Soft Absorbing States in Nonequilibrium Systems

• Soft absorbing states are defined as configurations where irreversible, constraint-violating events vanish due to the energetic suppression of particle overlaps and plastic rearrangements.
• They exhibit a nonequilibrium phase transition characterized by universal critical scaling laws, with activity decaying according to both reversible hysteretic loops and quantitative order parameters.
• Applications span soft condensed matter, granular materials, and neural manifold packing, linking mechanical rheology to machine learning loss landscapes.

A soft absorbing state is a configuration or phase in a driven, interacting many-body system—typically involving soft repulsive potentials—wherein the system ceases all irreversible, system-wide rearrangements due to the absence or energetic suppression of constraint-violating events (e.g., particle overlaps, or plastic rearrangements). This state arises through a nonequilibrium phase transition when external parameters such as control strain, density, or energy dissipation are tuned below a critical threshold. By contrast to "hard" absorbing states, where constraints are strictly enforced (e.g., hard-sphere exclusion), "soft" absorbing states emerge in systems with smoothly penalizing, finite-range interactions and may still exhibit reversible, hysteretic local dynamics and dissipation. Soft absorbing states govern dynamic criticality, mechanical response, and organization in soft condensed matter, machine learning loss landscapes, and granular or neural manifold systems.

1. Definitions and Model Formulations

A soft absorbing state is formally defined as a configuration in which any further dynamical evolution induces no activity—activity meaning irreversible or constraint-violating events such as particle overlaps or plastic rearrangements. In models employing soft-sphere potentials (e.g., $u(r_{ij}) = (2R - r_{ij})/2$ for $r_{ij}<2R$, $0$ otherwise), an absorbing configuration is one in which all interacting pairs satisfy the non-overlap condition ($r_{ij} \ge 2R$) (Zhang et al., 2024). This applies to both particulate models under shear or vibration and analogous problems in neural state-space packing.

Soft absorbing states generalize hard absorbing states by allowing finite, energetically suppressed overlaps, leading to an absorbing state transition that remains sharp and in the same universality class (typically Manna/conserved directed percolation) as the hard case (Zhang et al., 2024, Ness et al., 2020). In sheared amorphous solids, a soft absorbing state is attained when, under cyclic drive, particle trajectories become exactly periodic, and all net cycle-to-cycle displacements vanish, though closed, nontrivial hysteretic loops may persist (Otsuki et al., 2021).

2. Microscopic Dynamics and Order Parameters

In models such as the Biased Random Organization (BRO) or stroboscopically driven particle systems, the basic update involves identifying "active" elements (e.g., overlapping particles) and applying random, typically repulsive kicks or gradient-driven steps to resolve overlaps, with noise entering only where constraints are violated (Zhang et al., 2024, Wang et al., 8 Oct 2025). The system transitions to an absorbing state when no overlaps or rearrangements remain.

Quantitative characterization uses:

• Fraction of active particles $f_\infty(\phi)$: long-time fraction of constraint-violating (e.g., overlapping) particles (Wang et al., 8 Oct 2025).
• "Alive" fraction $A(t)$: number of particles irreversibly displaced per cycle (Ness et al., 2020).
• Activity field or steady-state fraction of irreversible particles $\Psi$ in rheological experiments (Nagamanasa et al., 2014).
• Onset and scaling of $f_\infty$, relaxation time $\tau_r$, and related observables near the transition points provide critical exponents.

3. Absorbing-State Phase Transitions and Scaling

The transition into a soft absorbing state is a genuine nonequilibrium phase transition, often continuous (second-order), with universal critical scaling. Near the critical point (e.g., critical packing $\phi_c \approx 0.64$ for soft spheres (Zhang et al., 2024)), the key scaling laws are:

• $f_\infty \sim (\phi - \phi_c)^\beta$ above $\phi_c$.
• Relaxation time $\tau_r \sim |\phi - \phi_c|^{-\nu_\parallel}$ on both sides.
• In $d=3$, $\beta \approx 0.84$, $\nu_\parallel \approx 1.08$, $\nu_\perp \approx 0.59$ for soft sphere/BRO/SGD systems (Zhang et al., 2024); consistent with Manna class scaling observed in classical and multicomponent driven suspensions (Ness et al., 2020, Nagamanasa et al., 2014).

These critical exponents and scaling functions describe the divergence of time/length scales, the decay of activity at criticality ($f_a(t) \sim t^{-\alpha}$), and the finite-size scaling behavior. Softness of the repulsive interaction is formally irrelevant to the universality class, though it shifts the position of the critical point and allows for richer micro- and meso-scale behaviors (Ness et al., 2020, Otsuki et al., 2021).

