An Equation of State for Turbulence in the Gross-Pitaevskii model

This paper reports the numerical observation of a universal far-from-equilibrium equation of state in the Gross-Pitaevskii model, demonstrating that in a regime of mixed turbulence, the momentum distribution amplitude scales with the energy flux to the power of approximately 0.67, a finding that extends the concept of quasi-static thermodynamic processes to non-equilibrium steady states.

The Gist

Imagine you have a giant, invisible bathtub filled with a special kind of "super-fluid" water. This isn't normal water; it's a cloud of atoms cooled down so much that they all start acting like a single, giant wave. Physicists call this a Bose-Einstein Condensate.

Now, imagine you start shaking this bathtub back and forth. You aren't just sloshing the water; you are creating a chaotic storm of waves and tiny whirlpools (vortices) inside this super-fluid. This is turbulence.

For decades, scientists have tried to write a "rulebook" (an Equation of State) to predict exactly how this turbulence behaves. They have two main rulebooks:

1. The Wave Rulebook: Good for when the fluid ripples like a calm ocean.
2. The Whirlpool Rulebook: Good for when the fluid spins like a tornado.

But in this experiment, the fluid is doing both at the same time. It's a "mixed" storm. The old rulebooks failed to predict what would happen.

The Experiment: Shaking the Quantum Bathtub

The researchers (Gevorg, Kazuya, and Nir) built a digital simulation of this quantum bathtub. They used a computer to model the Gross-Pitaevskii (GP) model, which is like a super-accurate video game engine for these quantum fluids.

They shook the virtual fluid with a rhythmic force (like a parent shaking a baby's crib) and watched what happened.

• The Cascade: Energy was pumped in at the "big wave" level. Instead of staying there, the energy broke down into smaller and smaller waves, traveling from big scales to tiny scales, until it vanished. This is called a cascade.
• The Steady State: After a while, the chaos settled into a steady rhythm. The fluid wasn't calm, but the pattern of the chaos was stable.

The Big Discovery: A New Rulebook

The team measured two things:

1. How much energy was flowing through the system (the "flux").
2. How strong the waves were at different sizes (the "amplitude").

They expected the relationship between these two to follow one of the old rulebooks. Instead, they found something completely new.

The Analogy of the Traffic Jam:
Imagine a highway where cars (energy) are flowing.

• Old Theory: If you double the number of cars entering the highway, you expect the traffic density to double in a predictable way.
• The Old Rulebooks: Predicted that if you double the energy flow, the wave strength should go up by a factor of roughly 1.5 or 2.
• The New Discovery: The researchers found that if you double the energy flow, the wave strength goes up by a factor of about 1.6.

It sounds small, but in the world of physics, this is a massive difference. It's like discovering that gravity doesn't pull things down at 9.8 m/s², but at 10.1 m/s². It means the "mixed" storm of waves and whirlpools follows a brand new law of nature that no one had written down before.

Why This Matters

1. Universal Truths: The paper shows that even in a chaotic, far-from-equilibrium state (like a storm), nature still follows simple, universal rules. You don't need to know every single atom to predict the big picture; you just need the right "Equation of State."
2. The "Quasi-Static" Magic: As the experiment ran, the fluid slowly lost a few atoms (like water evaporating from a cup). Surprisingly, even as the system changed, it stayed perfectly on this new rulebook line. It's as if the fluid was slowly sliding down a smooth, invisible slide, staying in perfect balance the whole time, even though it was technically a "mess."
3. Bridging Theory and Reality: The researchers compared their computer simulation to real experiments done in labs. The computer model got it mostly right, but not perfectly. This tells us that while the computer model is a great tool, there are still tiny quantum secrets we haven't fully unlocked yet.

The Bottom Line

This paper is like finding a new page in the universe's instruction manual. We thought we knew how turbulence worked (either as waves or whirlpools), but we discovered a third, "mixed" way that follows its own unique rhythm.

It's a reminder that even in the most chaotic, turbulent systems—whether in a quantum lab or a stormy ocean—there is an underlying order waiting to be found. The universe, it seems, loves to follow a pattern, even when it's shaking things up.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in the physics of far-from-equilibrium systems. While equilibrium thermodynamics provides a unifying framework for complex many-body systems, a similar framework for far-from-equilibrium steady states is lacking. Specifically, the authors investigate matter-wave turbulence in ultracold atomic gases, a regime sustained by energy injection and dissipation at distinct scales.

Recent experiments (e.g., Dogra et al., Nature 2023) measured a "far-from-equilibrium Equation of State" (EOS) relating the amplitude of the turbulent momentum spectrum to the energy flux. However, existing theoretical paradigms—specifically Weak Wave Turbulence (WWT) and Kolmogorov vortex turbulence—failed to predict the experimental EOS. The authors aim to determine if the Gross-Pitaevskii (GP) model, the standard classical-field description for these systems, can reproduce this experimental EOS and whether it reveals a new universal regime of turbulence.

2. Methodology

The authors performed large-scale numerical simulations of the time-dependent Gross-Pitaevskii equation (GP) in a 3D cylindrical box geometry, mimicking experimental conditions.

