🔬 condensed matter

Navier-Stokes hydrodynamics near the critical point with out-of-equilibrium modes

This paper investigates how slow out-of-equilibrium modes reshape the power spectrum of dynamical density fluctuations near the critical point, revealing that these modes introduce an extensivity constraint on velocity coupling and significantly reduce fluctuation decay rates, thereby manifesting the characteristic slowing down of the system.

Original authors: Md Hasanujjaman, Golam Sarwar, Mahfuzur Rahaman, Abhijit Bhattacharyya, Jane Alam

Published 2026-09-22

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CC0 1.0

Original authors: Md Hasanujjaman, Golam Sarwar, Mahfuzur Rahaman, Abhijit Bhattacharyya, Jane Alam

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Fluids, whether they are the water in a river or the air in a storm, usually behave in predictable ways when they are calm. Scientists have long used a set of rules called hydrodynamics to describe how these fluids flow, mix, and carry heat, but these rules work best when the fluid is in a state of balance, where its temperature and pressure are steady. However, nature has a way of throwing these rules into chaos near a critical point, a specific state where a substance is on the verge of changing its phase, such as a gas turning into a liquid. At this precise moment, the fluid becomes incredibly sensitive; tiny fluctuations in its density grow massive, and the usual laws of flow begin to break down because the fluid is no longer in equilibrium. Understanding what happens in this chaotic, critical zone is a major challenge, as it reveals how matter behaves when it is pushed to its limits.

A team of researchers has taken a fresh look at this problem by extending the standard rules of fluid motion to include a new, slow-moving element that appears only when the system is out of balance. In their study, they focused on a fluid near its critical point and introduced a concept called a slow mode, which acts like a lingering memory of the fluid's past state. While standard physics assumes that a fluid adjusts instantly to changes, this new approach acknowledges that near the critical point, some parts of the fluid take a long time to settle down. By adding this slow, out-of-equilibrium component to the equations, the researchers were able to track how the fluid's density fluctuates over time and how these fluctuations show up in the light or sound waves that pass through the material.

The researchers used a specific type of fluid, helium-4, and modeled its behavior at temperatures both far from and very close to its critical point, which occurs at 5.2 Kelvin. They looked at a measurable quantity known as the dynamic structure factor, which essentially acts as a fingerprint of how the fluid's density wiggles and vibrates. Under normal conditions, far from the critical point, this fingerprint typically shows three distinct peaks: a central spike caused by heat diffusion and two side spikes caused by sound waves traveling through the fluid. When the team included the slow mode in their calculations, they found that the overall shape of this fingerprint remained the same, but the details changed significantly. The side peaks, representing the sound waves, shifted closer to the center and became narrower, indicating that the sound was traveling slightly slower and losing energy more gradually. The central peak also became narrower, suggesting that the heat fluctuations were decaying more slowly than usual.

As the fluid approached the critical point, the behavior became even more dramatic. The side peaks representing sound waves vanished entirely, leaving only the central peak. This disappearance is a known hallmark of the critical point, where the fluid becomes so compressible that sound waves can no longer propagate effectively. However, the inclusion of the slow mode had a profound effect on the remaining central peak. The researchers found that as the influence of the slow mode increased, the height of this central peak grew much larger near the critical point than it did when the fluid was far away from it. This suggests that the slow mode acts as a reservoir, trapping and holding onto the energy of the fluid's fluctuations for a longer time. This phenomenon, known as critical slowing down, means that the fluid takes much longer to return to a calm state once it is disturbed.

The study also explored how the fluid reacts when the connection between this slow mode and the flow of heat is strengthened. They discovered that increasing this connection actually reduced the height of all the peaks in the fingerprint. This indicates that a stronger link between the slow mode and heat flow provides a more efficient path for the fluid to release its stored energy, causing the fluctuations to die out faster. In essence, the slow mode can either hold onto the fluid's energy, making the fluctuations last longer and more intense, or it can help dissipate that energy quickly, depending on how it is coupled to the rest of the system.

These findings offer a clearer picture of how fluids behave when they are teetering on the edge of a phase change. By incorporating the slow, out-of-equilibrium modes into the standard equations, the researchers showed that the fluid's response to disturbances is fundamentally reshaped. The work suggests that the critical point is not just a place where things break down, but a regime where new, slow dynamics take over, storing energy and delaying the return to equilibrium. This deeper understanding could help scientists better interpret experiments involving light or neutron scattering in critical fluids, providing a more accurate map of how matter behaves in its most extreme and fascinating states.

Technical Summary: Navier-Stokes Hydrodynamics Near the Critical Point with Out-of-Equilibrium Modes

Problem Statement
Standard hydrodynamic descriptions, including the Navier-Stokes (NS) equations, rely on the assumption that the system is in local thermal equilibrium and that the correlation length ( $\xi$ ) is much smaller than the characteristic wavelength ( $\lambda$ ) of the fluctuations ( $\xi \ll \lambda$ ). Near the critical point (CP) of a fluid, the correlation length diverges, violating this condition. Consequently, the fluid falls out of equilibrium, rendering standard hydrodynamic predictions invalid. While gradient expansion schemes (such as second-order Müller-Israel-Stewart theory) have been proposed to extend hydrodynamics, they often fail to converge or adequately describe fluids with large fluctuations near the CP. The "Hydro+" framework offers an alternative by introducing a parametric slow mode ( $\phi$ ) to account for these large, non-equilibrium fluctuations. However, the specific impact of this slow mode on the dynamic structure factor and the constraints imposed by thermodynamic consistency within a non-relativistic NS framework require further investigation.

