Fragmentation Effect in Complex Systems

• Fragmentation effect is a cross‐disciplinary phenomenon where the process or criterion of breaking a system into fragments decisively alters its macroscopic behavior.
• In astrophysical contexts, it governs star formation and disk stability by influencing cooling thresholds and local gravitational instabilities.
• In network, nuclear, and social systems, the effect emerges from clustering criteria or connectivity rules, shaping observable outcomes and evolutionary pathways.

Searching arXiv for the papers on arXiv to ground the article in current bibliographic records. arxiv_search({"query":"id:(Chiaki et al., 2013) OR id:(Gupta et al., 2011) OR id:(Pham et al., 2020) OR id:(Loshaj et al., 2011) OR id:(Standish, 11 Apr 2026) OR id:(Feng et al., 2021) OR id:(Mercer et al., 2016) OR id:(Matevosyan et al., 2013) OR id:(Böhme et al., 2012)","max_results":10,"sort_by":"submittedDate","sort_order":"descending"}) I'll also search for a few additional IDs from the data block that cover other disciplinary usages of the term. arxiv_search({"query":"id:(Orgeron et al., 6 Nov 2025) OR id:(Takahashi et al., 2016) OR id:(Esteves et al., 2021) OR id:(Clement et al., 2018) OR id:(Lotker et al., 2017) OR id:(Lukienko et al., 2020) OR id:(Ledentsov, 2021) OR id:(Khrennikov et al., 2020) OR id:(He et al., 2024)","max_results":10,"sort_by":"submittedDate","sort_order":"descending"}) “Fragmentation effect” is a cross-disciplinary label for situations in which fragmentation itself—or the rule by which fragments are defined—changes the macroscopic behavior inferred from a system. In the literature surveyed here, the term denotes physically distinct phenomena: dust-growth-enabled breakup in metal-poor star-forming clouds, radiative or gravitational control of disk fragmentation, apparent multifragment yields that depend on cluster-recognition criteria, balance-driven splitting of social networks, biodiversity changes induced by habitat fragmentation, strong dynamical disconnection of Hilbert space sectors, and cumulative damage probabilities from fragment clouds [(Chiaki et al., 2013); (Takahashi et al., 2016); (Gupta et al., 2011); (Pham et al., 2020); (Standish, 11 Apr 2026); (Feng et al., 2021); (Mazonka, 2013)]. The common thread is not a single mechanism, but the fact that the emergence, suppression, counting, or downstream consequences of fragments becomes the decisive variable.

1. Cross-disciplinary scope

The surveyed literature uses the term in several technically distinct senses [(Chiaki et al., 2013); (Gupta et al., 2011); (Pham et al., 2020); (Standish, 11 Apr 2026); (Feng et al., 2021); (Mazonka, 2013)].

Domain	What fragments	Operative effect
Low-metallicity star formation	Collapsing cloud into low-mass clumps	Dust growth enhances cooling and enables fragmentation
Heavy-ion transport	Nucleon distribution into clusters	Fragment yields depend on MST cutoff $R_{\text{clus}}$
Social dynamics	Network into disconnected communities	Rewiring or balance creates cohesive-to-fragmented transitions
Ecology	Landscape into isolated patches	Reduced migration alters diversity and coexistence
Fracton circuits	Symmetry sector into disconnected subsectors	Nonthermal dynamics and effective attraction emerge
Blast analysis	Casing into damaging fragments	Cumulative effect is the probability at least one fragment causes damage

A recurrent misconception is that “fragmentation effect” always denotes a physical breakup instability. The literature does not support that simplification. In some cases fragmentation is a material or dynamical process; in others it is a measurement, reconstruction, or post-processing dependence. The heavy-ion MST studies are explicit that a significant part of the observed fragmentation effect depends on the geometric criterion used to define a fragment, not only on the underlying transport evolution (Gupta et al., 2011). Conversely, in star formation and protoplanetary disks, fragmentation refers to an actual physical transition in a self-gravitating medium [(Chiaki et al., 2013); (Takahashi et al., 2016)].

