Gaussian Complexity in Information Geometry

• Gaussian Complexity is a framework that quantifies the structure and exploration volume of statistical manifolds via Gaussian distributions and the Fisher-Rao metric.
• It reveals how microcorrelations deform the metric, rapidly reducing effective macrostates and constraining a system's degrees of freedom.
• The analysis shows that geodesic flows decay in a power law, linking microscopic dependencies to the macroscopic compression of accessible states.

Gaussian complexity in the context of statistical models, information geometry, and dynamical systems quantifies the volume and structure of the statistical manifold explored by a family of Gaussian distributions under constraints such as microscopic correlations. It encapsulates how the geometry of parameter space, as measured by the Fisher-Rao metric, controls the effective macroscopic degrees of freedom and the rate at which a system's accessible macrostates, or high-probability configurations, contract along geodesics (most probable paths) over time. Microcorrelations among the microscopic degrees of freedom (microvariables) compress this exploration and thus reduce the effective "complexity" at the macroscopic level by tightening the dynamical statistical volume explored.

1. Information Geometric Complexity: Statistical Manifold and Fisher-Rao Metric

The core construct is the Information Geometric Complexity (IGC), a metric of temporal complexity in the Information Geometrodynamical Approach to Chaos (IGAC). For a finite-dimensional Gaussian model parameterized by a vector θ, the statistical manifold is endowed with the Fisher-Rao metric: $$g_{\mu \nu}(\theta) = \int dx\, P(x|\theta)\, [\partial_\mu \log P(x|\theta)][\partial_\nu \log P(x|\theta)]$$ This metric quantifies local distinguishability between probability distributions in parameter space. The geodesic flow on this manifold, defined by solutions to the geodesic equation,

$$\frac{d^2 \theta^k}{d\tau^2} + \Gamma^k_{\mu\nu} \frac{d\theta^\mu}{d\tau} \frac{d\theta^\nu}{d\tau} = 0$$

traces the evolution of the system’s macroscopic states (macrostates), with τ as a dynamical variable (e.g. time).

The IGC is then defined as the logarithm of the averaged dynamical statistical volume: $$\text{IGC}(\tau) = S_\text{math}(\tau) = \log\big[\text{vol}(\mathcal{D}_\text{geodesic}(\tau))\big]$$

$$\text{vol}(\mathcal{D}_\text{geodesic})(\tau) = \lim_{T \to \infty} \frac{1}{T} \int_0^T \left(\int_\mathcal{D} d\theta \sqrt{\det g_{\mu\nu}(\theta)}\right) d\tau'$$

Here, $\mathcal{D}_\text{geodesic}(\tau)$ is the region traced by the geodesic path up to τ and the volume element $\sqrt{\det g_{\mu\nu}(\theta)}$ arises from the Fisher density. Higher IGC indicates larger, more slowly compressing accessible macrostate regions; lower IGC indicates rapid contraction—information geometric compression.

2. Microcorrelations and Metric Deformation

Microcorrelations are statistical dependencies between microscopic variables $x_i$, $x_j$, characterized in the Gaussian ensemble by the correlation coefficient

$$r \equiv \frac{\langle (x_i - \langle x_i \rangle)(x_j - \langle x_j \rangle) \rangle}{\sigma_i \sigma_j}$$

with $n_i = (x_i - \langle x_i \rangle)/\sigma_i$ as the normalized form.

In the absence of microcorrelations, the Fisher-Rao metric is diagonal (variables are independent). Correlations introduce off-diagonal (covariance) terms, altering the geometry of the statistical manifold. The resulting deformation encodes not just diagonal local “spread” but also how changes in one variable propagate through the manifold, which in turn impacts the geodesic flows and hence the effective exploration of macrostate space.

The crucial observation is that as $r$ increases, the statistical volume (effective number of accessible macrostates) contracts more rapidly. Microcorrelations induce constraints that geometrically “compress” the set of macrostates, effectively reducing the complexity that can be generated at the macroscopic level.

3. Asymptotic Temporal Evolution and Power Law Decay

The central analytical result concerns the long-time, $\tau \to \infty$, scaling of the statistical volume and thus the IGC. In the presence of microcorrelations, the geodesic statistical volume decays as a power law: $$\text{vol}(\mathcal{D}_\text{geodesic})(\tau)_{\mathrm{corr}} \approx \frac{a^2 g'_2(r)}{200 A(r)} \frac{1}{\tau}$$ where $A(r)$ is a function of constants and the correlation parameter $r$, and $a$ relates to integration constants of the geodesic equations. This $1/\tau$ behavior reflects that the statistical region shrinks with increasing $\tau$, with the decay coefficient modulated by the correlation strength. Specifically, larger $r$ (in the range $0 < r < 1$) yields a faster rate of “information geometric compression.”

