Fragility Spectrum: From Glasses to Systemic Risk

• Fragility Spectrum is a continuum that ranks systems by how sharply their dynamics and failure thresholds degrade under cooling, perturbation, or enrichment.
• It quantifies phenomena such as the super-Arrhenius increase in viscosity in supercooled liquids using metrics like the Angell steepness index and Vogel-Fulcher-Tammann fits.
• Beyond glass physics, the concept organizes responses in fields like seismic engineering, systemic risk, and digital assets by comparing degradation thresholds and response curvatures.

Fragility spectrum denotes a family of quantitative orderings that place systems along a continuum according to how sharply their dynamics, failure thresholds, or formal properties deteriorate under cooling, perturbation, or enrichment. In glass physics—the domain in which the term is most established—it refers primarily to the strong-to-fragile continuum of supercooled liquids, that is, the degree of super-Arrhenius growth of viscosity or structural relaxation time near the glass transition. In later work, the same spectrum language is extended to relaxation-spectrum breadth, finite-amplitude failure landscapes, transportation disruption losses, block-tail systemic risk, memecoin ecosystems, and the resilience of model-theoretic properties under language expansions (Tarjus et al., 2014, Tah et al., 2021, Limkumnerd, 30 May 2026, Adilkhan, 13 Aug 2025).

1. Definition, measurement, and the strong-to-fragile continuum

In the classical glass-transition literature, fragility measures how quickly transport coefficients and relaxation times increase as temperature decreases toward the glass transition. The canonical Angell steepness index is

$$m=\left.\frac{d\log_{10}\tau}{d(T_g/T)}\right|_{T=T_g},$$

or the analogous viscosity form, and it orders liquids from strong, nearly Arrhenius systems to fragile, strongly super-Arrhenius ones. Representative values discussed in the literature span about $m\approx 20$ for silica, about $80\!-\!100$ for liquids such as ortho-terphenyl or toluene, and $m\gtrsim 150$ for some polymers, which is the empirical basis for speaking of a broad fragility spectrum rather than a binary taxonomy (Tarjus et al., 2014).

This spectrum is not quantified by a unique convention. In density-tunable harmonic-sphere glass formers, fragility is extracted from a Vogel-Fulcher-Tammann fit of the structural relaxation time,

$$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$

with $K_{VFT}$ identified explicitly as the kinetic fragility index. In that setting, $\tau_\alpha$ is defined by the self-overlap condition $\langle Q(\tau_\alpha)\rangle=1/e$, and $T_g$ is operationally set by $\tau_\alpha(T_g)=5\times10^6$ (Tah et al., 2021). In another study on the same family of density-tuned liquids, $m\approx 20$0 is instead defined by $m\approx 20$1, but the operational role of $m\approx 20$2 remains the same: larger $m\approx 20$3 means greater fragility (Tah et al., 2022).

Metallic-glass work often uses the VFT strength parameter $m\approx 20$4,

$m\approx 20$5

with the opposite monotonicity: the most fragile glass formers have $m\approx 20$6, while the strongest are of order $m\approx 20$7; within this convention, larger $m\approx 20$8 means stronger behavior and smaller $m\approx 20$9 more fragile behavior (Busch et al., 2014). This multiplicity of metrics is central to the concept’s history. Tarjus and Alba-Simionesco argued that fragility is useful as a descriptor of the degree of super-Arrhenius slowdown, but that “a precise quantitative measurement is not straightforward,” because $80\!-\!100$0-based measures are operational, can mix high-temperature activation scales with super-Arrhenius growth, and differ between isobaric and isochoric protocols (Tarjus et al., 2014).

2. Structural organization of the fragility spectrum in glass-forming liquids

A major line of work treats the fragility spectrum as structurally organized rather than merely phenomenological. In a three-dimensional 50:50 binary soft-repulsive harmonic-sphere mixture, density tunes the liquid continuously from nearly Arrhenius strong behavior at $80\!-\!100$1 to strongly super-Arrhenius fragile behavior at $80\!-\!100$2, while keeping particle composition and interactions fixed. Across this range, the fitted VFT fragility changes from $80\!-\!100$3 to $80\!-\!100$4, a factor of about 17, which the authors describe as a wide fragility spectrum (Tah et al., 2021).