2. Astrophysical gas fragmentation and disk breakup

In extremely metal-poor star-forming gas, the fragmentation effect is tied to dust thermal emission. Grain growth by accretion of gas-phase refractory elements onto pre-existing grains raises the dust mass density before the cloud becomes too optically thick, enhancing cooling in the density window where low-mass fragmentation is decided. In the fiducial MgSiO$_3$ case, efficient growth turns on at $n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$ for $Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$, and the extracted threshold is $Z_{\rm crit}\simeq10^{-5.5}Z_\odot$ for $r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$ and $Z_{\rm crit}\sim10^{-4.5}Z_\odot$ for $r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$ (Chiaki et al., 2013). The cooling channel is encoded through

$$\tau_{\rm cont}=(\kappa_{\rm g}\rho_{\rm g}+\kappa_{\rm d}\rho_{\rm d})l_{\rm BE}, \qquad \beta_{\rm cont}=\min\{1,\tau_{\rm cont}^{-2}\},$$

so grain growth matters mainly by increasing $\rho_{\rm d}$ in advance of the decisive high-density phase (Chiaki et al., 2013). The inference of fragmentation in that work is thermodynamic rather than a direct multidimensional instability calculation.

In self-gravitating protoplanetary disks, the surveyed literature rejects a universal cooling-time-only criterion. One revised formulation divides the process into two stages—spiral-arm formation and spiral-arm breakup—and proposes the local condition

$_3$0

as the relevant fragmentation threshold inside the spiral arm (Takahashi et al., 2016). This is explicitly contrasted with the standard $_3$1-cooling criterion: some rapidly cooling disks do not fragment, while sufficiently massive adiabatic disks can. The consequence is a shift from a disk-averaged to a local arm-stability criterion.

Protostellar disk fragmentation is also regulated by feedback from already formed secondaries. In 3D radiative hydrodynamic simulations, continuous radiative feedback from secondaries stabilizes the disk and suppresses further fragmentation, whereas episodic feedback stabilizes the disk only during outbursts and permits renewed fragmentation between them. The compared runs formed 7 secondary objects with no radiative feedback, 1 with continuous radiative feedback, and 3–4 with episodic radiative feedback (Mercer et al., 2016). This suggests that “fragmentation effect” can denote either the onset of breakup or the self-regulation that limits additional fragments once the first clump appears.

3. Collisional, atmospheric, and molecular fragmentation

In planetary-formation $_3$2-body work, the dynamical importance of collisional fragmentation is model-dependent rather than universal. Within the “Breaking the Chains” migration model for close-in super-Earths, imperfect accretion and fragmentation are common in impact classification, but only $_3$3 of the system mass is fragmented during a typical late instability phase, and most of that mass is reaccreted. The result is that final spacing, multiplicity, eccentricity, inclination, and mass distributions remain qualitatively similar to perfect-accretion runs; from a dynamical point of view, perfect accretion is therefore a suitable approximation in that regime (Esteves et al., 2021). By contrast, in the Early Instability scenario for terrestrial planet formation, including fragmentation keeps small bodies alive longer, increases dynamical friction, damps orbits, lengthens Earth’s accretion, and often truncates Mars’ growth. In that case, over 25% of instability systems reach AMD below the solar-system value, whereas fewer than 10% did in the non-fragmentation study, and the authors regard fragmentation as “supremely important” for successful inner-solar-system evolution (Clement et al., 2018).

For atmospheric entry of large iron bodies, fragmentation can be induced by surface shape rather than by bulk stagnation pressure alone. Deep frontal wedge- or cone-like recesses convert hypersonic aerodynamic loading into transverse forces. For a wedge, the paper gives

$_3$4

with the maximum at $_3$5 (Khrennikov et al., 2020). FEM-CFD calculations then track the first principal stress and compare it to a conservative tensile threshold of $_3$6 MPa for iron. At $_3$7, fragmentation is avoided if the notch-bottom curvature radius exceeds approximately 3 m for $_3$8 m, 7 m for $_3$9 m, and 12 m for $n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$0 m (Khrennikov et al., 2020). Here the fragmentation effect is a stress-concentration effect under asymmetric aerodynamic loading.

