Anomalous Criticality of Absorbing State Transition toward Jamming

By re-examining biased random organization models, this paper reveals that jamming transitions in dense athermal systems exhibit anomalous criticality deviating from the Manna universality class due to crystallization, active-glass states, and Griffiths effects, which are unified under a field theory with fractional time dynamics.

The Gist

Imagine a crowded dance floor. At first, everyone is moving freely, bumping into each other but finding space to wiggle around. This is like a fluid. Now, imagine the music stops, and the crowd gets so dense that no one can move an inch without pushing someone else. The floor has "jammed."

For a long time, scientists thought this jamming was just a simple geometry problem: "How many marbles can you fit in a jar?" But a new theory suggested it was actually a dynamic drama—a story about how particles stop moving when shaken repeatedly, similar to how a crowd might suddenly freeze.

This paper, titled "Anomalous Criticality of Absorbing State Transition toward Jamming," goes on a deep dive to test that story. The authors found that the story is much more complicated, and the "rules of the game" change depending on how crowded the dance floor is.

Here is the breakdown of their discovery using simple analogies:

1. The Original Theory: The "Manna" Game

Scientists previously thought that when particles get jammed, they behave like a specific type of game called the Manna model (named after a physicist, not the food).

• The Analogy: Imagine a game where if two people bump into each other, they both get pushed away randomly. If they stop bumping, they sit still.
• The Expectation: Scientists thought that as you pack the room tighter, the point where everyone stops moving (the "jamming point") would follow a universal, predictable pattern, like a standard recipe for a cake.

2. The Surprise: The Dance Floor Changes Rules

The authors ran massive computer simulations (like running the dance floor scenario millions of times) and found that the "Manna recipe" breaks down when the room gets very crowded. They discovered three different "acts" in the play, depending on how tight the crowd is:

Act I: The "Crystal" Interruption (3D Monodisperse Systems)

• What happened: When they used identical particles (everyone is the same size) in a 3D room, the particles didn't just jam; they suddenly organized themselves into a perfect, rigid crystal structure (like a stack of oranges).
• The Metaphor: It's like a chaotic mosh pit suddenly turning into a perfectly synchronized military march. The "jamming" transition got hijacked by this sudden order. The particles stopped moving not because of the usual jamming rules, but because they locked into a crystal lattice.

Act II: The "Active Glass" (Binary Mixtures)

• What happened: When they used a mix of big and small particles (so they couldn't form a perfect crystal), they found a brand new state of matter.
• The Metaphor: Imagine a crowd where people are stuck in a "cage" of their neighbors. They can't move far, but they are still jiggling and shuffling locally. They aren't fully frozen (solid), but they aren't flowing (liquid) either.
• The Discovery: This is called an "Active-Glass" state. It's a new type of transition that doesn't follow the old "Manna" rules. It's like the crowd is stuck in a traffic jam where everyone is still revving their engines and inching forward, rather than just stopping dead.

Act III: The "Griffiths Effect" (The Foggy Zone)

• What happened: As the crowd got even tighter (approaching the absolute limit of how many people can fit), the transition didn't happen at a single, sharp moment. Instead, it became "smeared out."
• The Metaphor: Think of a light switch. Usually, a switch is either ON or OFF. But in this "Griffiths phase," the switch is broken. As you flip it, the light doesn't just turn on; it flickers, dims, and glows in patches. Some parts of the crowd are frozen, while others are still moving, creating a patchwork of "frozen" and "active" zones.
• Why it matters: This "smearing" is caused by disorder. Because the crowd is so messy and uneven, the jamming happens in some spots before others, making the critical point fuzzy rather than sharp.

3. The Crystal Exception

The authors also looked at perfect crystals (no disorder). Here, the "fuzzy" effect disappeared, and the transition became sharp again. However, even then, it still didn't follow the old Manna rules. The geometry of the crystal itself changed the physics, proving that the old theory was too simple.

4. The New Theory: "Fractional Time"

To explain all these weird behaviors, the authors proposed a new mathematical framework involving "fractional time dynamics."

