Extended Coarsening Model Overview

• Extended coarsening model is a generalization of classical coarsening theory that relaxes assumptions like single-length scaling and diffusion-limited transport.
• It employs multi-length scaling to integrate disorder, jamming, substrate growth, and conversion-limited kinetics into coarsening frameworks.
• The approach spans systems from dry foams to molecular simulations, offering actionable insights for analyzing non-equilibrium and complex dynamics.

Searching arXiv for the specified paper and closely related "extended coarsening" usages across domains. “Extended coarsening model” is not a single canonical formalism in the arXiv literature. The expression is used for a family of generalizations of classical coarsening and phase-ordering frameworks in which the standard assumptions of a single growing length, fixed geometry, diffusion-limited transport, or structure-independent observables are relaxed. In the cited works, the extension may take the form of disorder-aware scaling, transport through additional compartments, substrate growth, jamming, occupancy constraints, inertia, or an energy-to-length map that remains meaningful outside a nearly periodic regime. The common theme is that coarsening remains the organizing phenomenon, but the dynamical variables, constraints, and asymptotic laws are broadened beyond the classical Allen–Cahn, Lifshitz–Slyozov–Wagner, or pure Family–Vicsek settings (Corberi et al., 2016, Khakalo et al., 2017, Lee, 2021, Ross et al., 2024, Ernst et al., 11 Sep 2025, Howard et al., 21 Jan 2026).

1. Classical baseline and the sense in which coarsening is “extended”

Several of the cited papers define their extension by contrast with a simpler baseline. In the pure Ising case, the domain size $L(t)$ and the interface roughening length $\xi(t)$ both grow as $t^{1/z}$, so that coarsening and roughening can be described by the same growing scale. In the pure roughening problem, the interface width obeys the Family–Vicsek form

$$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$

while in pure nonconserved coarsening $L(t,0)\propto t^{1/z}$ with $z=2$. In classical Ostwald ripening, the critical radius follows $R_c(t)\sim t^{1/3}$. Dry-foam coarsening likewise provides a reference exponent close to $1/2$ for mean bubble growth (Corberi et al., 2016, Lee, 2021, Khakalo et al., 2017).

Within that baseline, the cited literature extends coarsening in several distinct ways. One class introduces additional lengths or crossover variables, so that a single-scale description is no longer sufficient. Another modifies the transport bottleneck, replacing diffusion-limited exchange by conversion-limited or compartment-limited kinetics. A third alters the geometry itself, as on a uniformly growing surface or near a jamming transition. A fourth extends the notion of “coarseness” from an asymptotic pattern property to a quantity that can be assigned throughout the full evolution of a PDE solution. This suggests that “extended coarsening model” functions less as the name of one theory than as a recurrent strategy for embedding classical coarsening laws inside a richer dynamical setting (Corberi et al., 2016, Ross et al., 2024, Howard et al., 21 Jan 2026).

2. Multi-length scaling, disorder, and geometric crossover

A central explicit use of the term occurs in the disordered Ising work of Corberi, where the coarsening length $L(t,\epsilon)$ is promoted to the fundamental dynamical scale controlling interface roughening. Quenched disorder introduces an additional length $\lambda(\epsilon)$, with $\xi(t)$0, and the pure Family–Vicsek law is replaced by the two-parameter scaling form

$\xi(t)$1

For site dilution, two disorder lengths appear,

$\xi(t)$2

leading to a multi-length generalization. In weak disorder the roughening exponent remains $\xi(t)$3; in strong disorder the cited simulations support $\xi(t)$4 as a preasymptotic value, with $\xi(t)$5 expected asymptotically in some disordered interface problems (Corberi et al., 2016).

The related study of coarsening and percolation in a disordered ferromagnet introduces a different extended scaling structure. There, ordinary domain growth with length $\xi(t)$6 is supplemented by a second length $\xi(t)$7 associated with the approach to critical percolation, with $\xi(t)$8 and $\xi(t)$9. The spin-spin and connectivity observables are written as

$t^{1/z}$0

and, in disordered systems, a further crossover length $t^{1/z}$1 can enter. The main empirical conclusion is that disorder changes the growth law $t^{1/z}$2, but in the time window studied does not alter the percolation morphology once distances are rescaled by $t^{1/z}$3; the paper presents this as a form of superuniversality for connectivity, wrapping, and winding observables (Corberi et al., 2016).

