Configurational Entropy in Physical Systems

• Configurational entropy is an information-theoretic measure that maps spatial energy densities into momentum space to quantify the degree of order or randomness.
• It is used to diagnose phase transitions and critical phenomena by tracking modal distributions and scale-free, turbulent, and critical regimes.
• The method underpins analyses of stability in field theories, gravitational models, and complex systems by selecting the most structured, low-entropy configurations.

Configurational entropy (CE) is an information-theoretic measure that quantifies the spatial complexity of physical configurations, widely applied across condensed matter, field theory, statistical mechanics, and gravitational settings. CE generalizes Shannon entropy to field configurations and energy densities, providing a quantitative framework to characterize localization, structural order, phase transitions, and stability in systems where classical or quantum microstates do not suffice to describe all relevant degrees of freedom.

1. Mathematical Formulation and Foundational Principles

Configurational entropy is constructed by mapping spatially localized functions—typically energy densities, order parameters, or field profiles—into momentum space, and then analyzing the distribution of power across momentum modes. For a real, square-integrable function $\rho(\mathbf{x})$ in $d$ spatial dimensions, its Fourier transform is

$$\tilde{\rho}(\mathbf{k}) = \int d^d x\,\rho(\mathbf{x})\,e^{-i\mathbf{k}\cdot\mathbf{x}}$$

The modal fraction serves as the analogue of a probability distribution: $$f(\mathbf{k}) = \frac{|\tilde{\rho}(\mathbf{k})|^2}{\int d^d k\,|\tilde{\rho}(\mathbf{k})|^2}$$ The configurational entropy is then defined as the Shannon entropy of this modal fraction: $$S_C = -\int d^d k\,f(\mathbf{k})\,\ln f(\mathbf{k})$$ Alternatively, for discrete systems or finite domains, the sums and integrals are replaced by discrete indices and normalization is retained. In some applications, $f(\mathbf{k})$ is rescaled by its maximum value to ensure the positivity of the logarithm, but the informational content resides in the structure and breadth of the modal fraction itself (Sowinski et al., 2016).

Physically, CE quantifies how compressible or spread out a configuration is in momentum space. Sharply peaked spectra with few dominant modes yield low CE, indicating "ordered" or "compressible" spatial structures, while flat or equipartitioned (white-noise) spectra maximize the entropy and signal spatial randomness or disorder.

2. CE in Phase Transitions and Critical Phenomena

Configurational entropy provides a rigorous diagnostic for phase transitions, particularly continuous (second-order) transitions. In scalar field models with symmetry-breaking (such as Ginzburg–Landau systems), CE is computed from the order parameter field. As temperature approaches the critical point ($T_C$), long-range correlations increase and the low-momentum modes dominate; the CE exhibits a sharp minimum at $T_C$ (Sowinski et al., 2016, Gleiser et al., 2015).

Distinct scaling regimes are revealed in the CE density:

• Scale-free regime: For low $k$ far from criticality, $f(k)\approx$ const, and $dS_C/dk \sim k^0$.
• Turbulent regime: Around the inverse correlation length $\xi^{-1}$, $dS_C/dk \sim k^{-5/3}$, analogous to Kolmogorov turbulence, reflecting an information flux across scales.
• Critical regime: At high $k$ near $T_C$ (with critical exponent $\eta$), one finds $dS_C/dk \sim k^{-7/4}$ in $d=2$.

At $T_C$, CE attains its global minimum, signifying that the system is maximally "compressible"—the spatial configuration is highly structured and dominated by long-wavelength modes. This point is associated with maximal information storage in analogy with data compression (Sowinski et al., 2016).

