Fluidity Index: A Unified Metric

• Fluidity Index is a quantitative measure that parameterizes stress relaxation and adaptability across physical, biological, and computational systems.
• It integrates microscopic observables like velocity fluctuations, defect concentrations, and phase lags to relate dynamic mechanisms to macroscopic behavior.
• Applications include evaluating jamming in granular materials, rheological properties in cells and superfluids, and model adaptability in AI systems.

The Fluidity Index (FI) is a quantitative construct employed across a diverse range of physical, biological, and computational systems to parameterize the ability of a system to flow, adapt, or reorganize under external or internal stimuli. Defined in terms that reflect the context—be it rheological relaxation rates, atomic mobility, cell deformation, or model adaptability—the FI provides a rigorous means to relate microscopic mechanisms with macroscopic response functions. Recent literature demonstrates converging interest in FI as both an operationally measurable and theoretically grounded property, with explicit mathematical formulations and model-specific interpretations.

1. Definitions and Conceptual Framework

In granular materials, the FI is typically formalized as a time-dependent internal variable, specifically quantifying the stress relaxation rate. If $f$ is the fluidity variable, then %%%%1%%%% measures how rapidly internal stress is dissipated; high $f$ denotes rapid, fluid-like relaxation, low $f$ signals slow, glassy or jammed dynamics (Nguyen et al., 2011). In soft biological matter, such as single cells, the FI is a nondimensional measure of “hysteresivity” or the normalized tendency of the cell to behave like a viscous liquid as opposed to a perfectly elastic solid (Maloney et al., 2013). The FI in superfluid hydrodynamics is operationalized via the ratio of shear viscosity to entropy density, $\eta/s$, with low values indicating nearly perfect (high fluidity) and high values denoting “imperfect” or less fluid behavior (Boyack et al., 2014).

At the microscopic level, for granular and molecular fluids, FI is systematically built from physically observable quantities: velocity fluctuations, packing fraction, and defect concentrations, offering a direct bridge from particle-scale processes to continuum rheology (Zhang et al., 2016, Cockrell et al., 2023, Evans et al., 12 Oct 2025). In computational or artificial intelligence systems, FI quantifies response adaptability as the degree to which models update predictions in response to dynamically varying environment states (Ngoiya et al., 23 Oct 2025).

2. Mathematical Formalism Across Domains

The precise mathematical definition of FI varies with the system studied:

• Granular Materials: The viscoelastic model introduces equations for the evolution of shear stress and fluidity:

  $\partial_t \sigma = -f\, \sigma + G\, \dot{\gamma}$

  $\partial_t f = -a\, f^2 + r\, \dot{\gamma}^2$

where $G$ is shear modulus, $a$ and $r$ are parameters representing “ageing” and “rejuvenation,” and $\dot{\gamma}$ is the shear rate. For constant $\sigma$, fluidity decays according to:

$f(t) = \frac{f_0}{1 + a_\text{ex} f_0 t}$

with $$a_\text{ex} = a \left[1 - (\sigma/\sigma_D)^2\right]$$; the dependence of $f_0$ on packing fraction underlies the approach to jamming (Nguyen et al., 2011).

• Cellular Mechanics: The FI (denoted $a$) is extracted using either time-domain or frequency-domain rheological responses. For oscillatory loading, the phase lag $\varphi$ yields

  $a = \frac{2\varphi}{\pi}$

and the power-law relationships for creep compliance and complex modulus:

$$J(t) \propto t^a, \quad G^*(\omega) = g_0 (i\omega/\omega_0)^a$$

A frequency-independent $a$ validates the structural damping (power-law) model as the appropriate rheological framework (Maloney et al., 2013).

• Superfluid Hydrodynamics: Here, the ratio $\eta/s$ is central:

  $\frac{\eta}{s} \sim \frac{T}{\gamma}$

where $T$ is temperature and $\gamma$ is the quasi-particle inverse lifetime. This scaling persists across bosonic and fermionic superfluids, offering a unifying “fluidity” metric (Boyack et al., 2014).

• Granular Flow Models: Operational and microscopic definitions equate the fluidity field $g$ with observable quantities:

  $$g = \frac{\dot{\gamma}}{\mu} = \frac{\delta v}{d} F(\Phi)$$

where $\delta v$ is velocity fluctuation, $d$ is particle diameter, and $F(\Phi)$ captures the packing fraction dependence. This identity is theoretically grounded via kinetic and activated-process models (Zhang et al., 2016).

