Exact response theory and Kuramoto dynamics

Amadori, Debora; Colangeli, Matteo; Correa, Astrid; Rondoni, Lamberto

doi:10.1016/j.physd.2021.133076

Cited by 3 publications

(3 citation statements)

References 35 publications

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“…The above is in addition to the fact that many theoretical results can be derived from the exact response formula before they can be numerically computed in simulations, e.g., see [7,40,41]. Moreover, the linear response cannot treat phase transitions [42], and various studies show an impressively better performance of the TTCF formula compared to other averaging methods for time-independent perturbations [5,6]. We expect this to be the case with our approach when dealing with time-dependent (deterministic or stochastic) perturbations.…”

Section: Discussionmentioning

confidence: 99%

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Exact Response Theory for Time-Dependent and Stochastic Perturbations

Iannella,

Rondoni

2023

Entropy

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The exact, non perturbative, response theory developed within the field of non-equilibrium molecular dynamics, also known as TTCF (transient time correlation function), applies to quite general dynamical systems. Its key element is called the dissipation function because it represents the power dissipated by external fields acting on the particle system of interest, whose coupling with the environment is given by deterministic thermostats. This theory has been initially developed for time-independent external perturbations, and then it has been extended to time-dependent perturbations. It has also been applied to dynamical systems of different nature, and to oscillator models undergoing phase transitions, which cannot be treated with, e.g., linear response theory. The present work includes time-dependent stochastic perturbations in the theory using the Karhunen–Loève theorem. This leads to three different investigations of a given process. In the first, a single realization of the stochastic coefficients is fixed, and averages are taken only over the initial conditions, as in a deterministic process. In the second, the initial condition is fixed, and averages are taken with respect to the distribution of stochastic coefficients. In the last investigation, one averages over both initial conditions and stochastic coefficients. We conclude by illustrating the applicability of the resulting exact response theory with simple examples.

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Section: Discussionmentioning

confidence: 99%

“…Keeping the initial condition (Γ, θ, ϕ) ∈ M fixed, we denote by ⟨ • ⟩ (st) the averages made with respect to all realizations of the stochastic perturbation. Consider the equation of motion (42) and its first cumulant: ˙ Γ…”

Section: Stochastic Coefficients With Fixed Initial Conditionmentioning

confidence: 99%

Exact Response Theory for Time-Dependent and Stochastic Perturbations

Iannella,

Rondoni

2023

Entropy

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“…Linear response theory has been used to study sensitivity and transport properties of a plethora of physical problems and systems, including stochastic resonance [21], optical materials [22], simple toy models of chaotic dynamics [23][24][25][26], Markov chains [27][28][29], neural networks [30], turbulence [31], galactic dynamics [32], financial markets [33], plasma [34], interacting identical agents [35][36][37], optomechanical systems [38], and the climate system [39][40][41][42][43], just to name a few; see also the many examples discussed in [44], the theoretical developments presented in [5,[45][46][47][48][49], and the recent special issue edited by Gottwald [50].…”

Section: Introductionmentioning

confidence: 99%

On Some Aspects of the Response to Stochastic and Deterministic Forcings

Gutiérrez¹,

Lucarini²

2022

Preprint

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On some aspects of the response to stochastic and deterministic forcings

Gutiérrez¹,

Lucarini²

2022

J. Phys. A: Math. Theor.

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The perturbation theory of operator semigroups is used to derive response formulas for a variety of combinations of acting forcings and reference background dynamics. In the case of background stochastic dynamics, we decompose the response formulas using the Koopman operator generator eigenfunctions and the corresponding eigenvalues, thus providing a functional basis towards identifying relaxation timescales and modes in physically relevant systems. To leading order, linear response gives the correction to expectation values due to extra deterministic forcings acting on either stochastic or chaotic dynamical systems. When considering the impact of weak noise, the response is linear in the intensity of the (extra) noise for background stochastic dynamics, while the second order response given the leading order correction when the reference dynamics is chaotic. In this latter case we clarify that previously published diverging results can be brought to common ground when a suitable interpretation - Stratonovich vs. Ito - of the noise is given. Finally, the response of two-point correlations to perturbations is studied through the resolvent formalism via a perturbative approach. Our results allow, among other things, to estimate how the correlations of a chaotic dynamical system changes as a results of adding stochastic forcing.

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Exact response theory and Kuramoto dynamics

Cited by 3 publications

References 35 publications

Exact Response Theory for Time-Dependent and Stochastic Perturbations

Exact Response Theory for Time-Dependent and Stochastic Perturbations

On Some Aspects of the Response to Stochastic and Deterministic Forcings

On some aspects of the response to stochastic and deterministic forcings

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