The organizing structural variable proposed in that study is an order-agnostic many-body static amorphous order, quantified primarily by the point-to-set length $80\!-\!100$5. In the cavity protocol, particles outside a spherical cavity are frozen, the interior is re-equilibrated, and the overlap excess

$80\!-\!100$6

is fitted as

$80\!-\!100$7

with $80\!-\!100$8. The central result is that low-density strong liquids show little growth of $80\!-\!100$9 on cooling, whereas high-density fragile liquids show pronounced growth. The proposed mechanism is then RFOT-like: $m\gtrsim 150$0 with a common $m\gtrsim 150$1 producing a good collapse across all studied densities. On this interpretation, fragility differences arise because the same many-body amorphous order grows at different rates along the spectrum, not because distinct classes of activated dynamics are required (Tah et al., 2021).

A second structural program uses machine learning rather than cavity pinning. In the same density-tunable harmonic-sphere family, softness is defined as the signed distance to a linear-kernel support-vector-machine hyperplane in a 73-dimensional local-structure feature space. A combined training set drawn from the strongest and most fragile densities yields a single hyperplane with nearly uniform predictive accuracy across the full spectrum: $m\gtrsim 150$2, $m\gtrsim 150$3, $m\gtrsim 150$4, $m\gtrsim 150$5, and $m\gtrsim 150$6 for $m\gtrsim 150$7, respectively. The rearrangement probability at fixed softness obeys

$m\gtrsim 150$8

and the analysis decomposes fragility into three coupled ingredients: stronger temperature dependence of average structure $m\gtrsim 150$9, stronger structure-dependence of effective barriers, and a narrower reduced-temperature window over which nonlocal dynamical corrections become important. One especially distinctive conclusion is that in fragile liquids mobility heterogeneity is dominated by the softness dependence of energy barriers, whereas in the strongest liquid it is dominated mainly by entropic variations (Tah et al., 2022).

3. Dynamic heterogeneity, nonexponentiality, and relaxation-spectrum breadth

The fragility spectrum is also a spectrum of kinetic heterogeneity. In the harmonic-sphere family, the dynamic heterogeneity length $$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$0 grows in all systems, but more strongly in more fragile ones, even though the structural thesis of the work is that $$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$1 is not itself the primary explanatory variable. Near the mode-coupling crossover the dynamic length is reported to follow a critical-like growth with exponent $$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$2, close to the inhomogeneous mode-coupling prediction $$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$3 (Tah et al., 2021).

A more microscopic cage-jump analysis sharpens this point. In a density-tunable soft-repulsive binary mixture, increasing fragility broadens cage-lifetime distributions, increases the randomness parameter

$$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$4

and makes the survival probability increasingly nonexponential. The cage-state survival is fitted as

$$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$5

with $$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$6 at the lowest temperature in the strong regime $$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$7 and $$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$8 in the fragile regime $$\tau_\alpha(T)=\tau_0 \exp\left [\frac{1}{K_{VFT} \ (T/T_{VFT}-1)}\right],$$9. The paper argues that all densities share intermittent cage-breaking jumps, but that fragility changes the structural bottleneck controlling jump-rate fluctuations: localized neighbor-distance fluctuations suffice in the strong regime, more extended shells are needed in the intermediate regime, and in the fragile regime distance-based descriptors alone fail at the lowest temperature unless Voronoi free volume is added. Point-to-set correlations still grow with fragility, but the slow-variable extent inferred from jump dynamics exceeds the PTS length (Kumar et al., 6 Jul 2026).

Mechanical relaxation spectra add a further dimension. In organic ionic glass formers, fragility is measured in the Angell sense using

$K_{VFT}$0

often with rheological shift factors $K_{VFT}$1 as the proxy for $K_{VFT}$2. Relaxation broadness is measured by the KWW exponent $K_{VFT}$3 or by the loss-peak full width at half maximum $K_{VFT}$4, connected approximately by

$K_{VFT}$5

For non-charged materials the recovered trend is the conventional one: increasing fragility is associated with broader, more nonexponential relaxation. For charged systems—ionic liquids, polymerized ionic liquids, ionomers, and the new “compleximers”—the trend is reversed. Compleximers show $K_{VFT}$6, thus a relatively strong organic glass transition, together with unusually broad mechanical relaxation spectra, and the authors summarize the whole charged family as following an inverted fragility–broadness correlation relative to neutral glass formers (Lange et al., 7 Sep 2025). The fragility spectrum here is therefore not only strong-to-fragile in the Angell sense, but also a spectrum of how fragility couples—or fails to couple—to spectral broadness.