Consequently, the IGC, as the logarithm of this volume, manifests a linearly decreasing behavior with log-time scaling, and the slope of the decay is tuned by $r$.

4. Information Geometric Compression and Macrostate Accessibility

To quantitatively assess the effect of microcorrelations, the ratio

$$\mathcal{F}_{\mathrm{MS}}(r) = \frac{\text{vol}(\mathcal{D}_\text{geodesic})(\tau; r)}{\text{vol}(\mathcal{D}_\text{geodesic})(\tau; r=0)}$$

is shown to be a monotonically decreasing function of $r$. Thus, even moderate microcorrelations can substantially accelerate the shrinkage of the space of accessible macrostates. This translates, at the macroscopic level, to a reduction in the effective degrees of freedom: the dynamics are increasingly confined as more microscopic information is locked into statistical dependencies.

This phenomenon is referred to as “information geometric compression.” It has direct analogies to notions in quantum systems, where, for instance, entanglement (as a form of microcorrelation) restricts the possible macro-observables by coherently correlating otherwise independent subsystems.

5. Mathematical Summary

The relevant mathematical relationships are as follows:

Concept	Formula or Definition	Description
Fisher–Rao metric	$$g_{\mu\nu}(\theta) = \int dx\, P(x|\theta)\, [\partial_\mu \log P(x|\theta)][\partial_\nu \log P(x|\theta)]$$	Local geometry of the statistical manifold
Info. geometric entropy (IGE)	$$S_{\mathrm{MS}}(\tau) = \log \operatorname{vol}\big[\mathcal{D}_\mathrm{geodesic}(\tau)\big]$$	Logarithmic measure of visited macrostate volume
Statistical volume	$$\operatorname{vol}\big[\mathcal{D}_\text{geodesic}(\tau)\big] = \int_\mathcal{D} d\theta \sqrt{\det g_{\mu\nu}(\theta)}$$	Accessible region of parameter space under geodesic flow
Power law decay (asymptotics)	$$\operatorname{vol}(\mathcal{D}_\text{geodesic})(\tau)_{\mathrm{corr}} \sim \tau^{-1}$$	Volume shrinks at rate determined by microcorrelation
Microcorrelation coefficient	$$r = \frac{(x_i - \langle x_i \rangle)(x_j - \langle x_j \rangle)}{\sigma_i \sigma_j}$$	Quantifies pairwise correlation between microvariables

All significant temporal and geometric complexity features emerge from the interplay between the microlevel correlations (as encoded by $r$) and the induced geometry on the statistical manifold.

6. Microcorrelations and Reduced Macrocomplexity: Broader Implications

The findings rigorously demonstrate that the information geometric complexity of a Gaussian system is not static, but sensitive to statistical dependencies at the microscopic scale. As microcorrelations strengthen, the system rapidly loses macroscopic "degrees of freedom", with the geodesic flow on the statistical manifold quickly settling onto a narrow manifold (in the sense of Fisher information volume).

This mechanism provides a bridge from microlevel interactions—such as correlated noise, environmental coupling, or quantum entanglement—to emergent macrolevel simplicity or rigidity. It suggests that observing rapid decoherence or loss of macroscopic diversity in complex systems could be interpreted in terms of geometric compressions induced by hidden correlation structure. The analytical treatment also enables the tuning of model complexity by controlling microcorrelations, directly linking them to statistical prediction and inference efficiency.

Moreover, this paradigm has interpretations and applications in high-dimensional data analysis, quantum statistical models, and systems biology, wherever the emergence of effective macrostate reduction can be traced to statistical structure at the microscopic level.

7. Conclusion

The analysis of Gaussian complexity through information geometry shows that the evolution and reduction of accessible macrostatistical states in correlated Gaussian systems is precisely controlled by the Fisher-Rao metric and the microcorrelation structure. As microcorrelations increase, the information geometric complexity exhibits a robust power law decay, leading to a faster asymptotic compression of macrostates and a reduced effective macroscopic complexity. The results provide a quantitative, geometrically motivated framework for connecting microscopic statistical mechanics to macroscopic system behavior and complexity, with rigorous expressions for the dependence on microcorrelation parameters.