• The Analogy: In normal physics, time flows like a river: 1 second, 2 seconds, 3 seconds. In this new theory, time is more like a stuttering video. Sometimes the system moves fast, sometimes it gets stuck in a loop, and sometimes it moves in slow motion. This "stuttering" (fractional time) explains why the particles get trapped in cages and why the transition gets smeared out.

Why Should You Care?

This isn't just about sand or marbles.

• Real World: It helps us understand how to pack materials, how traffic jams form, and how granular materials (like grain or sand) behave under stress.
• Future Tech: The authors mention that these same mathematical rules might apply to Artificial Neural Networks (AI). Just like particles getting stuck in a jam, AI learning processes can get "stuck" in local patterns. Understanding how these systems transition from "learning" to "stuck" could help us build smarter, more efficient AI.

The Bottom Line

The old idea that "jamming is just a geometric puzzle" is incomplete. The reality is a complex dance between order and disorder, movement and trapping, and time and space. The transition to a jammed state isn't a single event; it's a rich landscape of new phases, including "active glasses" and "fuzzy" transitions, governed by a new set of rules that the authors are now beginning to decode.

Technical Summary

1. Problem Statement

The jamming transition, where disordered materials (like granular media or foams) transition from a soft, fluid-like state to a rigid, solid-like state, has traditionally been viewed as a geometric packing problem governed by static contact networks (Random Close Packing, RCP).

Recently, a conjecture emerged linking jamming to dynamic phase transitions in athermal particles under periodic shearing. Specifically, it was proposed that the jamming point corresponds to an absorbing-state transition within the Manna universality class (also known as Conserved Directed Percolation, CDP). This conjecture relies on the Biased Random Organization (BRO) model, where particles undergo repulsive displacements upon overlap. While the BRO model successfully generates RCP configurations at the critical point, the validity of mapping the jamming transition directly to the Manna universality class in high-density regimes (dimensions $d \le 4$) remains unverified, particularly given the violation of the Harris criterion ($d\nu_\perp > 2$) which suggests disorder could significantly alter criticality.

2. Methodology

The authors investigated two variants of the BRO model in dimensions $d=2$ to $4$:

• Conserved BRO: The center of mass (CMC) of the system is conserved; overlapping particles undergo pairwise repulsive displacements with equal magnitude.
• Non-conserved BRO: The CMC is not conserved; displacements are directed from the cluster center to the particle.

Simulation Details:

• System: $N$ spherical particles in a hypercubic box with periodic boundary conditions.
• Variables: Packing fraction ($\phi$) and displacement size ($\epsilon$, corresponding to shear amplitude).
• Dynamics: Particles are inactive if isolated; active (moving) if overlapping. Active particles move to resolve overlaps.
• Analysis:
  • Order Parameters: Survival activity ($f_a$), overall activity ($f_b$), and survival probability ($P_s$).
  • Scaling Analysis: Finite-size scaling of critical exponents ($\beta, \nu_\parallel, \nu_\perp, \alpha, z$) to determine universality classes.
  • Structural Analysis: Monitoring crystallization via order parameters and analyzing Mean Square Displacement (MSD) to distinguish diffusive vs. glassy dynamics.
  • Field Theory: Proposed a theoretical framework involving fractional time derivatives to explain observed anomalies.
  • Disorder Mapping: Introduced quenched disorder (spatially varying $\epsilon$) to test for Griffiths effects and mapped the jammed limit to a heterogeneous Directed Percolation (DP) model.

3. Key Contributions and Results

A. Anomalous Criticality at High Density

The study reveals that while the BRO model generates RCP configurations, the criticality of the transition is fundamentally reshaped by the high-density state, deviating from the Manna universality class in several regimes:

1. Crystallization in Monodisperse Systems (3D):

   • In 3D monodisperse systems at intermediate displacement sizes ($0.2\sigma < \epsilon < 0.7\sigma$), the absorbing transition is interrupted by crystallization. The system forms crystal grains before reaching the critical jamming point, preventing the observation of standard critical phenomena.

