Frozen States in Physics and Computing

• Frozen state is a condition where key degrees of freedom cease to evolve on observable timescales, occurring in cases of dynamic arrest and fixed modeling protocols.
• This concept spans multiple disciplines, appearing in statistical mechanics, active matter, machine learning, and ultracompact gravitational models.
• It is characterized by threshold parameters and diagnostic markers that highlight the suppression of selected internal motions without implying complete immobility.

“Frozen state” is a polysemous technical term used across contemporary research to denote a condition in which relevant degrees of freedom cease to evolve on the observation timescale, are held fixed by construction, remain trapped behind kinetic or energetic barriers, or become effectively inaccessible to an external observer. In current usage this includes fixed recurrent representations in machine learning, frozen operating points in LPV systems, absorbing and stripe states in coarsening and active matter, frozen superparaelectric polar domains, finite-size cascade arrest in wave turbulence, and quasi-horizon states in ultracompact gravitating objects (Wagh et al., 30 Apr 2026, Alkhoury et al., 2017, Blanchard et al., 2013, Reichhardt et al., 2014, 1803.02126, Zhao et al., 19 Feb 2025).

1. Terminological scope

A useful editorial grouping is to distinguish between three broad meanings. First, “frozen” may mean dynamically arrested: the system still exists far from equilibrium, but rearrangements or fluctuations are lost, as in absorbing active states, frozen stripe states after Ising coarsening, or superparaelectric polar domains below a freezing temperature (Reichhardt et al., 2014, Blanchard et al., 2013, 1803.02126). Second, it may mean fixed by modeling protocol: a backbone is not fine-tuned, a scheduling signal is held constant, a vertex is pinned, or virtual spinors are truncated and removed from the active space (Wagh et al., 30 Apr 2026, Alkhoury et al., 2017, Damron et al., 2015, Mukhopadhyay et al., 11 May 2025). Third, it may denote a critical redshifted limit: metric functions approach zero near a surface or quasi-horizon, so that the object becomes nearly indistinguishable from a black hole to an external observer (Brustein et al., 2023, Zhao et al., 19 Feb 2025, Tan et al., 29 Dec 2025, Tan et al., 11 Sep 2025).

Domain	Frozen entity	Operational meaning
Statistical and active matter	Fluctuations, interfaces, domain motion	Absorbing or metastable arrest
Control and computation	State, basis, backbone, pinned site	Fixed by construction or protocol
Gravitation	Interior dynamics near a critical horizon	Extreme redshift or quasi-horizon
Materials and ferroics	Local polar regions	Nonergodic, barrier-limited response

This breadth explains why the term resists a single definition. In one literature it marks the disappearance of local updates; in another it marks a constant-parameter reduction; in another it marks a limiting geometry rather than literal immobility.

2. Dynamical arrest in classical many-body systems

In nonequilibrium statistical mechanics, a frozen state commonly denotes an absorbing or infinitely long-lived configuration. In the two-dimensional ferromagnetic Ising model quenched from equilibrium at $T_c$ to $T=0$, the dynamics proceeds by local energy descent with $\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$, and the system can become blocked in stripe states whose straight interfaces have no flippable spins; these stripe states are infinitely long-lived under the single-spin-flip dynamics considered (Blanchard et al., 2013). The final-state topology is tied to the initial crossing probabilities of critical Ising spin clusters, so the frozen state is selected by large-scale spanning geometry rather than by local disorder alone.

Active-matter studies use “frozen” in a different but related sense. In a two-dimensional system of sterically repulsive disks with fixed run directions, a transition occurs from a fluctuating state of transient clusters to a dynamically frozen or quiescent state above a critical packing fraction $\phi_c = 0.462$, with transient time $\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$ and $\nu_\parallel = 1.21 \pm 0.06$ (Reichhardt et al., 2014). The frozen state does not imply immobility: a dense crystalline cluster drifts as a rigid body, particles outside it move on closed periodic orbits, yet velocity fluctuations vanish and the sixfold coordination fraction reaches $P_6 \approx 0.9$. In the same system, pinned obstacles shift freezing to lower densities, with $\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$ and $\beta = 1.9 \pm 0.4$.