At molecular scale, nearby nuclear reaction products can perturb Coulomb explosion. In the model study of $n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$1 in ammonia monohydrate, the chosen water channel

$n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$2

has a baseline KER of about $n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$3 eV, and the nuclear reaction broadens that distribution. For the $n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$4 MeV case, more than 95% of KER values lie in $n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$5–$n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$6 eV; for a reduced-velocity test with $n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$7, about 6% of events differ from the baseline by more than 1 eV (He et al., 2024). In that usage, the fragmentation effect is not the fact of molecular breakup itself, but the modification of fragment energy release by nearby charged nuclear products.

4. Nuclear, hadronic, and blast fragmentation observables

In intermediate-energy heavy-ion collisions, the fragmentation effect may reflect the clusterization prescription as much as the transport dynamics. In IQMD with minimum spanning tree (MST) reconstruction, nucleons are grouped if their coordinate-space separation satisfies a cutoff $n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$8, taken between 2 and 6 fm. The reported dependence is non-monotonic: smaller $n_{\rm H}=10^{10},10^{12},10^{14}\ {\rm cm^{-3}}$9 leaves more free nucleons and reduces light-mass-fragment production, larger $Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$0 merges substructures into heavy fragments, and the LMF multiplicity reaches a maximum at $Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$1 fm (Gupta et al., 2011). The same value gives the best agreement with ALADIN IMF data for central Au+Au events at 600 MeV/nucleon, while MST still fails in semi-peripheral and peripheral collisions (Gupta et al., 2011). The paper’s central caution is that apparent multifragmentation observables are not uniquely determined by the dynamics alone.

In heavy-ion jet quenching, the term appears in a different sense: “jet fragmentation scaling.” A quasi-Abelian Schwinger-model treatment of in-medium hadronization argues that the observed near equality

$Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$2

for moderate and large $Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$3 is compatible with strong quenching if the jet mean free path is short, $Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$4 fm (Loshaj et al., 2011). The mechanism is LPM-like interference: repeated color rotations in a strongly coupled medium suppress extra hard fragments and drive medium-induced hadrons to soft momenta $Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$5 GeV. The fragmentation effect here is therefore a coherence effect in hadronization, not a breakup of a medium into pieces.

Dihadron fragmentation functions provide yet another usage. For $Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$6, vector meson decays alter the DFF far more strongly than a single-hadron fragmentation function would suggest. In both the NJL-jet model and PYTHIA 8.1, the full-final-state DFF including strong decays of $Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$7 and $Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$8 is typically an order of magnitude larger than the direct-pion-only result, with a $Z=10^{-4},10^{-5},10^{-6}\,Z_\odot$9 structure near $Z_{\rm crit}\simeq10^{-5.5}Z_\odot$0 and a low-$Z_{\rm crit}\simeq10^{-5.5}Z_\odot$1 enhancement from $Z_{\rm crit}\simeq10^{-5.5}Z_\odot$2 (Matevosyan et al., 2013). In this context, the fragmentation effect is the sensitivity of hadron-pair observables to resonance feed-down and combinatorial multiplicity.

Blast analysis formalizes cumulative fragment effects probabilistically. If $Z_{\rm crit}\simeq10^{-5.5}Z_\odot$3 is the effect probability for a fragment of mass $Z_{\rm crit}\simeq10^{-5.5}Z_\odot$4 conditional on hit, $Z_{\rm crit}\simeq10^{-5.5}Z_\odot$5 is hit probability, $Z_{\rm crit}\simeq10^{-5.5}Z_\odot$6 is normalized mass density, and $Z_{\rm crit}\simeq10^{-5.5}Z_\odot$7 is total fragment number, then the cumulative probability that at least one fragment causes the effect is

$Z_{\rm crit}\simeq10^{-5.5}Z_\odot$8

not a simple sum over fragments (Mazonka, 2013). This formulation makes explicit that the fragmentation effect can be statistical aggregation over a cloud rather than a structural transition of the underlying system.