4. Mechanics, Rheology, and Hysteresis in Soft Absorbing States

Even in a soft absorbing state, where there is no long-term irreversible rearrangement, soft matter systems such as jammed particle assemblies exhibit nontrivial mechanical phenomena:

• The storage modulus $G'$ (in-phase elastic response) decreases, i.e., the material "softens" as the amplitude of cyclic drive increases, even before yield (Otsuki et al., 2021).
• The loss modulus $G''$ (out-of-phase, dissipative component) remains finite as frequency $\omega \to 0$, due to reversible but hysteretic rearrangements; $G''$ is directly tied to the area of nontrivial closed loops in stress-strain space (Otsuki et al., 2021).
• Yielding is defined as the critical drive amplitude $\gamma_c$ above which particles no longer return to their original state after each cycle—i.e., the system leaves the absorbing state and enters a regime of plastic deformation and finite, irreducible flow (Otsuki et al., 2021, Nagamanasa et al., 2014).

In oscillatory shear, closed non-affine loops in single-particle trajectories encode dissipation, and their average area directly controls the residual $G''$. Fourier analysis reveals that both storage and loss responses in the absorbing state are quantitatively attributable to properties of these reversible loops (Otsuki et al., 2021).

5. Universality Classes and Field-Theoretic Descriptions

Generic soft absorbing-state transitions fall into the conserved directed percolation (Manna) universality class under broad conditions (isotropy, conservation, local rules) (Zhang et al., 2024, Ness et al., 2020, Nagamanasa et al., 2014). The critical exponents observed in driven granular, colloidal, and soft glass systems ($\beta \approx 0.64$, $\nu_\parallel \approx 1.23$, $\nu_\perp \approx 0.80$ in 2D and 3D) support this mapping.

For cases with long-range mediated (elastic or hydrodynamic) interactions, a departure from Manna-class scaling is observed (Mari et al., 2021). The resulting nonanalytic terms ($\sim -\mu \rho A^{3/2}$) in the coarse-grained activity field equations generate new universality classes with distinct exponents (e.g., $\beta \approx 1.7$ in 2D; suppressed hyperuniformity).

At densities near jamming, particle polydispersity, crystallization, caging, and quenched disorder introduce further anomalies—e.g., a smeared, Griffiths-type transition, active-glass regimes, and fractional-time field-theoretic dynamics (Wang et al., 8 Oct 2025). The proposed coupled Langevin equations with a Caputo fractional derivative in the conserved density evolve from the Manna universality class ($\theta=1$) to subdiffusive or frozen-glass dynamics ($0 < \theta < 1$) as contacts become increasingly quenched.

6. Implications for Machine Learning and Manifold Packing

The analogy between soft absorbing states in driven particle systems and manifold separation in neural embedding spaces—mediated by stochastic gradient descent (SGD)—establishes a physics-informed mapping of learning dynamics to nonequilibrium absorbing-state criticality (Zhang et al., 2024). In this picture, classes correspond to manifolds, and overlaps to classification errors or ambiguities.

SGD dynamics in such energy landscapes show both the sharp absorbing transition (perfect manifold separation below $\phi_c$) and the phenomenon of flat-minima selection (bias toward local minima with low Hessian trace for small batch size). This suggests that the universal features of soft absorbing transitions directly inform high-dimensional learning and generalization (Zhang et al., 2024).

7. Extensions and Anomalies: Multicriticality, Active Glass, and Griffiths Effects

At high density and in complex, heterogeneous systems, soft absorbing-state transitions exhibit anomalies and regime changes:

• Monodisperse sphere packs show preempting crystallization, interrupting the absorbing transition entirely (Wang et al., 8 Oct 2025).
• Binary mixtures suppress crystallization; yet, a decoupling between steady activity and long-time diffusion marks a distinct absorbing-to-active-glass transition, with new critical exponents, outside both Manna and directed percolation universality (Wang et al., 8 Oct 2025).
• Near jamming, rare-region (Griffiths) physics emerges: local contact network heterogeneities smear critical scaling, resulting in power-law activity decay across a pseudo-critical window and finite-size reversal of relaxation times (Wang et al., 8 Oct 2025).
• Fractional-time field-theoretic models are required to synthesize these behaviors, with the order of the fractional derivative encoding the degree of dynamical freezing or caging.

These results emphasize the sensitivity of soft absorbing-state transitions to the interplay between particle-scale dynamics, interaction softness, global constraints, and disorder.

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Key references:

• (Zhang et al., 2024) Absorbing state dynamics of stochastic gradient descent
• (Ness et al., 2020) Absorbing-state transitions in granular materials close to jamming
• (Nagamanasa et al., 2014) Direct Evidence for an Absorbing Phase Transition Governing Yielding of a Soft Glass
• (Otsuki et al., 2021) Softening and residual loss modulus of jammed grains under oscillatory shear in an absorbing state
• (Mari et al., 2021) Absorbing phase transitions in systems with mediated interactions
• (Wang et al., 8 Oct 2025) Anomalous Criticality of Absorbing State Transition toward Jamming
• (Néel et al., 2014) Dynamics of a first order transition to an absorbing state