Dimensional Reduction: By exploiting scale invariance and exchange symmetries, the authors reduce the complex multi-dimensional integrals required for the collision rate into a manageable numerical form involving a dimensionless coefficient $C_\phi(z_\phi)$, where $z_\phi = e^{\beta \mu_\phi}$ is the phonon fugacity.
• Thermodynamic Constraints: The evolution of the fugacity is coupled with the conservation of total energy ($E_\phi$), allowing the elimination of the time-dependent temperature $T(t)$ in favor of the fugacity $z_\phi(t)$.

3. Key Contributions

• Quantitative Resolution of a Historical Problem: The study completes the qualitative work initiated by Khalatnikov in 1950 regarding sound absorption in superfluid helium. It provides the first fully quantitative derivation of the relaxation rate for the phonon chemical potential in the concave regime.
• Explicit Amplitude Derivation: The paper derives the explicit form of the $2\phi \to 3\phi$ scattering amplitude, including the numerical prefactor $C_\phi(z_\phi)$, which was previously unknown.
• Validation of the "Vacuum" Approximation: The authors rigorously demonstrate that for $2\phi \to 3\phi$ processes on a concave branch, the thermal occupation of intermediate states does not alter the transition amplitude at the dominant order, justifying the use of vacuum calculations in a thermal bath.
• Numerical Evaluation: The authors perform high-precision numerical integration to determine the behavior of the coefficient $C_\phi(z_\phi)$ across the entire range of fugacities ($z_\phi \in [0, 1]$).

4. Key Results

The paper derives a closed-form evolution equation for the phonon fugacity $z_\phi(t)$:
$$\frac{1}{\Gamma_{eq}^\phi} \frac{d z_\phi}{dt} = F(z_\phi)$$
where $\Gamma_{eq}^\phi \propto T^9$ is the asymptotic relaxation rate near equilibrium.

Relaxation Kinetics:

• Short Times ($z_\phi \to 0$): The system starts in a non-degenerate (Boltzmann) regime. The fugacity grows according to a non-integer power law:
  $$z_\phi(t) \propto t^{4/5}$$
  This behavior arises from the specific scaling of the collision integral at low fugacity.
• Long Times ($z_\phi \to 1$): As the system approaches complete equilibrium ($\mu_\phi \to 0$), the relaxation becomes exponential:
  $$z_\phi(t) \approx 1 - \exp(B - \Gamma_{eq}^\phi t)$$
  The rate is governed by $\Gamma_{eq}^\phi$, which scales as $T^9$.

Entropy Production:
The authors analyze the rate of entropy change $(d/dt)S_\phi$. They confirm that entropy always increases (Second Law of Thermodynamics). Crucially, they find that the rate of entropy production is asymptotically proportional to the square of the rate of change of the fugacity:
$$\frac{d S_\phi}{dt} \propto \left( \frac{d z_\phi}{dt} \right)^2$$
This result aligns with Landau's prediction for arbitrarily slow adiabatic transformations, validating the theoretical consistency of the approach.

Numerical Coefficients:
The dimensionless coefficient $C_\phi(z_\phi)$ was calculated numerically:

• $C_\phi(0) \approx 3.528 \times 10^4$
• $C_\phi(1) \approx 5.422 \times 10^4$
  The weak dependence of $C_\phi$ on $z_\phi$ simplifies the physical interpretation of the relaxation dynamics.

5. Significance and Experimental Relevance

• Theoretical Closure: This work closes a decades-old gap in the theory of superfluid hydrodynamics by fully characterizing the slowest relaxation mode in systems with concave dispersion.
• Experimental Verification: The authors propose two experimental platforms to verify these predictions:
  1. Cold Fermionic Atoms: Specifically, gases on the BCS side of the BEC-BCS crossover, where the collective excitation branch is known to be concave ($\gamma < 0$).
  2. Superfluid Helium-4: Under sufficiently high pressure, liquid helium-4 exhibits a concave acoustic branch.
• Implications for Transport: Understanding these relaxation timescales is critical for modeling thermal transport and sound propagation in quantum fluids at ultra-low temperatures, distinguishing between hydrodynamic regimes and kinetic regimes.

In summary, the paper provides a complete, quantitative, and experimentally testable theory for how a non-equilibrium phonon gas in a 3D superfluid with a concave dispersion relation relaxes to full thermodynamic equilibrium via rare five-phonon collisions.