🔬 condensed matter

Classical fractons with cosmological fixed points

This paper demonstrates that a specific scale-invariant, dipole-conserving fracton Hamiltonian naturally evolves toward stable fixed points that reproduce the key features of a flat, matter-dominated cosmology—including Einstein-de Sitter expansion, homogeneous particle distributions, and a bidirectional arrow of time—without requiring fine-tuning.

Original authors: Akash Singh, Dileep P. Jatkar, S. L. Sondhi, Abhishodh Prakash

Published 2026-08-11

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CC BY 4.0

Original authors: Akash Singh, Dileep P. Jatkar, S. L. Sondhi, Abhishodh Prakash

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

Imagine the universe as a giant, invisible dance floor where particles are the dancers. In most physics stories we know, these dancers move based on their own momentum; if you push one, it glides across the floor until it hits something else. But there is a strange, newer idea in physics called "fractons." In the world of fractons, a single dancer cannot move at all on their own. They are stuck in place, like a dancer glued to the floor. To move, they need a partner. In fact, their ability to dance depends entirely on the crowd around them. If the whole group shifts, the individual can finally take a step. This "Machian" behavior—where motion is a collective property rather than an individual one—creates a very different kind of physics, one where the rules of movement are written by the group, not the person.

Scientists have been fascinated by how these fractons behave, especially because they seem to break the usual rules of how energy and order work. Usually, in a closed system, things just bounce around forever without settling down into a specific pattern. But fractons are weird: they seem to have a way of organizing themselves into attractors, like a magnet pulling iron filings into a specific shape. The big question is: can we find a simple set of rules that explains how these particles organize themselves, and could this organization look like the way our actual universe expands and forms galaxies? This paper dives into that question, looking for a mathematical "toy model" that might explain the grand structure of the cosmos using these quirky, glued-together particles.

The Paper's Discovery: A Cosmic Toy Model

The authors of this paper, Akash Singh and his team, decided to test a specific family of fracton rules. They imagined a system of particles interacting through a special formula that depends on two numbers, which they call $\alpha$ and $\beta$ . Think of these numbers as the "dial settings" on a cosmic machine. By turning these dials, they could change how the particles pull or push on each other. Their goal was to find a specific setting where the particles would naturally settle into a pattern that looks exactly like our expanding universe.

They found that for most settings, the particles do some interesting things, but for one very special setting—where $\alpha = -2$ and $\beta = 1$ —the system becomes a perfect mimic of a flat, matter-dominated universe (like the one we live in, filled with stars and galaxies but no dark energy).

Here is what happens in this "distinguished model":

The Expansion: The particles start moving apart. As they do, the distance between them grows in a very specific way: the size of the system grows like the cube root of time squared ( $t^{2/3}$ ). This is the exact same rate at which our real universe expands when it is dominated by matter.
The Shape: As the universe expands, the particles don't just scatter randomly. They settle into a specific shape. For a large number of particles, they form a giant, expanding ball that is evenly filled, just like a smooth cloud of gas.
The Clusters: When the team simulated this with a lot of particles (around 1,000), something even cooler happened. While the whole group expanded, small groups of particles (clusters) formed tight little knots. These knots stayed the same physical size, even as the space between them grew. Inside these knots, the particles danced around each other, but the knots themselves drifted apart. This is exactly how galaxies and galaxy clusters behave in our universe: they are bound together by gravity, but the space between them is stretching.

The "Janus Point" and the Arrow of Time

One of the most playful and profound findings in the paper is about time. Usually, we think of time as having a direction: it moves forward, and things get more messy (entropy increases). In this fracton model, the authors found a "Janus point." Imagine a moment in the middle of the simulation where the system is at its smallest and simplest. If you run the simulation forward from this point, the universe expands and gets complex. If you run it backward from the same point, the universe also expands and gets complex in the other direction.

It's like a ball rolling down a hill in both directions from the bottom. In both directions, the "arrow of time" points away from the center, and the universe gets bigger and more structured. This happens without the scientists having to set up any special "low entropy" starting conditions. The attractor nature of the fracton rules forces the system to behave this way naturally.