3. Applications in Field Theory, Gravity, and Brane Models

Configurational entropy has become a central tool in analyzing stability and phase structure of spatially localized solutions in nonlinear field theories, brane models, and holography. Applications include:

• Brane-world scenarios: CE is used to distinguish dynamically stable solutions and constrain model parameters, selecting "most ordered" thick brane configurations even when multiple solutions have identical energy (Chinaglia et al., 2017, Correa et al., 2015, Cruz et al., 2017).
• Black holes and AdS/QCD: For AdS–Schwarzschild black holes, CE reproduces the Hawking–Page transition (thermal AdS to black hole dominance), and quantitatively tracks the stability landscape: $S_C$ decreases (greater stability) as temperature rises above $T_C$, consistent with thermodynamic dominance of the black hole phase (Braga et al., 2016, Braga et al., 2021). CE growth also signals increasing instability in rotating and highly excited states (Braga et al., 2023).
• Spectroscopic discrimination: CE of bulk energy profiles in AdS/QCD models discriminates between conventional and exotic hadronic states (e.g., multiquark, glueball, hybrid), with higher-dimensional operator duals yielding higher CE values (Colangelo et al., 2018).
• Topological defects and solitons: CE provides an information-theoretic selection rule for kink-like solitons and energy-degenerate solutions, choosing spatial profiles with minimal entropy as physically preferred even when energetic criteria are inconclusive (Correa et al., 2014).

4. CE in Glass-formers, Disordered Lattices, and Complex Systems

In glassy materials, supercooled liquids, and classical assemblies with a rugged energy landscape, configurational entropy quantifies the multiplicity of accessible minima ("basins") at given thermodynamic parameters. For a system with metastable states $k$ with probabilities $p_k$,

$S_{\rm conf} = -\sum_{k} p_k \ln p_k$

and the effective number of states is $N_{\rm eff} = e^{S_{\rm conf}}$. This measure is robust against rare, low-probability configurations and is central to the theoretical framework describing the glass transition, the Adam–Gibbs relation, and point-to-set correlations (N et al., 2022, Berthier et al., 2019, Radzvilavičius et al., 2012).

In hydrogen-disordered ice polymorphs, the residual configurational entropy quantifies the macroscopic degeneracy of ground states subject to network constraints (ice rules). Monte Carlo integration combined with high-precision thermodynamic integration yields values sensitive to topology (e.g., mean ring size, connective constant), with broader topological networks reducing entropy due to enhanced constraint propagation (Herrero et al., 2014).

5. Dynamical Diagnostics, Scaling Laws, and Information Flow

Configurational entropy is sensitive to dynamical processes, stability thresholds, and information transfer across scales:

• Oscillon lifetimes: In relativistic field theories, CE-based measures correlate with the longevity of oscillons. A decreasing fluctuation amplitude in CE tracks increasing stability and, in some instances, infinite-lifetime conjectures (Gleiser et al., 2020).
• Scaling relations: Power-law relations between CE and physical observables (e.g., decay rates in atomic systems, plateau energy and radiation rates in oscillons, Regge trajectories of glueballs) have been established, supporting CE as a predictive tool for dynamics and spectroscopy (Gleiser et al., 2017, Rodrigues et al., 2020).
• Cosmological evolution: In cosmology, CE density for radiation and matter components exhibits comoving scaling laws—$a^{-3}$ for radiation and $a^{-3/2}$ for nonrelativistic matter—paralleling the behavior of thermodynamic entropy densities with cosmic expansion (Bernardini et al., 2019).

6. Methodological Frameworks and Broader Implications

Computational recipes for CE depend on the context but share a common structure: specification of a localized field or energy density, Fourier analysis, normalization to obtain the modal fraction, and integration (or summation) of $-f(k)\ln f(k)$. For large or complex systems, accurate sampling of configurations and convergence studies are essential (e.g., in glass formers and Wigner crystals) (N et al., 2022, Radzvilavičius et al., 2012).

Configurational entropy acts as an information-theoretic organizing principle, complementing energetic and dynamical approaches. Minima of CE select preferred physical configurations, parameter bounds, and phase transitions, even in degenerate or non-standard settings. The framework applies broadly—from quantum field theory and general relativity to active matter, complex networks, and nonequilibrium systems—whenever the distribution of spatial or structural modes carries critical physical information.

A plausible implication is that CE provides a unifying quantitative handle for studying information storage, transfer, and structural organization in systems across physics, with ongoing extensions to new domains such as active media, cosmological backgrounds, and strongly correlated quantum phases.