Model Compressibility: Theory & Applications

• Model compressibility is a measure that quantifies how a model’s spectral, thermodynamic, or structural properties respond to parameter variations, external perturbations, or encoding constraints.
• It bridges diverse domains by examining spectral rigidity, phase transitions, and information redundancy, with applications in random matrices, lattice models, and turbulence corrections.
• Analytical and numerical methods in studying compressibility enable effective tuning and prediction of critical behaviors in disordered systems, non-equilibrium field theories, and complex networks.

Model compressibility is a concept that quantifies how the structural, spectral, or thermodynamic properties of a mathematical or physical model respond to parameter variations, external perturbations, or encoding constraints. In statistical physics, condensed matter, network theory, and turbulence modeling, compressibility serves as a unifying metric for understanding spectral rigidity, phase transitions, information-theoretic redundancy, and nonequilibrium response. The concept acquires distinct mathematical formulations depending on context: spectral compressibility in random matrices, negative electron compressibility in interacting electron systems, rate-distortion compressibility in networks, and compressibility corrections in turbulence models. Across these domains, compressibility is intimately linked to emergent phenomena such as criticality, multifractality, universality classes, and resource-efficient representations.

1. Spectral Compressibility in Random Matrix Models

In disordered quantum or random matrix systems, compressibility (denoted $\chi$) characterizes the fluctuations in the number of eigenlevels within a spectral window. It is rigorously defined via the asymptotic linear behavior of the level number variance:

$$\langle \delta n^2(E) \rangle = \langle n^2(E)\rangle - \langle n(E)\rangle^2 \approx \chi\, \langle n(E)\rangle \;\;\;\;\; (n(E)\to\infty).$$

The value of $\chi$ serves as a phase diagnostic:

• $\chi = 1$ for Poisson (localized) statistics,
• $\chi = 0$ for Wigner-Dyson (metallic) correlations,
• $0 < \chi < 1$ for critical/multifractal regimes.

In the two-dimensional random matrix model with power-law-decaying matrix elements parameterized by $b$, a perturbative analysis yields $\chi(b) = 1 - (\pi^2 b^2)/\sqrt{2} + O(b^4)$ in the multifractal regime ($b\ll 1$), and a mapping to the non-linear $\sigma$-model in the metallic regime ($b\gg 1$) yields $\chi=0$. Numerically, the transition between these regimes is found at a critical value $b_c = 5.2\pm 0.2$. This transition is accompanied by different eigenfunction scaling:

$I_q \propto L^{-2(q-1)} \ln^{\nu_q(q-1)}L,$

for $b\gg b_c$, reflecting a metallic-critical phase with logarithmic corrections. The scaling relation $\chi + (d_1/d) = 1$ (where $d_1$ is the fractal dimension, $d$ the system dimension) generalizes this spectral-eigenstate correspondence (1111.3520).

2. Model Compressibility and Universality in Stochastic Dynamical Systems

In field-theoretic models of non-equilibrium critical phenomena—exemplified by the Ashkin-Teller-Potts model randomly advected by a compressible velocity field—compressibility directly informs universality class structure and crossover behavior. The velocity field is specified by a correlator

$D_{ij}(r) \propto P_{ij}(k) + a Q_{ij}(k),$

where $a>0$ quantifies compressibility via the longitudinal projection.

Renormalization group analysis yields $\beta$-functions for the coupling constants (e.g., $u$ for order parameter self-interaction, $w$ for advective mixing), with compressibility entering as:

$$\begin{aligned} \beta_u &= u\Bigl[-\varepsilon + R\,u + \frac{5+a}{2}w - w\,a\,f(a)\Bigr], \ \beta_w &= w\Bigl[-\xi + \frac{R_1}{6}u + \frac{5+a}{6}w\Bigr], \qquad f(a) = 2a^2+(a-1)^2. \end{aligned}$$

Here, $\varepsilon=6-d$ and $\xi$ are expansion parameters for space dimension and velocity field spectrum. Four fixed points are identified, including a novel full-scale non-equilibrium class (fixed point IV) where both self-interaction and turbulent mixing contribute. Increasing the compressibility parameter $a$ induces crossovers between universality classes or can destroy criticality altogether, depending on symmetry group invariants $R_{1,2}$. The critical dimensions and phase stability boundaries emerge as functions of $(\varepsilon, \xi, a)$, demonstrating the direct and nontrivial impact of compressibility on the critical phenomena landscape (Antonov et al., 2012).

3. Compressibility and Interactions in Lattice Models

Model compressibility exhibits unconventional behavior in interacting electron systems on lattices, notably in the Harper model with repulsive nearest-neighbor potential. Here, the compressibility is assessed via the discrete second difference of ground-state energies,

$$\Delta_2(N) = \mathcal{E}(N) - 2\mathcal{E}(N-1) + \mathcal{E}(N-2).$$

Contrary to classical expectations, increasing the repulsive interaction $U$ leads to enhanced compressibility (decreased $\Delta_2$) due to an interaction-induced flattening of the central band. This effect arises from the background density in filled bands modifying the potential landscape for the nearly flat band near half-filling, leading to increased localization (reduced wavefunction width $\xi$) and suppressed bandwidth. Mean-field theory captures the renormalized effective on-site potential and hopping,

$$\lambda_{\text{eff}} = \lambda + 2U \bar{n}(\lambda/2t),\qquad t_{\text{eff}} = t + U \bar{p}(\lambda/2t),$$

resulting in a decreased $t_{\text{eff}}/\lambda_{\text{eff}}$ and thus decreased $\xi$.

