🔬 condensed matter

Phase Separation in Fractonic Fluids: Coarsening, Interfaces, Nucleation, and Hyperuniformity in and out of Equilibrium

This paper develops a theory of phase separation in fractonic fluids with higher-order multipole conservation laws, demonstrating that such conservation slows relaxation dynamics while nonequilibrium violations of the fluctuation-dissipation theorem suppress fluctuations and hinder nucleation, predictions that are supported by simulations.

Original authors: Raphael Maire

Published 2026-10-01

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CC BY 4.0

Original authors: Raphael Maire

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

In the vast landscape of physics, there is a special class of materials where movement is not free. Imagine a crowded room where people are not allowed to walk in straight lines; instead, to move from one side to the other, they must coordinate a complex, collective shuffle with their neighbors. This is the world of "fractonic" systems. In these materials, the basic building blocks are so restricted in their mobility that they cannot move individually. Any transport of matter requires a group effort, a rearrangement of many particles acting in unison. This behavior, first discovered in exotic quantum models, has since been found to describe a broader range of constrained dynamics, including how defects move inside solid crystals. The rules governing these systems are strict: they conserve not just the total amount of matter, but also higher-order properties like the center of mass and even more complex geometric arrangements of that matter.

When scientists study how these materials change from one state to another—such as when a fluid separates into two distinct phases like oil and water—they expect to see familiar patterns of growth and relaxation. However, the rigid rules of fractonic systems seem to rewrite the script. A recent study by Raphaël Maire at the University of Barcelona investigates exactly how these strict conservation laws alter the process of phase separation. The research explores how these fluids form patterns, how they grow, and how they fluctuate, comparing what happens when the system is in a calm, balanced state versus when it is being actively driven out of balance. The findings reveal that these constraints do more than just slow things down; they fundamentally change the statistical nature of the material, suppressing the very fluctuations that usually drive change.

Maire's work builds a theoretical framework to describe these fluids, treating them as a continuous field rather than a collection of individual particles. The core of the theory focuses on a specific type of fluid that conserves higher-order "multipole moments." In simple terms, this means the fluid must preserve not only its total mass but also the balance of its mass distribution, much like a seesaw that must remain level even as weights shift. The study examines two scenarios: one where the fluid is in thermal equilibrium, obeying the standard laws of heat and energy exchange, and another where the fluid is driven out of equilibrium, breaking the usual symmetry of time. By simulating these conditions on a computer, the researcher could observe how the fluid evolves over time without the interference of real-world experimental noise.

The simulations show that the stricter the conservation laws, the slower the system relaxes. When a fluid separates into two phases, it typically forms domains or patches that grow larger over time. In ordinary fluids, this growth follows a predictable speed. In fractonic fluids, however, the growth is significantly delayed. The more complex the conservation rule—meaning the more geometric properties the fluid must preserve—the slower the domains expand. This slowing effect is observed in both equilibrium and non-equilibrium settings, but the reasons differ. In the balanced, equilibrium state, the system is forced to move slowly because the rules of thermodynamics link the noise (random jiggling) to the friction (resistance to movement). In the non-equilibrium state, where the system is driven by external forces, the noise behaves differently. Here, the random fluctuations that usually help the system explore new configurations are suppressed at large scales.

This suppression of fluctuations leads to a phenomenon known as hyperuniformity. In a normal fluid, density varies randomly across large distances. In these fractonic fluids, the density becomes remarkably uniform over long ranges, as if the material is actively hiding its irregularities. This happens because the random noise that would normally create large-scale bumps and dips in the density vanishes faster than the system can relax. The result is a material that is unusually ordered, even when it is not in a crystal state. The study confirms that this hyperuniformity is a direct consequence of the mismatch between the way the system relaxes and the way the noise acts on it.

The research also looks at how new phases form, a process called nucleation. In a standard fluid, a small droplet of a new phase can appear spontaneously due to random fluctuations, and if it gets big enough, it will grow. In the fractonic fluids studied here, this process is much harder. The suppression of large-scale fluctuations means that the random kicks needed to form a critical droplet are much weaker. Consequently, the energy barrier to start nucleation is higher, making it more difficult for the new phase to appear. The simulations show that as the conservation laws become more complex, this barrier increases, effectively freezing the system in its current state for longer periods.