Taken together, these works show one major meaning of extension: classical coarsening is retained, but the morphology and observables are organized by more than one length scale. The extra variables are not perturbative decorations; they determine crossover between weak and strong disorder, between pre-percolative and percolative structure, and between pure and disorder-dominated roughening regimes.

3. Transport-limited extensions: wet foams, conversion bottlenecks, and chromosomal droplets

Another major use of the phrase concerns extensions of the transport law itself. In the wet-foam simulations of Sanyal and Karmakar, the Durian bubble model is extended by adding gas-diffusion-driven bubble growth and shrinkage to the overdamped soft-sphere dynamics. The baseline force law is

$t^{1/z}$4

and coarsening is introduced through the Gardiner-type diffusion law

$t^{1/z}$5

Near jamming, the simulations reach a self-similar scaling state after an induction period, with mean radius growth exponent $t^{1/z}$6, a broad size distribution containing many small rattler bubbles, and a characteristic coarsening time

$t^{1/z}$7

which diverges as jamming is approached. The same work couples coarsening to viscoelastic aging through the competition between $t^{1/z}$8 and the mechanical relaxation time $t^{1/z}$9, obtained from the linear response equation

$$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$0

Here extension means that coarsening is no longer only a growth law for bubble sizes; it reorganizes contact networks, shifts the effective jamming point upward, and changes the relaxation spectrum (Khakalo et al., 2017).

The later 3D mean-field foam theory of Höhler and coauthors extends classical foam coarsening in a complementary way. Their border-blocking growth law,

$$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$1

adds a nontrivial size dependence through the film-coverage factor, rather than merely multiplying Lemlich’s law by a $$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$2-dependent constant. This allows the scaling-state distribution itself to depend on liquid fraction up to the unjamming transition $$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$3. The resulting scaling law still gives $$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$4, but the predicted distribution is finite at $$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$5 for any $$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$6, implying a large population of small bubbles; the paper attributes discrepancies with previous simulations and experiments mainly to the absence of rattlers in the mean-field model (Morgan et al., 1 Aug 2025).

In conversion-limited phase separation, the extension is more radical: dilute-phase diffusion is assumed fast, but the conversion of a constituent molecule between dilute and condensate states is slow. The regime criterion is

$$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$7

and the drop radius evolves as

$$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$8

This is mapped to Hillert grain growth in three dimensions, yielding the late-stage law $$W(t,\ell)=A(T)\,\xi(t)^{\alpha}{\cal V}\!\left(\frac{\ell}{\xi(t)}\right),$$9 instead of the LSW result $L(t,0)\propto t^{1/z}$0, together with the universal distribution

$L(t,0)\propto t^{1/z}$1

The paper presents this as a universal coarsening theory for a transport mechanism in which the interface, rather than the bulk, is rate-limiting (Lee, 2021).

A biologically distinct but structurally analogous extension appears in the coarsening model of chromosomal crossover placement. Here the compartments are the nucleoplasm $L(t,0)\propto t^{1/z}$2, the synaptonemal complex $L(t,0)\propto t^{1/z}$3, and droplets $L(t,0)\propto t^{1/z}$4, with thermodynamically consistent flux

$L(t,0)\propto t^{1/z}$5

On a single synaptonemal complex, for $L(t,0)\propto t^{1/z}$6, the mean droplet volume and droplet count scale as

$L(t,0)\propto t^{1/z}$7

while nucleoplasmic exchange weakens crossover interference and assurance. In the absence of the synaptonemal complex, positional interference disappears and assurance becomes an occupancy problem. The extension here lies in adding material exchange not only along the chromosome axis but also through the nucleoplasm, thereby bridging wild type and the zyp1 mutant within one coarsening framework (Ernst et al., 11 Sep 2025).

4. Geometry, substrate growth, confinement, and jamming

A different branch of extended coarsening theory concerns changes in geometry or kinematics. In the growing voter model on an expanding circular surface, the substrate radius $L(t,0)\propto t^{1/z}$8 itself becomes a control parameter through the displacement field

$L(t,0)\propto t^{1/z}$9

For uniform stretch, $z=2$0, the system is described as critical: interface density decays algebraically,

$z=2$1

the interface fractal dimension is

$z=2$2

and the cluster-size distribution obeys $z=2$3. The paper argues that uniform growth is a critical point separating regimes with different macroscopic morphology, so that growth protocol itself becomes the control variable of coarsening (Ross et al., 2024).