• Defect-Mediated Thin Film Fluidity: FI is given via local inverse viscosity as a linear combination of vacancy and interstitial concentrations:

  $$\frac{1}{\eta(x,z,t)} = \frac{1}{\eta^*} + \frac{C_I(x,z,t)}{\eta_I} + \frac{C_V(x,z,t)}{\eta_V}$$

Coupling defect kinetics to viscosity establishes FI as a field variable responsive to irradiation and kinetic processes (Evans et al., 12 Oct 2025).

• AI Adaptability (Fluidity Index): FI is calculated as an average “accuracy adaptation” (AA) score for model prediction in a dynamic environment:

  $$\text{AA}_i = 1 - \frac{|\text{New Prediction}_i - \text{Old Prediction}_i|}{\Delta \text{Initial Environment State}_i}$$

  $$\text{FI}(t) = \frac{\sum_{i=1}^n \text{AA}_i}{\text{NC}}$$

Extended to higher-order integrals for multi-level adaptability, measuring both immediate and self-sustaining fluidity (Ngoiya et al., 23 Oct 2025).

3. Physical Interpretation and Mechanistic Origins

Across contexts, FI quantifies relaxation, adaptability, or flow capacity at a fundamental level:

• In granular and amorphous materials, a high FI signals rapid stress dissipation, indicating proximity to flowing states, while low FI (approaching zero) marks the jamming threshold where flow ceases (Nguyen et al., 2011, Zhang et al., 2016).
• In molecular liquids, FI hinges on the statistical properties of transit atoms, with enhanced FI linked to non-Maxwellian velocity distributions, indicative of high-dimensional dynamic sub-ensembles facilitating rapid particle rearrangement and flow (Cockrell et al., 2023).
• In cellular mechanics, FI reflects the spectrum from solid-like (elastic) to fluid-like (viscous) behavior, modulated by cytoskeletal organization, crosslinking, and temperature, but surprisingly insensitive to ATP-driven active processes under linear deformations (Maloney et al., 2013).
• In superfluids, FI ($\eta/s$) characterizes the balance of dissipative and entropy-driven processes, with near-minimum values evidencing “perfect” fluids at unitarity, while much larger values denote highly “imperfect” fluids (Boyack et al., 2014).
• Membrane physics contrasts local versus global formulations of FI, with local models (zero local shear modulus) producing isotropic stress and global models (zero integrated shear modulus) leading to finite, anisotropic local stress, critically affecting the calculation of elastic moduli in curved membranes (Pinigin, 23 Oct 2024).
• In computational benchmarks, FI operationalizes model flexibility to context change, aggregating real-time prediction accuracy and self-sustaining resource reallocation, providing a multi-order metric for super-intelligence (Ngoiya et al., 23 Oct 2025).

4. Measurement, Parameter Dependence, and Experimental Signatures

Methodologies to determine FI are tailored to system type:

• Granular Packings: Experimental creep deformation under constant shear below yield is analyzed with FI extracted from strain evolution, revealing logarithmic creep governed by the decay of $f(t)$ and its dependence on packing fraction $\phi$. FI extrapolates to zero at random close packing ($\phi \approx 0.635$), signifying jamming (Nguyen et al., 2011).
• Cells: FI is obtained via creep compliance or dynamic rheometry, with consistent values across time and frequency domains under the structural damping model. Chemical crosslinking suppresses FI, while ATP depletion does not, and temperature elevation linearly increases FI (Maloney et al., 2013).
• Superfluids: FI ($\eta/s$) is inferred via transport measurements or theoretical Kubo formulas, with values scaling as $T/\gamma$ and exhibiting sharp decrease towards the perfect fluid bound in strongly interacting regimes (Boyack et al., 2014).
• Particle Mobility in Liquids: FI is extracted by fitting transit atom velocity distributions to

  $\rho(v) \propto v^\alpha \exp(-mv^2/2k_BT)$

with $\alpha > 2$ signifying non-Maxwellian, high-fluidity regimes. $\alpha$ varies nonmonotonically with pressure and temperature, reverting to Maxwellian ($\alpha=2$) in solid and gas states (Cockrell et al., 2023).