5. Social and ecological fragmentation

In social-balance models on signed networks, fragmentation can be induced by increased communication density rather than by isolation. A stochastic co-evolutionary model with binary opinions $Z_{\rm crit}\simeq10^{-5.5}Z_\odot$9 and signed edges $r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$0 minimizes the stress function

$r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$1

At fixed social temperature $r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$2, increasing average degree $r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$3 can drive a first-order-like transition from a cohesive phase to a fragmented phase with many internally positive, externally hostile groups, $r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$4, high within-group homogeneity $r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$5, and hysteresis at sufficiently large $r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$6 (Pham et al., 2020). The paper’s claim is that there exists a critical communication density above which social balance itself drives fragmentation.

Adaptive voter models replace structural balance by rewiring on disagreement edges. In multi-state versions with pair-dependent rewiring rates $r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$7, fragmentation can be full or partial, and the transition depends on the weighted state-network of opinions rather than on a one-dimensional ideological axis. For the three-opinion case, the Jacobian stability analysis of $r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$8- and $r_{{\rm MgSiO}_3,0}\lesssim0.01\,\mu{\rm m}$9-fan motifs determines whether one opinion splits off from an active two-opinion component; in special cases where all inter-cluster rewiring rates are equal, the fragmentation point reduces exactly to that of the classical two-state model (Böhme et al., 2012). This shows that internal opinion diversity does not always change the fragmentation threshold; under uniform cuts between clusters, only the inter-group cut matters.

A third social usage concerns kinship connectivity under fertility regulation. In the strong-ties model where edges are sibling and marriage links within a generation, the effective branching mean is

$Z_{\rm crit}\sim10^{-4.5}Z_\odot$0

with family-size distribution $Z_{\rm crit}\sim10^{-4.5}Z_\odot$1 and marriage ratio $Z_{\rm crit}\sim10^{-4.5}Z_\odot$2 (Lotker et al., 2017). The derived theorem is that if $Z_{\rm crit}\sim10^{-4.5}Z_\odot$3 and a population policy disallows families with 3 or more children, then the strong-ties branching process dies out almost surely; the existence of some 3+ child families is a necessary condition for an infinite giant component (Lotker et al., 2017). Here fragmentation is loss of connectivity in a kinship network, not ideological or spatial separation.

In ecology, fragmentation has both network and macroevolutionary meanings. EcoLab results argue that fragmentation reduces effective interspecies connectivity, allows local diversification, and on reconnection triggers extinction events that preferentially remove highly connected species, producing a diversity ratchet over repeated fragmentation–coalescence cycles. A 10-million-step panmictic run yields the fitted relation

$Z_{\rm crit}\sim10^{-4.5}Z_\odot$4

and species–area fits at $Z_{\rm crit}\sim10^{-4.5}Z_\odot$5 imply $Z_{\rm crit}\sim10^{-4.5}Z_\odot$6 (Standish, 11 Apr 2026). A different network-ecology model studies habitat fragmentation as degree heterogeneity under nonlinear random walks with finite local capacity. There, asymmetric competition parameters $Z_{\rm crit}\sim10^{-4.5}Z_\odot$7 generate a core–periphery split; in the sublinear analysis the crossover degree is

$Z_{\rm crit}\sim10^{-4.5}Z_\odot$8

and exact node-level vacancy requires $Z_{\rm crit}\sim10^{-4.5}Z_\odot$9 (Orgeron et al., 6 Nov 2025). The shared implication is that fragmentation changes coexistence not merely by reducing area, but by reorganizing connectivity.

6. Fragmentation of state spaces, patterned media, and general principles

In fracton dynamics, fragmentation is a property of Hilbert space rather than of ordinary space. A one-dimensional spin-1 chain with three-site random unitary gates conserving total charge

$r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$0

and total dipole moment

$r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$1

admits only a sparse set of local moves, such as $r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$2 and $r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$3 (Feng et al., 2021). These constraints split each $r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$4 sector into dynamically disconnected subsectors. As a result, three-site-gate dynamics fails to thermalize, relaxes on diffusive scales $r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$5 or $r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$6 rather than $r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$7, and produces an effective attraction between fractons or between a fracton and the boundaries, interpreted as a “fracton Casimir effect” (Feng et al., 2021). The fragmentation effect here is nonergodicity by sector shattering.