A related absorbing-state mechanism appears in high-density motility assays of actin, myosin II, and fascin. There, active transport couples to irreversible growth into rings or coherently moving fibres; mechanical stiffening progressively suppresses curvature fluctuations until the assembly becomes frozen in (Schaller et al., 2011). The critical fascin concentration is $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$, and the order parameter for the ring state is $T=0$0. Closed and open rings coexist, with a robust double-exponential radius distribution.

Surface premelting and freezing of ice introduces a metastability-based sense of freezing. On basal ice in the ML-mW model, the premelting onset is $T=0$1, while the frozen state corresponds to a surface whose first two bilayers remain largely crystalline (Cui et al., 2022). The solid surface remains metastable up to $T=0$2 and the premelted surface down to $T=0$3 in $T=0$4 tests. Holes in the top bilayer promote premelting by biasing nucleation in the second bilayer, but the same inhomogeneity hinders refreezing because shrinking holes cap solid nuclei; complete freezing can also stall at stacking-order mismatches between Ih- and Ic-like domains.

Granular media provide a gravitationally compacted variant. In a two-dimensional vibrating bed of hard disks, a frozen portion forms below a critical temperature $T=0$5, and the number of frozen layers is $T=0$6 (Koser et al., 2010). Here the frozen state is a condensed bottom region created when the local packing reaches the square-packing onset and later approaches hexagonal packing.

Multiferroic manganites add a nonergodic dielectric example. In GdMn$T=0$7O$T=0$8 and Gd$T=0$9Ce$\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$0Mn$\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$1O$\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$2, local polar phase-separation domains enter a frozen superparaelectric state below a direction-dependent freezing temperature $\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$3, while the long-range exchange-striction ferroelectric transition remains at $\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$4 (1803.02126). In GMO, remanent polarization along $\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$5 persists up to $\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$6 and vanishes near $\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$7; along $\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$8, it diminishes near $\Delta E = 2J s_i \sum_{j\in \mathrm{nn}(i)} s_j$9.

3. Frozen transport, cascades, and topological transitions

Some frozen states are defined not by local arrest alone but by the failure of a transport process to span the available state space. For surface gravity waves in a finite discrete wavenumber domain $\phi_c = 0.462$0, energy transfer is restricted to exact four-wave resonances satisfying

$\phi_c = 0.462$1

If the initially excited region is too small, the cascade stalls before reaching the boundaries of $\phi_c = 0.462$2, producing a frozen turbulence state (Zhang et al., 2022). For direct cascade at $\phi_c = 0.462$3, $\phi_c = 0.462$4 gives $\phi_c = 0.462$5 and freezing, while $\phi_c = 0.462$6 yields $\phi_c = 0.462$7 and an unlimited cascade. The transition is sharp, with $\phi_c = 0.462$8 below the threshold and favorable scale-resonance connectivity emerging for $\phi_c = 0.462$9.

Directional solidification of complex emulsions yields a topological freezing phenomenon. A water-in-oil-in-water droplet can convert to a frozen oil-in-water single-emulsion configuration when the encapsulating oil shell freezes from the outside inward, develops tension, and expels the inner water drop (Meijer et al., 2023). The transition is suppressed below a critical inner radius $\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$0. The observed ring-like front expands as $\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$1, and the expelled pocket later migrates through ice at $\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$2.

Integrable quantum transport presents a different case. High-energy electrons injected into a chiral quantum Hall edge with finite-range interactions relax into a near-frozen non-equilibrium state: after a short transient, the energy distribution retains a peak near the injection energy and evolves only very slowly with distance (Fischer et al., 2021). In the continuum model, the peak position and height drift as

$\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$3

which indicates extremely slow relaxation rather than thermalization.

4. Quasi-horizon and ultracompact frozen configurations

In gravitational physics, “frozen state” typically refers to a configuration in which metric functions approach zero near a critical surface, causing extreme redshift. The frozen star model uses an anisotropic fluid with

$\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$4

in the idealized limit, and a defrosted deformation parameter $\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$5 leads to internal fluid modes with

$\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$6

(Brustein et al., 2023). In the strict frozen limit $\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$7, all $\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$8 and the object becomes ultra-stable.