2. New Universality Class: Absorbing to Active-Glass Transition:

   • In binary mixtures (where crystallization is suppressed), a distinct dynamic phase transition is identified between the absorbing state and an active-glass state.
   • Regime II ($0.3\sigma < \epsilon < 1.0\sigma$): Critical exponents ($\nu_\perp, \alpha, z$) are non-monotonic and do not match the Manna class.
   • Mechanism: As $\epsilon$ decreases, particles become "caged" by neighbors. The transition is no longer a non-local search for a non-overlapping configuration (Manna) but a local rearrangement process within a glassy landscape.
   • Evidence: Mean Square Displacement (MSD) shows sub-diffusion and early-time plateaus characteristic of glass-forming liquids. The decoupling between the active fraction ($f_a^\infty$) and the diffusion coefficient ($D$) confirms the active-glass nature.

3. Griffiths Effects in the Pre-Jamming State (Regime III):

   • As $\epsilon \to 0$ (approaching the RCP limit), the system enters a pre-jamming state.
   • Phenomenon: The sharp dynamic critical point is smeared out into a broad Griffiths phase due to spatial heterogeneity in the contact network.
   • Signatures:
     • Relaxation follows a continuous power law $f_b(t) \sim t^{-d/\kappa}$ over a wide density range, rather than a single critical point.
     • The index $\kappa$ depends on the distance to the critical density.
     • Finite-size reversal: Small systems stay active longer than large systems near the transition, a hallmark of Griffiths phases in disordered systems.
   • Cause: Quenched disorder in the contact network creates "frozen" regions that hinder relaxation, leading to strong dynamic heterogeneity.

4. Mapping to Heterogeneous Directed Percolation (DP):

   • In the limit $\epsilon \to 0$, the particle configuration becomes static (frozen). The dynamic field theory reduces to Directed Percolation with quenched disorder (Heterogeneous DP).
   • Simulations of a contact model on the BRO-generated network confirm that the critical behavior matches Heterogeneous DP, not the Manna class.

5. Crystal Phase Anomalies:

   • Even in defect-free close-packed crystals (where Griffiths effects are absent due to lack of quenched disorder), the dynamic criticality still deviates from the Manna class.
   • Measured exponents differ from Manna values, attributed to the intrinsic anisotropy of the crystal lattice, which prevents the reduction of the field equations to the standard Manna form.

B. Theoretical Framework

The authors propose a field theory with fractional time dynamics to unify these phenomena:

• Standard Manna: Described by a conserved field $\psi$ with normal diffusion ($\theta=1$).
• Active-Glass State: The caging effect induces sub-diffusion in the conserved field, modeled by a fractional time derivative ($0 < \theta < 1$). This alters the coupling between the activity field $\rho$ and the conserved field $\psi$, changing the universality class.
• Pre-Jamming/Griffiths: The frozen topology acts as quenched disorder, leading to the Heterogeneous DP description.

4. Significance

• Refutation of Direct Mapping: The work challenges the direct mapping of the jamming transition to the Manna universality class for disordered systems in $d \le 4$. It demonstrates that geometric rigidity (RCP) and disorder fundamentally alter dynamic criticality.
• New Universality Classes: It identifies a new dynamic universality class (Absorbing $\to$ Active-Glass) and clarifies the role of Griffiths effects in the pre-jamming regime.
• Interplay of Static and Dynamic: The study highlights the complex interplay between static packing constraints (geometry) and dynamic critical processes.
• Broader Implications: The findings offer insights into:
  • Ultra-stable glasses and the Gardner transition.
  • Artificial Neural Networks: The learning processes of models like the spherical negative perceptron, which share similar absorbing state dynamics and jamming-like transitions.
  • Disordered Systems: Provides a framework for understanding dynamic criticality in systems with strong spatial disorder and memory effects.

Conclusion

The paper concludes that while the BRO model can generate RCP configurations, the criticality of the transition is not universal but is heavily dependent on the specific dynamic constraints (caging, crystallization, disorder) present at high densities. The transition evolves from Manna-like behavior at low density to an Active-Glass transition, and finally to a Griffiths-smeared Heterogeneous DP transition as the system approaches the jamming point.