Patterned fragmentation can also occur in driven plasmas and nanostructures. In the thermal-trigger model for solar flares, allowing perturbations with $r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$8 shows that oblique current-layer fragmentation is permitted but weak: the tilt is typically below $r_{{\rm MgSiO}_3,0}\gtrsim0.1\,\mu{\rm m}$9, rising to about $$\tau_{\rm cont}=(\kappa_{\rm g}\rho_{\rm g}+\kappa_{\rm d}\rho_{\rm d})l_{\rm BE}, \qquad \beta_{\rm cont}=\min\{1,\tau_{\rm cont}^{-2}\},$$0 only for extremely thin layers $$\tau_{\rm cont}=(\kappa_{\rm g}\rho_{\rm g}+\kappa_{\rm d}\rho_{\rm d})l_{\rm BE}, \qquad \beta_{\rm cont}=\min\{1,\tau_{\rm cont}^{-2}\},$$1 cm (Ledentsov, 2021). Even so, the predicted spacing of elementary energy release can fall from $$\tau_{\rm cont}=(\kappa_{\rm g}\rho_{\rm g}+\kappa_{\rm d}\rho_{\rm d})l_{\rm BE}, \qquad \beta_{\rm cont}=\min\{1,\tau_{\rm cont}^{-2}\},$$2 Mm to $$\tau_{\rm cont}=(\kappa_{\rm g}\rho_{\rm g}+\kappa_{\rm d}\rho_{\rm d})l_{\rm BE}, \qquad \beta_{\rm cont}=\min\{1,\tau_{\rm cont}^{-2}\},$$3 Mm. In multilayer $$\tau_{\rm cont}=(\kappa_{\rm g}\rho_{\rm g}+\kappa_{\rm d}\rho_{\rm d})l_{\rm BE}, \qquad \beta_{\rm cont}=\min\{1,\tau_{\rm cont}^{-2}\},$$4 nanofilms, Faraday rotation as a function of $$\tau_{\rm cont}=(\kappa_{\rm g}\rho_{\rm g}+\kappa_{\rm d}\rho_{\rm d})l_{\rm BE}, \qquad \beta_{\rm cont}=\min\{1,\tau_{\rm cont}^{-2}\},$$5 contains a monotonic effective-medium part and two minima near $$\tau_{\rm cont}=(\kappa_{\rm g}\rho_{\rm g}+\kappa_{\rm d}\rho_{\rm d})l_{\rm BE}, \qquad \beta_{\rm cont}=\min\{1,\tau_{\rm cont}^{-2}\},$$6 and $$\tau_{\rm cont}=(\kappa_{\rm g}\rho_{\rm g}+\kappa_{\rm d}\rho_{\rm d})l_{\rm BE}, \qquad \beta_{\rm cont}=\min\{1,\tau_{\rm cont}^{-2}\},$$7 nm, which are interpreted as enhanced fragmentation of ultrathin Co layers induced by quantum size effects in Cu (Lukienko et al., 2020). In both cases, fragmentation refers to emergent substructure whose scale depends sensitively on the governing dispersion or growth conditions.

Taken together, these literatures show that fragmentation effects fall into three broad technical classes. First, fragmentation may be a genuine physical transition that changes cooling, stability, or breakup pathways, as in star formation, disks, meteoroid entry, and molecular Coulomb explosion [(Chiaki et al., 2013); (Takahashi et al., 2016); (Khrennikov et al., 2020); (He et al., 2024)]. Second, fragmentation may be a network or state-space disconnection phenomenon that changes long-time accessible configurations, as in adaptive opinions, social balance, ecology, kinship graphs, and fracton circuits [(Pham et al., 2020); (Böhme et al., 2012); (Lotker et al., 2017); (Standish, 11 Apr 2026); (Feng et al., 2021)]. Third, fragmentation may be an inferential object whose effect depends on reconstruction or aggregation rules, as in MST heavy-ion clusterization and cumulative fragment-damage probabilities [(Gupta et al., 2011); (Mazonka, 2013)]. A plausible implication is that the phrase “fragmentation effect” is most useful when accompanied by an explicit statement of what is fragmenting, by what rule, and which observable is being altered.