Finally, the study examines the boundaries between the two phases, known as interfaces. In ordinary fluids, these boundaries ripple and wave due to thermal fluctuations, a behavior described by the capillary wave theory. In fractonic fluids, these ripples are also suppressed. The interface becomes much stiffer, and the long-wavelength waves that usually dance along the boundary are damped out. The study finds that the height of the interface fluctuates far less than expected, especially in non-equilibrium conditions. This rigidity is another manifestation of the same hyperuniformity that affects the bulk fluid. The researchers note that measuring this effect requires careful definition of where the interface actually is, as the residual density fluctuations can sometimes mask the true position of the boundary.

The paper concludes that the conservation of higher multipole moments qualitatively changes the statistical mechanics of these fluids. It is not merely a matter of slowing down transport; the very nature of how density fluctuations behave is altered. The findings suggest that in these constrained systems, the rules of equilibrium and non-equilibrium diverge significantly. While the equilibrium system follows a predictable path dictated by thermodynamics, the non-equilibrium system exhibits unique behaviors where fluctuations are actively suppressed, leading to a state of matter that is more ordered and less prone to change than its conventional counterparts. The work provides a unified theory that explains these phenomena across different types of fractonic fluids, offering a clear picture of how strict conservation laws reshape the dynamics of phase separation.

Technical Summary: Phase Separation in Fractonic Fluids

Problem Statement
Fractonic phases are characterized by elementary excitations with restricted mobility, requiring collective rearrangements for transport. These constraints arise from the conservation of higher-order multipole moments of a scalar density (e.g., mass and center of mass). While fractonic behavior has been studied in quantum models and specific nonequilibrium lattice systems, a unified coarse-grained field theory describing phase separation in fluids conserving generic higher multipole moments remains underdeveloped. Specifically, it is unclear how these higher-order conservation laws, particularly when combined with nonequilibrium driving (which breaks the fluctuation-dissipation theorem), modify fundamental statistical-mechanical phenomena: coarsening dynamics, nucleation barriers, and interfacial fluctuations.

Methodology
The author develops a coarse-grained stochastic field theory for a scalar density $\rho(\mathbf{r}, t)$ governed by a generalized stochastic gradient flow equation:
$\partial_t \rho = -\Gamma(-\nabla^2)^m \frac{\delta F}{\delta \rho} + \eta_n$
where $F$ is a Landau free energy allowing for phase separation. The dynamics are defined by two integer exponents, $m$ and $n$ :

$m$ determines the order of the deterministic relaxation operator.
$n$ determines the order of the noise covariance, $\langle \eta_n \eta_n \rangle \sim k^{2n}$ .

The theory enforces the conservation of multipole moments up to order $P = \min(2m-1, n-1)$ . The author distinguishes between two regimes:

Equilibrium: The fluctuation-dissipation theorem (FDT) holds, requiring $n = m$ and $D = \Gamma T$ . This forces the deterministic term to have a higher order ( $m = P+1$ ) to match the noise allowed by conservation.
Nonequilibrium: FDT is violated. The author chooses the minimal deterministic scaling compatible with isotropy and conservation, leading to $m = \lceil (P+1)/2 \rceil$ while maintaining the leading noise allowed by conservation ( $n = P+1$ ).

The study employs analytical scaling arguments, linear stability analysis, and numerical simulations (spectral solvers and geometric minimum action methods) to investigate three main phenomena: homogeneous phase relaxation, phase separation (spinodal decomposition and nucleation), and interfacial fluctuations.

Key Contributions and Results

1. Homogeneous Phase and Hyperuniformity

Relaxation: Higher-order conservation laws ( $P$ ) significantly slow the relaxation of long-wavelength density perturbations. The relaxation time scales as $\tau_k \sim k^{-2m}$ . In the nonequilibrium case, where $m < P+1$ , relaxation is faster than in the equilibrium case for a fixed $P$ , as lower-order deterministic terms are permitted.
Hyperuniformity: In nonequilibrium systems, the mismatch between the noise scaling ( $k^{2n}$ ) and the deterministic relaxation ( $k^{2m}$ ) leads to the suppression of long-wavelength density fluctuations ( $S(k \to 0) \sim k^\alpha$ with $\alpha > 0$ ). This results in hyperuniformity, a property where density fluctuations vanish at large scales. The degree of suppression increases with the order of the conserved multipole $P$ .