4. Material-specific microscopic control parameters

Beyond generic many-body order, several studies propose material-specific organizers of the fragility spectrum. In metallic liquids, the proposed microscopic control parameter is the effective repulsive steepness $K_{VFT}$7 extracted from the low-$K_{VFT}$8 flank of the first peak of the pair distribution function through

$K_{VFT}$9

Across ten metallic alloy liquids, stronger liquids exhibit steeper repulsive interactions: the strongest systems, $\tau_\alpha$0 and $\tau_\alpha$1, have $\tau_\alpha$2 and $\tau_\alpha$3, whereas the most fragile, $\tau_\alpha$4, has $\tau_\alpha$5 and $\tau_\alpha$6. The authors therefore interpret the metallic fragility spectrum through nearest-neighbor cage hardness encoded in the potential of mean force (Pueblo et al., 2017).

Vitrimeric polymers furnish a different variant. In a coarse-grained star-polymer vitrimer model, decreasing bulk density tunes the system from fragile to strong and even superstrong behavior. The operational glass transition is defined by $\tau_\alpha$7, fragility is extracted from Angell plots, and the low-density regime $\tau_\alpha$8 reaches $\tau_\alpha$9, which the paper interprets as superstrong. Microscopic and schematic MCT both reproduce the trend and attribute it to the temperature sensitivity of the main peak of the static structure factor $\langle Q(\tau_\alpha)\rangle=1/e$0: high-density fragility is associated with strong cooling-induced growth of $\langle Q(\tau_\alpha)\rangle=1/e$1, whereas low-density superstrong behavior occurs when $\langle Q(\tau_\alpha)\rangle=1/e$2 changes only weakly with temperature. Bond-swap kinetics shift vitrification boundaries but do not control fragility itself (Ciarella et al., 2019).

A more theoretical proposal derives the spectrum from elasticity. In an aperiodic frozen-stress ansatz mapped to a six-component anisotropic spin model, the controlling variable is the anisotropy ratio

$\langle Q(\tau_\alpha)\rangle=1/e$3

which relates directly to the Poisson ratio $\langle Q(\tau_\alpha)\rangle=1/e$4. The framework predicts a continuous range between frozen-in shear and uniform compression/dilation regimes and suggests that fragility depends on the Poisson ratio in a non-monotonic way, especially near the crossover where $\langle Q(\tau_\alpha)\rangle=1/e$5 (Bevzenko et al., 2010). Taken together, these studies suggest not a single universally accepted microscopic order parameter, but several competing structural coordinates that organize different portions of the material fragility spectrum.

5. Generalizations beyond glass physics

Outside glass science, the phrase fragility spectrum is used for several distinct but structurally related objects. In nonlinear dynamical failure theory, the spectrum is the cumulative distribution of directional failure thresholds: $\langle Q(\tau_\alpha)\rangle=1/e$6 where $\langle Q(\tau_\alpha)\rangle=1/e$7 is the smallest perturbation amplitude in direction $\langle Q(\tau_\alpha)\rangle=1/e$8 that reaches a failure boundary. The paper shows that this nonlinear spectrum is predicted, before failure, by the tail of the boundary-normalized fragility gain

$\langle Q(\tau_\alpha)\rangle=1/e$9

Its central point is that finite-amplitude fragility depends not only on the worst perturbation direction but also on the breadth of near-dangerous directions. In a high-dimensional non-normal network, two systems were constructed with the same strongest directional gain $T_g$0, yet the one with broader response-channel breadth had a larger nonlinear fragility curve, with mean difference $T_g$1 and maximum difference $T_g$2 (Limkumnerd, 30 May 2026).

Civil engineering uses the same term differently. A seismic fragility curve is a conditional failure probability as a function of ground-motion intensity,

$T_g$3

with $T_g$4 the maximal drift ratio and $T_g$5 a prescribed threshold. In that setting, the paper compares classical lognormal parametric fits with nonparametric binned Monte Carlo simulation and kernel density estimation, and finds that the accuracy of lognormal fragility curves depends on the intensity measure, the failure criterion, and the estimation method, especially at higher drift thresholds (Mai et al., 2017). This usage shifts the spectrum idea from strong-to-fragile dynamics to a family of failure-probability curves indexed by damage threshold.

