Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1103.0237

Stochastic perturbations to dynamical systems: a response theory approach

Valerio Lucarini
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the impact of stochastic perturbations to deterministic dynamical systems using the formalism of the Ruelle response theory and explore how stochastic noise can be used to explore the properties of the underlying deterministic dynamics of a system. We find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the case of additive and multiplicative noise. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. Using Kramers-Kronig theory, it is then possible to derive the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide a example of application of our results by considering the Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations.

Explore related subjects

cond-mat.stat-mechnlin.AOnlin.CDphysics.ao-phphysics.comp-phphysics.flu-dyn

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Valerio Lucarini. 2011-09-12. Stochastic perturbations to dynamical systems: a response theory approach. https://doi.org/10.1007/s10955-012-0422-0

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Breakdown of Adiabatic Scaling and Noise-induced Functional Synchronization in Deeply Quiescent Excitable Systems

Coherence resonance (CR) characterizes noise-induced regularity in excitable systems, yet its evaluation in quiescent biological media is often obscured by flattened energy landscapes and complex nonlinear dynamics. In this study, we investigate the stochastic dynamics of a 3D Sherman-Rinzel-Keizer (SRK) model driven by multiplicative Feller noise. We show that traditional extremal evaluations of CR encounter a "bathtub effect", a broad resonance valley that can lead to statistical inaccuracies. To address this, we propose a logarithmic centroid extraction method, which filters out stochastic jitter and recovers the underlying adiabatic Kramers scaling with high linearity. Furthermore, we identify the physical boundary where this adiabatic approximation breaks down under the strong-noise limit. Extending our analysis to gap-junction coupled systems, we observe a noise-induced transition from sub-threshold physiological shivering (characterized by statistical correlation but negligible functional output) to macroscopic functional synchronization. Our results provide a mathematical framework for extracting optimal noise intensities in broad energy valleys and offer insights into how quiescent biological systems utilize stochastic fluctuations for functional recovery.

cond-mat.stat-mech↗
arXiv2026

Quantum Stochastic Walks on the Permutation Group

How rapidly does order give way to randomness, and can quantum coherence accelerate this process? We address these questions through the paradigmatic problem of card shuffling, formulated as a random walk on the symmetric group $S_n$. We first recast the random-transposition walk studied by Diaconis and Shahshahani, as well as more general walks generated by conjugacy classes of $S_n$, in continuous time. We then identify the transition matrix of each classical walk with a permutation Hamiltonian generating a corresponding unitary quantum walk. Purely unitary evolution, however, does not generically converge to the uniform distribution in the classical sense of mixing: coherence preserves information rather than erasing it. We therefore embed the problem into a quantum stochastic walk, where coherent dynamics competes with the dissipative process responsible for classical mixing. In this setting, quantum coherence assists randomization. We prove that it can only decrease the distance from the uniform distribution in the computational basis and can therefore accelerate mixing. An analysis of the slowest mode yields a criterion for the coupling strength required to produce an appreciable speedup. Finally, numerical results reveal a scaling collapse of the ratio between quantum and classical mixing times onto a simple one-parameter form. Our results illustrate how coherence and dissipation can cooperate in the emergence of randomness in walks on permutation groups.

cond-mat.stat-mech↗
arXiv2026

Exact Nonperturbative Equilibrium Mode Statistics in Nonlinear Wave and Lattice Systems

We derive exact finite-size nonperturbative representations of equilibrium modal occupations and related statistics for three representative nonlinear systems: the Majda-McLaughlin-Tabak dispersive-wave model, the Fermi-Pasta-Ulam-Tsingou beta anharmonic chain, and the discrete nonlinear Schrodinger lattice field. Independent simulations confirm the predictions from weak to strong nonlinearity. For DNLS, the theory remains accurate across the weak-coupling quasicondensation crossover, where large low-mode occupations and long-range coherence amplify interaction effects even when the bare nonlinear coefficient is small. The finite-ring DNLS occupations are further resolved into a positive sum of Rayleigh-Jeans channels with distinct correlation lengths, explaining when a single Rayleigh-Jeans law applies and why it fails near quasicondensation. In MMT and DNLS, the exact occupations also determine the mean modal frequencies even when the dynamical spectra broaden or split. The nonperturbative results allow a direct assessment of two representative perturbative approaches. Treating the mean interaction appropriately yields accurate low-order approximations, including at strong nonlinearity. At higher orders, however, the corrections cease to decrease and successive approximations oscillate with increasing amplitude; both finite-order approaches also fail near weak-coupling quasicondensation. Thus neither low-order agreement nor a small bare coupling guarantees a reliable perturbative description. The results establish nonperturbative equilibrium theory for widely used nonlinear wave and lattice models and provide a quantitative basis for modal distributions of energy, particles, and optical power in nonlinear optics, dispersive waves, anharmonic lattices, and cold-atom systems.

cond-mat.stat-mech↗