In the DDFT study of aggregating hard particles in a narrow channel, geometry enters through confinement and periodic corrugation. The model distinguishes two coarsening routes: Ostwald ripening, in which cluster centers remain essentially fixed while particles diffuse between them, and cluster translation/coalescence, in which whole clusters move and merge. In a smooth channel, the dominant mode crosses over with attraction strength, cluster size, and separation; in a corrugated channel with nonzero hard-core diameter $z=2$4, sufficiently strong pinning can arrest both translation and Ostwald ripening, with the final stable cluster size selected by the corrugation period $z=2$5. The paper interprets the two modes as symmetry modes associated with translation and mass/volume perturbations of steady clusters (Pototsky et al., 2014).

These cases broaden coarsening by changing where and how domains live. Instead of evolving on a static Euclidean background with unconstrained mobility, the structures coarsen on a growing substrate, in a jammed packing, or under geometrical pinning. The growth law then reflects not only interfacial curvature or mass exchange, but also the kinematics of the space in which interfaces move.

5. Nonequilibrium, active, and fluctuation-dominated realizations

In several nonequilibrium models, extension means adding microscopic ingredients that produce coarsening where a simpler exclusion or copying process would not. In the extended persistent exclusion process, a one-dimensional lattice gas of run-and-tumble particles is generalized by allowing a tunable maximum occupancy $z=2$6. The standard $z=2$7 limit yields a gas of clusters with exponential size distribution and no coarsening. For $z=2$8 and small tumble rate $z=2$9, a coarsening phase appears in which a few large clusters contain a macroscopic fraction of the particles. The model with real-valued $R_c(t)\sim t^{1/3}$0 and probabilistic hopping,

$R_c(t)\sim t^{1/3}$1

shows that the gas-of-clusters to coarsening transition is continuous, whereas the coarsening-to-gas transition can be continuous or discontinuous, with a critical point near

$R_c(t)\sim t^{1/3}$2

Here the extension is purely excluded-volume based: no attraction or alignment is added, but finite capacity is enough to generate true phase separation and coarsening (Sepulveda et al., 2016).

In condensation models such as the zero range process and conserved mass aggregation model, the cited work shows that the coarsening state is not a scaled-down steady state. A growing coarsening length $R_c(t)\sim t^{1/3}$3 is observed, with $R_c(t)\sim t^{1/3}$4 in one-dimensional asymmetric ZRP and CMAM, and local condensates have both mean and standard deviation proportional to $R_c(t)\sim t^{1/3}$5. The local-maximum distribution is bimodal: a condensate contribution scales as $R_c(t)\sim t^{1/3}$6, while the fluid contribution follows a Fréchet form. The paper’s main conceptual point is that coarsening is governed by a pre-asymptotic fluctuating regime with anomalously large fluctuations, not by the final condensate statistics (Iyer et al., 2022).

The long-range Persistent Voter Model extends voter kinetics by introducing a binary confidence variable $R_c(t)\sim t^{1/3}$7, with zealot sites retaining their opinion and normal voters copying neighbors chosen with

$R_c(t)\sim t^{1/3}$8

The resulting kinetics fall in the universality class of the long-range Ising model quenched to a small nonzero temperature. In one dimension, the asymptotic growth exponent is $R_c(t)\sim t^{1/3}$9 for $1/2$0 and $1/2$1 for $1/2$2; for $1/2$3, the correlation function has complementary-error-function scaling,

$1/2$4

The paper interprets zealotry as suppressing interfacial noise and reinstating curvature-driven Ising-like coarsening (Arenzon et al., 15 Mar 2026).

The granular review places these examples in a wider athermal context. In granular gases, the Homogeneous Cooling State obeys Haff’s law $1/2$5, while growing structures such as vortices and clusters have $1/2$6. In horizontally vibrated mixtures, stripe width grows as $1/2$7; in electrostatically driven monolayers, the number of solid clusters decays as $1/2$8 and average cluster area grows as $1/2$9 in quasi-2D; in compaction models, domain width may cross over from $L(t,\epsilon)$0 to $L(t,\epsilon)$1. The review’s overarching claim is that phase-ordering language can be extended to driven, dissipative, nonthermal media despite the absence of equilibrium thermodynamics (Baldassarri et al., 2015).