Bardeen boson stars exhibit a frozen-state limit when the magnetic charge exceeds a critical value $\tau(\phi) \sim A (\phi - \phi_c)^{-\nu_\parallel}$9 and the scalar frequency tends to zero, $\nu_\parallel = 1.21 \pm 0.06$0 (Zhao et al., 19 Feb 2025). The geometry develops a critical horizon $\nu_\parallel = 1.21 \pm 0.06$1 defined by a near-zero minimum of $\nu_\parallel = 1.21 \pm 0.06$2, and this radius lies between the inner and outer magnetic Reissner–Nordström horizons:

$\nu_\parallel = 1.21 \pm 0.06$3

The ADM mass of frozen Bardeen boson stars is independent of $\nu_\parallel = 1.21 \pm 0.06$4, and all frozen solutions possess light rings whose outer radius is independent of $\nu_\parallel = 1.21 \pm 0.06$5 and smaller than the magnetic RN value.

Hayward-boson stars show an analogous limit. In the Einstein–Hayward–scalar model, frozen states arise for sufficiently large magnetic charge, with critical values $\nu_\parallel = 1.21 \pm 0.06$6, $\nu_\parallel = 1.21 \pm 0.06$7, and $\nu_\parallel = 1.21 \pm 0.06$8 for the ground, first excited, and second excited states, respectively (Yue et al., 2023). At $\nu_\parallel = 1.21 \pm 0.06$9 and $P_6 \approx 0.9$0, the ground and excited frozen states have the same mass $P_6 \approx 0.9$1 and the same critical horizon radius, which indicates that the quasi-horizon geometry is fixed by the NED sector rather than by the scalar node number.

Ordinary-matter compact stars also admit frozen endpoints. In four-dimensional non-polynomial gravities, solving the modified Tolman–Oppenheimer–Volkoff equations for BSk19, SLy4, and AP4 shows that sufficiently large $P_6 \approx 0.9$2 drives neutron stars toward a universal frozen endpoint, where $P_6 \approx 0.9$3 at a critical horizon just inside the surface and $P_6 \approx 0.9$4 in the interior (Tan et al., 29 Dec 2025). The minimum $P_6 \approx 0.9$5 needed to reach such states within the causal density window is $P_6 \approx 0.9$6 for BSk19, $P_6 \approx 0.9$7 for SLy4, and $P_6 \approx 0.9$8 for AP4 in the rational model. The corresponding observationally allowed ranges still accommodate frozen neutron stars.

An Einstein–nonlinear electrodynamics realization produces a similar result. With Bardeen or Hayward magnetic monopoles, neutron stars reach a frozen state at a critical magnetic charge $P_6 \approx 0.9$9, where $\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$0 at the stellar surface and $\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$1 throughout the interior (Tan et al., 11 Sep 2025). Representative Bardeen values at $\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$2 are $\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$3 for BSk19, $\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$4 for SLy4, and $\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$5 for AP4. In these solutions the ADM mass becomes NED-dominated near freezing.

5. Frozen variables, representations, and observables

A separate family of meanings concerns quantities held fixed by protocol. In sequence modeling with Mamba, “frozen state” refers to using the pretrained recurrent state and token-level outputs as-is, with the backbone kept entirely fixed and only lightweight probes trained on top (Wagh et al., 30 Apr 2026). The selective state-space recurrence is

$\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$6

Under strict frozen-feature probing, raw final states show severe anisotropy, with mean pairwise cosine $\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$7 and standard deviation $\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$8, and the final-state readout collapses on CoLA with $\tau \sim B |\phi^p - \phi_c^p|^{-\beta}$9. Mean pooling over $\beta = 1.9 \pm 0.4$0 is generally more reliable than patch-boundary readout.

In LPV theory, the frozen model is the LTI realization obtained by holding the scheduling signal constant, $\beta = 1.9 \pm 0.4$1 (Alkhoury et al., 2017). Two LPV models can be frozen-equivalent, meaning that every frozen operating point has the same transfer function, yet still differ for non-constant scheduling. The output mismatch is bounded by

$\beta = 1.9 \pm 0.4$2

with $\beta = 1.9 \pm 0.4$3, which makes explicit that slow scheduling variation and coherent bases suppress global discrepancies.