• Governing Equation: The system is described by the nonlinear Schrödinger equation (GP equation) for the classical field $\psi(\mathbf{r}, t)$:
  $$i\hbar\frac{\partial\psi}{\partial t} = \left( -\frac{\hbar^2}{2m}\nabla^2 + g|\psi|^2 + V_{\text{drive}} + V_{\text{diss}} + V_{\text{box}} \right)\psi$$
• Driving Protocol: Energy is injected via a time-periodic potential gradient $V_{\text{drive}}(r, t) = U_s \sin(\omega t) z/L$, resonant with the lowest Bogoliubov excitation.
• Dissipation: To achieve a steady state, a high-momentum dissipation mechanism was implemented. This mimics experimental evaporative losses where atoms with energy exceeding a threshold $U_D$ leave the trap.
• Simulation Parameters: The study covered a wide range of interaction strengths (scattering lengths $a$ from $25a_0$ to $400a_0$) and drive strengths ($U_s$). The system was evolved until a steady state was reached, characterized by a constant energy flux.
• Analysis: The authors calculated:
  • The momentum distribution $n(k)$ and its cascade exponent $\gamma$.
  • The scale-resolved energy flux $\Pi_\epsilon(k)$ and particle flux $\Pi_N(k)$.
  • The relationship between the spectral amplitude $n_0$ and the energy flux $\epsilon$.
  • The decomposition of energy into compressible (wave) and incompressible (vortex) components.

3. Key Contributions and Results

A. Validation of the Direct Energy Cascade

The simulations confirmed the formation of a direct energy cascade.

• Steady State: The system evolves into a steady state where the mode occupation number follows a power law $N_k \propto k^{-\gamma}$ with $\gamma \approx 3.5$.
• Flux Invariance: The energy flux $\Pi_\epsilon(k)$ is constant (scale-invariant) across the inertial range, transporting energy from the injection scale to the dissipation scale $k_D$.
• WWT Agreement: The momentum distribution exponent $\gamma$ aligns remarkably well with 4-wave Weak Wave Turbulence (WWT) theory, provided the effective injection scale is identified as the interaction scale $k_0 \propto 1/\xi$ (where $\xi$ is the healing length) rather than the physical driving scale.

B. Discovery of a New Universal Equation of State (EOS)

The primary finding is the derivation of a universal EOS relating the spectral amplitude $n_0$ and the energy flux $\epsilon$:
$$\frac{n_0}{n} = C \left( \frac{\tau_t \epsilon}{n \zeta_t} \right)^b$$

• Exponent: The authors found the exponent $b \approx 0.67(2)$ (i.e., $n_0 \propto \epsilon^{2/3}$).
• Universality: This relationship holds across different interaction strengths and drive parameters when variables are rescaled by the instantaneous natural scales of the GP model (density $n$, healing length $\xi$, and chemical potential $\zeta$).
• Theoretical Discrepancy: This exponent contradicts all existing turbulence theories:
  • WWT: Predicts $b \leq 0.5$ (specifically $1/3$ for 4-wave interactions).
  • Vortex Turbulence: While the exponent $2/3$ matches the Kolmogorov $K41$ scaling for incompressible energy spectra ($E_k \propto \epsilon^{2/3}$), the authors show their system is not purely incompressible, and the momentum distribution is distinct from the velocity field spectrum.

C. The "Mixed" Turbulence Regime

The authors decomposed the energy spectrum into compressible (wave) and incompressible (vortex) parts.

• They found that both components are of comparable magnitude in the relevant momentum range.
• The observed EOS ($b \approx 0.67$) is likely a result of the interplay between compressible and incompressible excitations. No existing theory describes this "mixed" quantum turbulence regime, explaining why standard WWT or pure vortex theories fail to predict the result.

D. Comparison with Experiment

• The numerical EOS ($n_0 \propto \epsilon^{0.67}$) agrees well with recent experimental measurements for large energy fluxes.
• The authors note that the experimental data required an empirical scaling with the gas parameter ($na^3$) to collapse, which is outside the pure GP framework. However, as $na^3 \to 0$ (the limit where the GP model is most valid), the experimental data approaches the numerical results, suggesting the GP model captures the essential physics of the high-flux regime.

E. Quasi-Static Far-from-Equilibrium Processes

The study demonstrates that the concept of quasi-static thermodynamic processes extends to far-from-equilibrium steady states. As the system slowly loses particles (due to evaporation), the state variables ($\epsilon$ and $n_0$) evolve slowly, tracing a path along the universal EOS curve. This implies that even in a highly non-equilibrium state, the system can be described by a trajectory of steady states linked by a universal law.

4. Significance

• New Theoretical Paradigm: The discovery of an EOS with an exponent $b \approx 0.67$ challenges the dichotomy between wave turbulence and vortex turbulence, suggesting a new universality class for "mixed" quantum turbulence.
• Validation of Classical-Field Simulations: The work provides a critical benchmark for the validity of the Gross-Pitaevskii equation in describing far-from-equilibrium quantum fluids, showing it can capture complex turbulent scaling laws despite being a classical field theory.
• Bridge to Experiment: The results offer a theoretical explanation for recent experimental observations that were previously unexplained by standard turbulence theories.
• Thermodynamics of Non-Equilibrium: The finding that quasi-static processes apply to far-from-equilibrium steady states opens new avenues for defining thermodynamics in driven-dissipative quantum systems.

In conclusion, the paper establishes that the Gross-Pitaevskii model supports a robust, universal equation of state for turbulence that defies existing theoretical predictions, arising from a complex interplay of wave and vortex dynamics.