What the Paper Rules Out and How Sure They Are

The authors are very careful to distinguish between what they have proved, what they have simulated, and what they suspect.

Proved: They mathematically proved that for this specific model, the expansion rate is exactly $t^{2/3}$ and that the particles form a "central configuration" (a specific balanced shape) that matches the rules of Newtonian gravity. They also proved that for small numbers of particles (like 3), these shapes are stable.
Simulated: For larger numbers of particles (up to 1,500), they ran computer simulations. These simulations showed that random starting points naturally evolve into the expanding, clustered state. The simulations strongly suggest that the "Janus point" behavior and the formation of bound clusters are real features of the model.
Suggested (Conjectured): The authors propose a "conjecture" (a strong guess based on evidence) that as the number of particles gets huge, the clusters act like single heavy particles with their own effective mass, and the whole system behaves like a universe with unequal masses. They haven't mathematically proved this for infinite particles yet, but the simulations make it look very likely.

They explicitly rule out the idea that this is a full replacement for Einstein's theory of relativity. Their model lives in a fixed, flat space with a single clock; it doesn't have the bending of space-time or light cones that real gravity has. Instead, they present it as a "toy model"—a simplified playground to see how complex cosmic features (like expansion, homogeneity, and structure formation) can emerge from simple, local rules without needing to be "fine-tuned" by a creator.

Why It Matters

The beauty of this work is that it takes the "fine-tuning" problem of cosmology—the mystery of why our universe started in such a perfect, smooth state to expand the way it does—and solves it with a simple attractor. In this fracton world, you don't need to set the initial conditions perfectly. No matter how you start the dance (as long as you start randomly), the music of the fracton rules pulls the dancers into the cosmic waltz. The expansion, the formation of galaxies, and even the direction of time emerge naturally as the system settles into its attractor state. It suggests that the grand structure of the universe might not be a lucky accident, but a natural consequence of the underlying rules of how things move and interact.

Technical Summary: Classical Fractons with Cosmological Fixed Points

Problem Statement
The paper addresses a fundamental limitation in Hamiltonian dynamics: the Liouville theorem forbids asymptotically stable attractors in the full phase space of smooth Hamiltonian systems, as they would require phase-space volume contraction. However, classical fractons—systems with higher-moment conservation laws (specifically dipole conservation)—exhibit unusual collective mobility where isolated particles are immobile. Previous work suggested that while the full phase-space flow is volume-preserving, the projection onto configuration space or relational variables could exhibit attractor behavior, clustering, and an emergent arrow of time. The specific problem tackled here is to analytically characterize these attractor fixed points for a general class of scale-invariant, dipole-conserving fracton Hamiltonians and to determine if any specific model within this class reproduces the salient dynamical features of a flat, matter-dominated cosmology (Einstein-de Sitter expansion, homogeneity, and structure formation) without requiring fine-tuned initial conditions.

Methodology
The authors study a two-parameter family of scale-invariant fracton Hamiltonians:
$H_{\alpha,\beta} = \sum_{i<j} |p_i - p_j|^\alpha |x_i - x_j|^\beta$
where $\alpha, \beta \in \mathbb{R}$ . The methodology proceeds through several analytical and numerical steps:

Scale-Shape Separation: Utilizing the mechanical similarity of the Hamiltonian, the authors perform a dynamical similarity reduction. They separate the system into an overall scale variable $R(t)$ and dimensionless shape variables (positions $s_i$ and momenta $u_i$ ). This reduction yields autonomous equations for the shape dynamics, which are no longer constrained by the symplectic volume preservation of the original system, thereby allowing for attractor fixed points.
Fixed Point Analysis: The authors identify radial fixed points where the shape variables are time-independent. They demonstrate that these fixed points correspond to central configurations of a power-law Riesz potential with exponent $q = \alpha + \beta$ .
Stability Analysis: A conditional local stability theorem is derived for the expanding branch of these fixed points. The stability depends on the positivity of an auxiliary quadratic form $C_q$ and the sign of the constrained Hessian of the shape potential. Analytical proofs are provided for the equilateral three-body case, while numerical spectral analysis is used for larger $N$ .
Newtonian Duality: The paper investigates the conditions under which the fracton dynamics admits a dual description as a zero-energy Newtonian gravitational system. This requires matching the fracton fixed-point equation to the Newtonian central-configuration equation.
Large- $N$ Simulations: Numerical simulations are conducted for moderate and large particle numbers ( $N$ ) to observe the dynamical evolution from random initial conditions. These simulations test the convergence to fixed points, the emergence of homogeneity, and the formation of bound clusters.
Entropy and Time Arrows: The authors calculate the Boltzmann entropy associated with the scale variable and analyze the growth of shape complexity to define a bidirectional arrow of time originating from a "Janus point" (a minimum scale).