6. Coarseness as an observable, and computational meanings of coarsening

Some recent work extends coarsening not by modifying the physical kinetics but by redefining what it means to measure coarseness. For the one-dimensional Cahn–Hilliard equation,

$L(t,\epsilon)$2

Jang and collaborators introduce a coarseness measure that maps the solution energy $L(t,\epsilon)$3 to the period $L(t,\epsilon)$4 of a periodic stationary solution through a pseudoinverse of the energy–period relation $L(t,\epsilon)$5. Because $L(t,\epsilon)$6 is not globally invertible, the coarseness is defined by

$L(t,\epsilon)$7

which assigns a length scale to any admissible state without requiring near-periodicity. The authors emphasize three advantages: a length scale can be assigned to any $L(t,\epsilon)$8 with admissible energy; the measure is directly linked to stationary solutions of the equation; and it is readily computed by the pseudoinverse. They use it to compare direct numerics with Langer’s asymptotic model and Howard’s eigenvalue-based model, describing this as the first direct check they are aware of for the efficacy of these methods (Howard et al., 21 Jan 2026).

In incompressible multiphase flow, coarsening is treated as an interface-aware filtering operation. The phase index $L(t,\epsilon)$9 is smoothed by pseudo-time diffusion,

$\lambda(\epsilon)$0

while the interface contour is explicitly tracked and moved by

$\lambda(\epsilon)$1

A second diffusion equation with modified coefficient $\lambda(\epsilon)$2, set to zero at the interface, embeds small-scale structures left behind by interface motion into the opposite phase without diffusion across the interface, and a pressure-like projection enforces incompressibility of the coarse velocity field. The result is an interface-retaining coarse representation rather than a fully smeared filter (Chen et al., 2021).

A broader terminological extension appears in computational mathematics and molecular simulation. In geometric multigrid for Poisson’s equation, Liu and Henshaw study nonstandard coarsening factors $\lambda(\epsilon)$3, low-order coarse-grid approximations for fourth- and sixth-order fine discretizations, and red-black coarsening with a rotated-grid interpretation in 2D. Their conclusions are that second-order coarse approximations can be very effective and that coarsening factors near $\lambda(\epsilon)$4 are the most efficient overall (Liu et al., 2020). In machine-learning coarse-grained molecular dynamics, graph coarsening is used to construct CG mappings by edge contraction under a local spectral variation cost,

$\lambda(\epsilon)$5

with coarse Laplacian

$\lambda(\epsilon)$6

followed by MACE-based force matching on the coarse graph (Mondal et al., 22 Jul 2025).

These computational works use “coarsening” in a different but related sense: not late-stage domain growth, but principled reduction of degrees of freedom. A plausible implication is that the modern literature now treats coarsening both as a physical kinetic phenomenon and as a methodology for constructing reduced descriptions that preserve the structures most relevant to the large-scale dynamics.

7. Conceptual synthesis and recurring themes

Across these domains, several motifs recur. First, the single growing length of classical coarsening is frequently replaced by two- or multi-length scaling: $\lambda(\epsilon)$7 and $\lambda(\epsilon)$8 in disordered roughening, $\lambda(\epsilon)$9 and $\xi(t)$00 in percolative ferromagnets, or $\xi(t)$01 and $\xi(t)$02 in wet foams (Corberi et al., 2016, Corberi et al., 2016, Khakalo et al., 2017). Second, transport pathways matter: exchange through films, through the nucleoplasm, along a substrate, or across a conversion barrier changes both exponents and scaling functions (Lee, 2021, Morgan et al., 1 Aug 2025, Ernst et al., 11 Sep 2025). Third, geometry can itself induce criticality or arrest: uniformly growing surfaces produce scale-free domains, while corrugated confinement or near-jamming contact networks qualitatively alter the coarsening trajectory (Ross et al., 2024, Pototsky et al., 2014, Khakalo et al., 2017).

A further common feature is that extensions often expose quantities that are not visible in the simplest asymptotic picture. Examples include rattlers in wet foams, pre-asymptotic fluctuating condensates in aggregation–fragmentation systems, a coarseness measure valid in the early irregular Cahn–Hilliard regime, and a sharp large-scale interface retained during multiphase filtering (Khakalo et al., 2017, Iyer et al., 2022, Howard et al., 21 Jan 2026, Chen et al., 2021). The repeated pattern is not the abandonment of coarsening theory, but its enrichment: classical domain growth remains the backbone, while the extended model specifies which additional fields, scales, or constraints must be retained for the problem at hand.

In that sense, the phrase “extended coarsening model” names a research program rather than a unique equation set. It denotes the systematic broadening of coarsening theory so that it can accommodate disorder, jamming, growth geometry, active persistence, athermal driving, compartmental transport, and reduced-order representation, while still retaining scaling, universality, and late-stage selection as its central organizing ideas.