These results, obtained by DMRG and mean-field techniques, expose a counterintuitive regime where interaction-driven localization enhances compressibility. This mechanism offers insight into tuning electronic properties for quantum dot and optical lattice systems (Kraus et al., 2013).

4. Negative Compressibility and Correlation Effects

In strongly correlated multi-orbital models, such as the Hubbard–Kanamori or two-band Hubbard model with Hund's coupling, negative electronic compressibility (NEC) signals regions of thermodynamic instability and charge phase separation. Compressibility is defined as

$\kappa = x^{-2} \frac{dx}{d\mu},$

with $x$ the electron concentration and $\mu$ the chemical potential. NEC ($\kappa<0$) arises near critical values of $\mu$ where atomic ground state transitions occur (e.g., $\mu=0$, $\mu=U-3J$, etc.), causing rapid changes in electronic occupation. In these regimes, spectral functions display Mott or Slater insulating gaps, sharp spin-polaron peaks, or metallic features depending on $U$, $J$, and doping.

Slave-boson and BEG model approaches decompose $\kappa^{-1}$ into fermionic (quasiparticle) and bosonic (multiparticle, incoherent) contributions, with Hund's coupling favoring parallel-spin multi-occupancies and modifying both spectral response and phase diagram topology:

$$a^2 \kappa_f^{-1} \simeq -2 n^2 \frac{\partial^2 z^2}{\partial n^2} \frac{W}{\pi} \cos\left(\frac{\pi \delta}{4}\right).$$

Crucially, NEC allows for quantum capacitance enhancement in correlated-material-based devices:

$$(C_0/C) = 1 + \frac{2\varepsilon_0\varepsilon a^2}{e^2 d}\left(\frac{\partial\mu}{\partial n}\right),$$

enabling tunable high-capacitance heterostructures and memory devices through small density variations or resistive switching between Mott and metallic states (Sherman, 2020, Fresard et al., 2022).

5. Compressibility in Complex Network Models

The compressibility of a complex network quantifies the extent to which informative structure can be removed through clustering, subject to a rate-distortion trade-off. Modeling the network as an information source via random walks, the full "entropy" of the network is

$H(x) = \frac{1}{2E}\sum_i k_i \log k_i,$

where $k_i$ is the node degree. The compressibility $C$ is defined by the area between this full entropy and the optimal rate-distortion curve $R(S)$, computed as

$C = H(x) - \frac{1}{N} \sum_S R(S),$

for all clustering scales $S$. Explicit approximations for $R(S)$ in $k$-regular graphs are

$$\bar{R}(S) \simeq (1-S)^2 \log k + S(1-S)\log N - S\log S.$$

Empirical and analytic studies show that hierarchical organization—i.e., high clustering/transitivity, and degree heterogeneity—is the decisive network property for high compressibility. Real-world networks such as brains, language, or protein interactomes exhibit such properties, facilitating efficient, resource-constrained encoding and evolutionary persistence. As such, compressibility serves not only as a summary of modular/hierarchical structure but as a quantitative tool for designing community detection and coarse-graining algorithms (Lynn et al., 2020).

6. Compressibility Corrections in Turbulence Modeling

In turbulence modeling, specifically for Reynolds-averaged Navier–Stokes (RANS) approaches, compressibility corrections are essential for capturing variable-density and high Mach number effects. Recent developments leverage semi-local velocity scaling in the inner layer and Van Driest transformations in the outer layer to derive modifications for the turbulence kinetic energy (TKE) and eddy viscosity. The corrected TKE equation includes a compressibility source term,

$$\Phi_k = \frac{S_y}{\bar{\mu}} \frac{d}{dy}\left\{(\bar{\mu} + \sigma_k \mu_t)\frac{S_y}{\bar{\mu}} \frac{d(\bar{\rho} k)}{dy}\right\} - \frac{d}{dy}\left[(\bar{\mu} + \sigma_k \mu_t)\frac{dk}{dy}\right],$$

where $S_y$ is a wall-normal stretching function. Intrinsic compressibility corrections modify the near-wall damping of eddy viscosity using the turbulence Mach number $M_t$ and a damping function $D(R_t, M_t)$. Consistent energy equation modeling further requires careful treatment of the turbulent Prandtl number and turbulent diffusion terms.

Application of these corrections yields significant fidelity improvements in predicting velocity and temperature profiles across varied turbulent regimes, including high-speed cooled-wall boundary layers and supercritical fluids (Hasan et al., 18 Oct 2024).

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In summary, model compressibility is a unifying construct that captures (i) the rigidity of spectra in disordered/critical systems, (ii) universality class structure in non-equilibrium field theory, (iii) unconventional interaction-induced phenomena in condensed matter models, (iv) information-theoretic redundancy and structure in complex networks, and (v) correction strategies for turbulence models. Its mathematical instantiations and empirical signatures are domain-specific but share an overarching relevance to phase classification, criticality, and optimal representation under environmental or resource constraints.