• Granular Nonlocality: Kinematic FI ($g$) is measured using DEM simulations, relating local velocity fluctuations and packing, verified by collapse of $g d/\delta v$ data across shear geometries (Zhang et al., 2016).
• Thin Films Under Irradiation: FI is calculated directly from depth-resolved defect concentrations and varied with ion energy, angle, flux, and temperature, aligning theoretical predictions of surface evolution with experimental nanopatterning phenomena (Evans et al., 12 Oct 2025).
• Membrane Stress Analysis: Local stress tensor components from molecular dynamics simulations differentiate between isotropic (local fluidity) and anisotropic (global fluidity) models, identifying finite, sign-changing lateral shear stresses in curved geometries (Pinigin, 23 Oct 2024).
• AI Benchmarks: FI is computed across dynamic prediction trajectories, integrating over context switches and measuring both immediate and higher-order adaptability to environmental state changes (Ngoiya et al., 23 Oct 2025).

5. Applications, Implications, and System-Specific Impact

The FI framework finds broad application in elucidating mechanism, guiding material selection, and benchmarking system performance:

Context	Role of Fluidity Index (FI)	Key Parameters/Indicators
Granular materials	Quantify jamming, aging, and flow transition	$f_0(\phi)$, $a$, $r$, $\sigma_r$
Cellular biomechanics	Differentiate cell states and mechanics	Power-law exponent $a$, $\varphi$
Superfluid physics	Grade fluid perfection/imperfectness	$\eta/s$, $T/\gamma$
Structural fluids	Capture flow via rare, rapid configurations	$\alpha$ in $\rho(v) \propto v^\alpha$
Thin films	Model ion-enhanced relaxation/morphological stability	$C_V, C_I$, $\eta(x,z)$, irradiation
Lipid membranes	Determine contributions to bending/twisting modulus	$\lambda_S(z)$, local stress anisotropy
AI/Computation	Benchmark system adaptability and resource renewal	FI(t), AA, environment state offsets

Implications extend to predicting the jamming transition, diagnosing or sorting biological cells, optimizing mechanical or nanofabrication processes, modeling the elasticity of complex membranes, and establishing criteria for artificial general intelligence in dynamic computational environments.

6. Model Selection, Theoretical Consistency, and Limitations

Theoretical consistency and model choice critically underpin meaningful interpretation of the FI:

• The structural damping (power-law) model is empirically validated for cell rheology, whereas lumped-component models introduce spurious time constants inconsistent with measurements (Maloney et al., 2013).
• Choice between local and global fluidity models in membranes is resolved in favor of the latter by direct observation of finite, anisotropic stresses incompatible with strictly local fluidity (Pinigin, 23 Oct 2024).
• In granular flows, kinetic theory and Eyring-type activated process models both rationalize the kinematic foundation and functional form of the FI, corroborated by DEM simulation (Zhang et al., 2016).
• AI FI benchmarking introduces higher-order, closed-loop open-ended settings as necessary for capturing true adaptivity, beyond conventional, fixed-problem benchmarks (Ngoiya et al., 23 Oct 2025).

Limitations generally arise from experimental access to microscopic variables (e.g., transit atom statistics, local stress tensor profiles), uncertainties in parameterization (e.g., temperature dependence of defect kinetics), and model-specific artifacts when employing empirically inconsistent theoretical descriptions.

7. Future Directions and Open Questions

The scope of FI continues to expand as measurement, simulation, and theory converge:

• Improved experimental resolution of local stress and mobility in both granular and biological systems will sharpen model discrimination.
• Extension of FI measurement to non-equilibrium systems may clarify the roles of active and passive mechanics, particularly in living matter.
• In artificial intelligence and computational systems, inclusion of higher-order resource management and long-term adaptability metrics may enable more granular assessment of “super-intelligent” performance under open-ended, closed-loop conditions.
• Cross-domain generalization of the FI concept may uncover deeper unifying principles in the paper of flow, adaptation, and structural rearrangement in complex systems.

The Fluidity Index thus serves as a unifying metric that translates microstructural kinetics, collective relaxation, and dynamical adaptability into operationally meaningful, theoretically grounded, quantitatively comparable terms across diverse scientific domains.