Quantum-information theory uses “frozen” for an observable that remains constant over a finite interval. For Bell-diagonal states under local dephasing, the discord can remain constant until a switching time $\beta = 1.9 \pm 0.4$4 determined by

$\beta = 1.9 \pm 0.4$5

with $\beta = 1.9 \pm 0.4$6 (Yi, 2020). In the Ornstein–Uhlenbeck example discussed there, the frozen interval can be strongly extended by detuning, although it cannot be preserved indefinitely in that classical-noise model.

In electronic-structure theory, “freezing” denotes basis truncation. State-specific frozen natural spinors are obtained by diagonalizing a state-specific virtual–virtual density,

$\beta = 1.9 \pm 0.4$7

and discarding spinors with occupations $\beta = 1.9 \pm 0.4$8 (Mukhopadhyay et al., 11 May 2025). This substantially reduces $\beta = 1.9 \pm 0.4$9 in relativistic EE-EOM-CCSD; for Zn, the method converges at about $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$0 of the virtual space, and with the ADC(2)-based perturbative correction around $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$1, with errors at $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$2 of at most $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$3 across the four reported states.

A mathematically distinct use appears in stochastic coarsening with a pinned degree of freedom. In the nearest-neighbor zero-temperature coarsening model on $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$4, freezing the origin to $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$5 for all time does not induce fixation elsewhere: every other site still flips infinitely often almost surely (Damron et al., 2015). Here “frozen” refers purely to external pinning, not to emergent arrest.

6. Diagnostics, thresholds, and recurrent motifs

Across these literatures, frozen states are identified by sharply different observables but remarkably similar logical structure. One recurrent motif is a threshold parameter. Examples include the direct-cascade radius window $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$6 in wave turbulence, the active-matter threshold $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$7, the fascin threshold $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$8, the inner-drop stability threshold $c_c = 0.075 \pm 0.025\,\mu\mathrm{M}$9, direction-dependent freezing temperatures $T=0$00 in multiferroics, critical magnetic charges $T=0$01 in boson stars and NED neutron stars, and minimum gravitational modification scales $T=0$02 in non-polynomial gravities (Zhang et al., 2022, Reichhardt et al., 2014, Schaller et al., 2011, Meijer et al., 2023, 1803.02126, Zhao et al., 19 Feb 2025, Tan et al., 29 Dec 2025).

A second motif is the diagnostic of lost response. In active matter, velocity time series flatten and low-frequency noise collapses; in Ising stripes, there are no $T=0$03 moves; in premelting, the first two bilayers recover crystalline order; in frozen superparaelectricity, remanent $T=0$04 loops persist because domain reorientation is blocked; in Mamba probing, representational collapse appears as near-constant vectors and uniform predictions (Reichhardt et al., 2014, Blanchard et al., 2013, Cui et al., 2022, 1803.02126, Wagh et al., 30 Apr 2026).

A third motif is that “frozen” rarely means “completely static.” Dynamically frozen active clusters still drift, actin rings still rotate at about $T=0$05, ordinary superparaelectric domains can unfreeze above $T=0$06, and compact stars in frozen states remain regular configurations rather than literal event horizons (Reichhardt et al., 2014, Schaller et al., 2011, 1803.02126, Tan et al., 11 Sep 2025). This suggests that the term typically marks the suppression of a chosen set of internal degrees of freedom, not the absence of all motion.

A common misconception is therefore to equate frozen states with thermodynamic equilibrium. The cited work does not support that equation. Many frozen states are explicitly absorbing, metastable, nonergodic, or protocol-defined. Another misconception is to treat all freezing as universal arrest; the opposite is often true. In LPV theory and computational chemistry, “frozen” is a modeling choice. In machine learning, frozen backbones can expose geometric pathologies rather than robust representations. In gravitation, frozen states can be the endpoint of a parameter sequence and yet remain horizonless. The term is unified less by ontology than by operational effect: some relevant transformation, update, transport channel, or observational access effectively stops.