Diffusion as a Signature of Chaos

Abstract: While classical chaos is defined via a system's sensitive dependence on its initial conditions (SDIC), this notion does not directly extend to quantum systems. Instead, recent works have established defining both quantum and classical chaos via the sensitivity to adiabatic deformations and measuring this sensitivity using the adiabatic gauge potential (AGP). Building on this formalism, we introduce the ``observable drift" as a probe of chaos in generic, non-Hamiltonian, classical systems. We show that this probe correctly characterizes classical systems that exhibit SDIC as chaotic. Moreover, this characterization is consistent with the measure-theoretic definition of chaos via weak mixing. Thus, we show that these two notions of sensitivity (to changes in initial conditions and to adiabatic deformations) can be probed using the same quantity, and therefore, are equivalent definitions of chaos. Numerical examples are provided via the tent map, the logistic map, and the Chirikov standard map.

• The paper establishes observable drift diffusion as a unifying signature of chaos by linking classical mixing and quantum adiabatic sensitivity measures.
• It introduces a diffusion exponent (γ) to classify dynamics into dissipative, regular, strong, or weak chaos, thus overcoming the limitations of Lyapunov-based diagnostics.
• Numerical experiments on models such as the logistic and Chirikov standard maps validate the method, offering practical insights for complex and non-Hamiltonian systems.

Diffusion as a Signature of Chaos: Unifying Classical and Quantum Diagnostics

Introduction and Motivation

The paper "Diffusion as a Signature of Chaos" (2507.18617) critically re-examines chaos detection in dynamical systems, addressing long-standing limitations in both classical and quantum domains. Classical chaos is traditionally defined via sensitive dependence on initial conditions (SDIC), typically measured using maximal Lyapunov exponents. However, this approach neither captures late-time statistical properties nor extends naturally to quantum systems, which lack classical phase-space trajectories and do not manifest SDIC in the same manner. Recent research has proposed redefining chaos—especially in quantum contexts—through the sensitivity of stationary states to adiabatic deformations, operationalized via the adiabatic gauge potential (AGP). This work rigorously formalizes and generalizes these perspectives, introducing the "observable drift" as a unifying probe for both classical and quantum chaos, applicable to non-Hamiltonian systems.

Weak Mixing, Lyapunov Exponents, and Limits of Classical Diagnostics

The paper demonstrates the inadequacy of short-time SDIC metrics, such as Lyapunov exponents, for several classes of systems. While a positive Lyapunov exponent reliably indicates initial exponential separation of trajectories, it does not encode late-time dynamical behavior, fails in the presence of nontrivial attractors or sticky phase-space regions, and is inherently observable-independent. Examples such as the symmetric tent map and solar system dynamics show that different observables can thermalize or destabilize on vastly different timescales, which the Lyapunov exponent cannot resolve. The authors advocate adopting measure-theoretic chaos, i.e., weak mixing, as a foundational criterion: an ensemble-based framework where observables' correlations decay, and localized distributions spread over phase-space irreversibly.

Chaos Detection via Sensitivity to Adiabatic Deformations

Quantum chaos has been defined through spectral statistics and, more recently, through the response of stationary states to adiabatic parameter changes. The generator of this response, AGP, quantifies the norm of the system's sensitivity, with chaotic systems exhibiting large AGP fluctuations. This framework, initially specialized for Hamiltonian quantum systems, is shown to generalize to classical contexts via Wigner-Weyl transformations. Crucially, the AGP's fluctuation statistics map directly onto autocorrelation decay in observables, providing a quantitative bridge between statistical (mixing-based) and adiabatic definitions of chaos.

Observable Drift: A Universal Probe for Chaos

To transcend the limitations of AGP (which is not universally defined for non-Hamiltonian systems), the authors introduce the observable drift: the cumulative deviation of an observable from its long-time average as one evolves along a trajectory. The mean squared displacement of this drift, ensemble averaged, serves as a chaos diagnostic. The growth rate of drift variance — quantified via a diffusion exponent $\gamma$ — classifies system behavior:

• $\gamma < 0$: Dissipative regime (contracting distributions, fixed-point attractors)
• $\gamma = 0$: Regular regime (periodic, quasi-periodic motion)
• $\gamma = 1$: Strong chaos (normal diffusion, strongly mixing/ergodic systems)
• $1 < \gamma \leq 2$: Weak chaos (anomalous diffusion, slow correlation decay)

This classification intrinsically connects observable-dependent statistical properties, transient ensemble dynamics, and equilibrium (natural) distributions—features inaccessible to Lyapunov-based analysis. The formalism yields fluctuation-dissipation identities relating drift variance to decay of autocorrelation functions, unifying ergodic theory with modern chaos probes.

Implications and Numerical Verification

Numerical experiments with the tent map, logistic map, and the Chirikov standard map showcase the method's ability to delineate dissipative, regular, and chaotic regimes, and to pinpoint transitions between them. Notably, the drift variance diagnostic detects regions of stability within globally chaotic parameter regimes (such as period-3 windows in the logistic map) and resolves mixed phase-space structures in multi-dimensional maps. The analysis reveals that crossover times between dissipative/transient behavior and chaos are generally orders of magnitude larger than Lyapunov times, an observation with implications for high-dimensional and many-body systems.

Beyond practical classification, the drift framework sharpens the observable dependency of chaos diagnostics, attributes anomalies in thermalization, and interfaces directly with quantum systems—potentially illuminating phenomena in open/dissipative quantum dynamics. The method’s relationship with the 0-1 test of chaos is noted; while both rely on diffusion of observable-driven random walks, the drift approach allows for a higher-resolution hierarchy (distinguishing weak from strong chaos) and is more straightforward to apply to general time-series data from experiments or simulations.

Theoretical and Practical Consequences

The equivalence established between mixing-driven SDIC and adiabatic sensitivity has major implications. It implies that the disordering and unpredictability typically identified in classical systems via phase-space mixing are fundamentally linked to the sensitivity diagnostics (AGP norm growth) now used in quantum chaos, as both are determined by the decay of two-point correlation functions. This connection opens several future directions: application to complex systems (including high-dimensional, many-body, and open/dissipative setups), investigation of higher moments of drift to resolve ergodic hierarchies, and systematic comparison with binary tests like the 0-1 method across broader model classes.

Conclusion

The paper rigorously demonstrates that observable drift diffusion serves as a robust, universal signature of chaos across classical and quantum dynamical systems. By connecting statistical (mixing) and adiabatic (response) definitions, it offers a principled framework for classifying system dynamics—resolving the limitations of Lyapunov diagnostics and enabling chaos detection in general non-Hamiltonian, quantum, and multi-observable contexts. The approach provides both theoretical clarity and practical computational tools, supporting experimental chaos identification and enriching understanding of thermalization and ergodicity in physical systems. Future work leveraging these insights may advance both fundamental and applied research in nonlinear dynamics, statistical physics, and quantum information.