Systemic-risk work in multivariate extremes defines a block fragility index,

$T_g$6

where $T_g$7 counts the number of blocks in a partition $T_g$8 containing at least one exceedance. The resulting object is a partition-indexed fragility spectrum rather than a single scalar. In the financial application reported, the USA subsystem had estimated $T_g$9, the global system $\tau_\alpha(T_g)=5\times10^6$0, and the system divided into three regional blocks $\tau_\alpha(T_g)=5\times10^6$1 (Ferreira et al., 2011). Taleb and Douady proposed a more abstract cross-domain definition: fragility is the sensitivity of left-tail shortfall to left semi-deviation, robustness is bounded left-tail sensitivity, and antifragility combines left-side robustness with positive right-tail sensitivity (Taleb et al., 2012).

Transportation and digital-asset studies extend the same logic. In road transportation systems, fragility is identified with convexity of performance loss under disruption: positive second derivatives of Average Time Spent or Total Time Spent with respect to disruption magnitude certify fragile behavior under standard FD and MFD models. The same work proposes a skewness-based indicator to compare network fragility from MFD parameters alone (Sun et al., 2024). In memecoin markets, fragility is modeled as a three-dimensional ecosystem profile—Volatility Dynamics Score, Whale Dominance Score, and Sentiment Amplification Score—with political tokens such as TRUMP, MELANIA, and LIBRA occupying the high-fragility tier, established memecoins the middle, and benchmark assets ETH and SOL the resilient end (Xiang et al., 29 Nov 2025). Model theory pushes the idea in a formally different direction: the fragility spectrum of a theory $\tau_\alpha(T_g)=5\times10^6$2 is measured by

$\tau_\alpha(T_g)=5\times10^6$3

the least language expansion needed to destroy a property such as stability, NIP, or decidability; the paper proves exact stratification results, including NIP theories $\tau_\alpha(T_g)=5\times10^6$4 with $\tau_\alpha(T_g)=5\times10^6$5 for every $\tau_\alpha(T_g)=5\times10^6$6 (Adilkhan, 13 Aug 2025).

6. Debates, limitations, and interpretive significance

Across these literatures, fragility spectrum is best understood as a family of related organizing principles rather than a single universal invariant. Even within glass physics, Tarjus and Alba-Simionesco argue that fragility is a real descriptor of super-Arrhenius slowdown but not a uniquely defined intrinsic scalar: isobaric measures mix thermal and density effects, $\tau_\alpha(T_g)=5\times10^6$7-based indices depend on operational conventions, and the relation between fragility magnitude and cooperativity is not straightforward (Tarjus et al., 2014). A plausible implication is that any encyclopedia treatment must distinguish the empirical spectrum itself from the metric used to parametrize it.

The same caution applies to mechanistic claims. The point-to-set analysis is extensive but simulation-based and performed within one density-tunable model family (Tah et al., 2021). The machine-learning softness analysis establishes a common structural direction across the same family, yet its barrier interpretation is still indirect (Tah et al., 2022). The metallic-liquid result is explicit that its conclusions are demonstrated for metallic liquids and may not transfer unchanged to other classes (Pueblo et al., 2017). The ionic broadness inversion is presented as consistent across the surveyed charged materials, but not as a strict universal master curve with a single fitted law (Lange et al., 7 Sep 2025). The vitrimeric superstrong regime is observable within the accessible temperature window, while the accompanying schematic MCT still retains a lower-temperature critical divergence (Ciarella et al., 2019).

A further source of ambiguity is that the same phrase can name formally different objects. In glass science, it often denotes a strong-to-fragile ordering of relaxation laws. In seismic engineering it denotes conditional exceedance curves (Mai et al., 2017). In block-tail risk it denotes a partition-dependent expected number of extreme blocks (Ferreira et al., 2011). In nonlinear failure theory it is a distribution over directional failure amplitudes (Limkumnerd, 30 May 2026). In model theory it is the minimum language expansion required to degrade a property (Adilkhan, 13 Aug 2025). The shared core is not a common formula but a common comparative structure: fragility spectra rank systems by how much deterioration is triggered by cooling, perturbation, concentration, or enrichment.

The enduring significance of the concept lies in that comparative structure. It allows strong and fragile glasses, broad and narrow relaxation spectra, localized and delocalized network modes, resilient and whale-dominated token ecosystems, or robust and expansion-sensitive theories to be placed on calibrated continua rather than in coarse dichotomies. This suggests that “fragility spectrum” is most productively viewed as a spectrum of thresholds and response curvatures: the less perturbation, cooling, or enrichment required to produce disproportionate degradation, the higher the fragility.