Key Contributions and Results

Identification of the Distinguished Model: The analysis singles out a unique model on the Einstein-de Sitter (EdS) branch ( $\alpha = -2\beta$ ) with parameters $(\alpha, \beta) = (-2, 1)$ . This model is unique because:
- Its fixed-point equation is exactly the equal-mass Newtonian central-configuration equation.
- Its large- $N$ regular extremum corresponds to a homogeneous solid ball (unlike the shell structure found in other models on this branch).
- Its homothetic trajectories admit an exact dual description as a zero-energy Newtonian gravitational system, where the effective gravitational constant is dynamically generated by the fracton initial conditions rather than imposed.
Stability and Attractor Dynamics:
- Moderate $N$ : Random initial conditions converge to an expanding, regular fixed point with a shape distribution approaching a homogeneous ball. The scale evolution follows the EdS law $R(t) \propto |t|^{2/3}$ .
- Large $N$ : Simulations reveal a richer class of attractors. Instead of a single homogeneous distribution, the system forms tightly bound clusters with internal motion. The coarse-grained centers of these clusters evolve as an unequal-mass Newtonian system, approaching unequal-mass central configurations and separating as $t^{2/3}$ . The clusters retain a fixed physical size while the inter-cluster distances expand.
Emergence of Cosmological Features: The distinguished model reproduces several features of flat matter-dominated cosmology as emergent attractor properties:
- Einstein-de Sitter Expansion: The scale factor grows as $t^{2/3}$ .
- Flatness and Critical Energy: The system naturally selects the zero-energy (flat) Newtonian dual without fine-tuning the initial energy.
- Homogeneity: While small-scale structures (clusters) form, the large-scale distribution remains homogeneous, consistent with the cosmological principle.
- Bound Structures: Local bound subsystems (clusters) emerge dynamically, decoupling from the global expansion, analogous to galaxies in an expanding universe.
Machian Dynamics and Arrow of Time:
- The formation of bound clusters is shown to be Machian: isolated pairs cannot support bound orbits in this model; bound motion only emerges when embedded in a many-body background.
- The system exhibits a bidirectional arrow of time. Both the scale $R(t)$ and the shape complexity (measured by the Riesz potential) grow away from a minimum scale point (Janus point) in both time directions. The Boltzmann entropy grows logarithmically with the scale, $S_B \propto \log R$ , despite the microscopic time-reversal invariance.

Significance and Claims
The paper claims that the distinguished fracton model serves as a "toy model" for cosmological dynamics. Its primary significance lies in demonstrating that features typically requiring fine-tuned initial conditions in standard Newtonian cosmology—specifically homothetic expansion, spatial flatness (zero energy), large-scale homogeneity, and the alignment of scale and structure growth arrows—can arise as robust late-time attractor properties of a specific Hamiltonian system.

The authors emphasize that this is not a replacement for relativistic cosmology (lacking dynamical spacetime, light cones, or radiation), but rather a conceptual demonstration of how attractor mechanisms can resolve fine-tuning problems. The model suggests that the "specialness" of the early universe (low entropy, homogeneity) might be an emergent consequence of dynamical attraction rather than a fundamental constraint on initial data. The work bridges classical fracton physics, relational particle mechanics, and cosmological dynamics, offering a new perspective on the emergence of cosmological structure from microscopic many-body interactions.

The Paper's Discovery: A Cosmic Toy Model

The "Janus Point" and the Arrow of Time

What the Paper Rules Out and How Sure They Are

Why It Matters

Technical Summary: Classical Fractons with Cosmological Fixed Points

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