Methodology
The authors extend the non-relativistic Navier-Stokes hydrodynamic framework by incorporating a slow, non-hydrodynamic scalar mode, $\phi$ , into the definition of entropy. The methodology proceeds as follows:

Thermodynamic Formulation: The entropy density is generalized to $s_+(\epsilon, \rho, \phi)$ , where $\epsilon$ is energy density and $\rho$ is mass density. The Gibbs-Duhem relation is modified to include the conjugate force $\pi = (\partial s_+ / \partial \phi)_{\epsilon, \rho}$ .
Governing Equations: The authors derive a closed set of linearized hydrodynamic equations based on mass, momentum, and energy conservation, augmented by an evolution equation for the slow mode $\phi$ . The evolution equation is of the relaxation type:
$\frac{\partial \phi}{\partial t} + \nabla \cdot (\phi \mathbf{v}) = -\Gamma_\phi \pi - A_\phi (\nabla \cdot \mathbf{v})$
where $\Gamma_\phi$ is the relaxation rate and $A_\phi$ represents the susceptibility of the mode to compression/expansion.
Thermodynamic Constraints: A critical step in the derivation involves enforcing the second law of thermodynamics ( $\partial_\mu S^\mu_+ \geq 0$ ). The authors demonstrate that the extensivity condition imposes a strict constraint on the coupling between the slow mode and the velocity divergence. Specifically, the coefficient $S_n$ associated with the $(\nabla \cdot \mathbf{v})$ term in the entropy production rate must vanish ( $S_n = 0$ ) to ensure positive-definite entropy production for arbitrary flows. This leads to a specific relation between thermodynamic variables and the susceptibility $A_\phi$ .
Linear Mode Analysis: The governing equations are linearized around a homogeneous equilibrium state. The authors solve for the density fluctuation $\delta \rho(k, \omega)$ in Fourier space to derive the dynamic structure factor, $S_{\rho\rho}(k, \omega)$ , which characterizes the power spectrum of density fluctuations.
Numerical Evaluation: The dynamic structure factor is numerically evaluated for Helium-4 ( $^4$ He) using an equation of state from Ref. [22] and scaling laws for transport coefficients (viscosity, thermal conductivity) and response functions from Ref. [23]. Calculations are performed both far from the critical point ( $T = 20$ K) and near the critical point ( $T = 6$ K, with $T_c = 5.2$ K).

Key Contributions and Results

Constraint on Coupling: The study establishes that the requirement of thermodynamic extensivity imposes an extra constraint on the coupling of the slow mode $\phi$ with the four-divergence of velocity. This constraint is necessary to maintain the positivity of entropy production and distinguishes the Hydro+ formulation from arbitrary extensions of NS theory.
Modification of the Dynamic Structure Factor:
- Away from the Critical Point: In the absence of the slow mode ( $\phi=0$ $ϕ = 0$ ), $S_{\rho\rho}(k, \omega)$ $S_{ρρ} (k, ω)$ exhibits the standard three-peak structure: a central Rayleigh peak (thermal diffusion) and two symmetric Brillouin peaks (sound propagation). The inclusion of a non-zero $\phi$ $ϕ$ preserves the symmetry of the distributions but results in:
  - A shift of the Brillouin peaks toward zero frequency, indicating a reduction in the effective speed of sound due to slower pressure response.
  - A reduction in the widths of all peaks, signifying a reduced decay rate of fluctuations.
  - An asymmetry in the heights of the Brillouin peaks.
- Near the Critical Point: As the system approaches the CP, the Brillouin peaks vanish entirely due to the divergence of compressibility and the vanishing of the sound speed ( $c_s \to 0$ ). The Rayleigh peak remains, but its width is substantially reduced in the presence of $\phi$ . This narrowing indicates "critical slowing down," where the decay of fluctuations is significantly delayed.
Role of the Slow Mode ( $\phi$ ) and Coupling ( $b$ ):
- Increasing the equilibrium value of the slow mode ( $\bar{\phi}$ ) leads to a monotonic increase in the height of both the Rayleigh and Brillouin peaks. The authors attribute this to the slow mode acting as a reservoir for fluctuation energy, temporarily trapping energy before releasing it, thereby enhancing the spectral weight.
- The effect of $\phi$ is significantly more pronounced near the critical point than far from it, consistent with the phenomenon of critical slowing down.
- The coupling parameter $b$ (representing the interaction between the slow mode and heat flux) suppresses the peak heights. A stronger coupling introduces an additional relaxation channel, increasing the damping of both thermal and acoustic fluctuations.

Significance and Claims
The paper claims to demonstrate that the inclusion of out-of-equilibrium modes (OEM) fundamentally reshapes the power spectrum of dynamical density fluctuations near the critical point. The primary significance lies in showing that the Hydro+ framework, when constrained by the second law of thermodynamics, provides a consistent hydrodynamic description that captures the slowing down of fluctuations and the redistribution of spectral weight.

Specifically, the authors claim that:

The extensivity condition is a non-trivial constraint that dictates the coupling of the slow mode to hydrodynamic gradients.
The presence of the slow mode $\phi$ reduces the decay rates of fluctuations (narrowing peak widths), offering a hydrodynamic explanation for critical slowing down.
The slow mode acts as a source that redistributes fluctuation energy, enhancing the magnitude of peaks in the dynamic structure factor, particularly the Rayleigh peak near the critical point.

The work concludes that while the slow mode does not create new hydrodynamic modes, it significantly modifies the dynamics of existing modes, providing a theoretical basis for understanding critical phenomena within an extended hydrodynamic framework. The authors suggest future work could explore nonlinear fluctuations, multiple slow modes, and direct comparisons with experimental light or neutron scattering data.

Technical Summary: Navier-Stokes Hydrodynamics Near the Critical Point with Out-of-Equilibrium Modes

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