2. Coarsening Dynamics

Growth Laws: Following spinodal decomposition, the characteristic domain size $L(t)$ $L (t)$ grows as a power law $L(t) \sim t^{1/(2m+1)}$ $L (t) \sim t^{1/ (2 m + 1)}$ .
- Equilibrium: $L(t) \sim t^{1/(2P+3)}$ .
- Nonequilibrium (odd $P$ ): $L(t) \sim t^{1/(P+2)}$ .
- Nonequilibrium (even $P$ ): $L(t) \sim t^{1/(P+3)}$ .
  Higher conservation orders universally slow the coarsening process.
Structure Factor: The self-similar structure factor $S(k, t)$ exhibits a low-wavenumber scaling $S(k) \sim k^{4m}$ induced by the phase-separation patterns. This is distinct from the hyperuniformity of the homogeneous state. While noise can introduce a crossover to a lower-power tail at very small $k$ in equilibrium, the deterministic $k^{4m}$ scaling dominates in the late-time limit for $d \ge 2$ .

3. Nucleation

Activation Cost: Nucleation from a metastable state is analyzed using a projected radius dynamics equation.
Equilibrium: The nucleation barrier is determined by the reversible work of formation, $\Delta V_c \propto W(R_c)$ , consistent with Gibbs statistics.
Nonequilibrium: The barrier is significantly enhanced. Due to the suppression of large-scale fluctuations (hyperuniformity), the noise required to supply mass to a growing droplet is weaker. The activation cost scales as a higher power of the critical radius $R_c$ (e.g., $\Delta V_c \propto R_c^{(3P+1)/2}$ for odd $P \ge 3$ ) compared to the linear dependence in equilibrium. This indicates that nucleation is more difficult in nonequilibrium fractonic fluids.

4. Interfacial Fluctuations

Relaxation: Capillary waves on a flat interface relax with a time scale $\tau_q \sim |q|^{-(2m+1)}$ , which is slower than bulk modes ( $|q|^{-2m}$ ).
Fluctuation Suppression: The height structure factor $S_h(q)$ $S_{h} (q)$ at equilibrium follows the standard capillary wave result $S_h \sim |q|^{-2}$ $S_{h} \sim ∣ q ∣^{- 2}$ . In nonequilibrium systems, the mismatch between noise and dissipation suppresses long-wavelength fluctuations, yielding $S_h \sim |q|^{P-2}$ $S_{h} \sim ∣ q ∣^{P - 2}$ .
- For $P=1$ , fluctuations diverge as $|q|^{-1}$ .
- For $P=2$ , fluctuations become constant.
- For $P \ge 3$ , fluctuations vanish as $|q| \to 0$ .
  This suppression implies that nonequilibrium driving stabilizes interfaces against thermal roughening more effectively than equilibrium dynamics.

Significance and Claims
The paper claims to provide a systematic field-theoretic framework that generalizes previous results on center-of-mass conserving systems to fluids conserving arbitrary higher multipole moments. The primary significance lies in demonstrating that:

Conservation Laws: Higher-order multipole conservation universally slows down transport and coarsening dynamics.
Nonequilibrium Effects: Violating the fluctuation-dissipation theorem in these constrained systems does not merely alter transport rates but qualitatively changes the statistical mechanics of phase separation. Specifically, it suppresses density and interfacial fluctuations (hyperuniformity) and increases the energetic cost of nucleation.
Universality: These effects are robust across different conservation orders and apply to both equilibrium and driven fractonic fluids, suggesting that center-of-mass (and higher moment) conservation fundamentally alters the statistical mechanics of density fluctuations and phase-separated states.

The author notes that their simplified field theory does not capture phase-space fragmentation (a feature of some microscopic fracton models) and that future work could explore crystalline order and nonequilibrium melting in these systems.

Technical Summary: Phase Separation in Fractonic Fluids

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