Théorie de la réponse linéaire

Résumé

En physique statistique hors d'équilibre, la théorie de la réponse linéaire permet de définir les susceptibilités et les coefficients de transport d'un système au voisinage de l'équilibre thermique indépendamment des détails du modèle. La théorie de la réponse linéaire a été développée dans les années 1950 par Melville Green, Herbert Callen et Ryōgo Kubo. Formalisme général Le hamiltonien du système Dans la théorie de la réponse linéaire, on suppose que le système considéré est décrit par un certain hamiltonien d'équilibre H_0, perturbé par un hamiltonien perturbatif dépendant du temps H_1(t), qu'on peut expliciter sous la forme: H_1(t)=\sum_l \lambda_l(t)A_l ou les \lambda_i(t) sont les facteurs perturbatifs et les opérateurs hermitiens A_i sont des observables du système, de sorte que le hamiltonien total du système est: H=H_0+H_1(t)=H_0+\sum_l \lambda_l(t) A_l Il en resso

Source officielle

https://fr.wikipedia.org/wiki/Th%C3%A9orie_de_la_r%C3%A9ponse_lin%C3%A9aire

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Predictors and predictands of linear response in spatially extended systems

Umberto Maria Tomasini

The goal of response theory, in each of its many statistical mechanical formulations, is to predict the perturbed response of a system from the knowledge of the unperturbed state and of the applied perturbation. A new recent angle on the problem focuses on providing a method to perform predictions of the change in one observable of the system using the change in a second observable as a surrogate for the actual forcing. Such a viewpoint tries to address the very relevant problem of causal links within complex system when only incomplete information is available. We present here a method for quantifying and ranking the predictive ability of observables and use it to investigate the response of a paradigmatic spatially extended system, the Lorenz '96 model. We perturb locally the system and we then study to what extent a given local observable can predict the behaviour of a separate local observable. We show that this approach can reveal insights on the way a signal propagates inside the system. We also show that the procedure becomes more efficient if one considers multiple acting forcings and, correspondingly, multiple observables as predictors of the observable of interest.

SPRINGER HEIDELBERG2021

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Fluctuation-dissipation relations in the absence of detailed balance: formalism and applications to active matter

Sara Dal Cengio, Ignacio Pagonabarraga Mora

We present a comprehensive study about the relationship between the way detailed balance is broken in non-equilibrium systems and the resulting violations of the fluctuation-dissipation theorem. Starting from stochastic dynamics with both odd and even variables under time-reversal, we derive an explicit expression for the time-reversal operator, i.e. the Markovian operator which generates the time-reversed trajectories. We then exploit the relation between entropy production and the breakdown of detailed balance to establish general constraints on the non-equilibrium steady-states (NESS), which relate the non-equilibrium character of the dynamics with symmetry properties of the NESS distribution. This provides a direct route to derive extended fluctuation-dissipation relations, expressing the linear response function in terms of NESS correlations. Such framework provides a unified way to understand the departure from equilibrium of active systems and its linear response. We then consider two paradigmatic models of interacting self-propelled particles, namely active Brownian particles and active Ornstein-Uhlenbeck particles. We analyze the non-equilibrium character of these systems (also within a Markov and a Chapman-Enskog approximation) and derive extended fluctuation-dissipation relations for them, clarifying which features of these active model systems are genuinely non-equilibrium.

IOP PUBLISHING LTD2021

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We address the question of how interacting active systems in a nonequilibrium steady state respond to an external perturbation. We establish an extended fluctuation-dissipation theorem for active Brownian particles (ABP), which highlights the role played by the local violation of detailed balance due to activity. By making use of a Markovian approximation we derive closed Green-Kubo expressions for the diffusivity and mobility of ABP and quantify the deviations from the Stokes-Einstein relation. We compute the linear response function to an external force using unperturbed simulations of ABP and compare the results with the analytical predictions of the transport coefficients. Our results show the importance of the interplay between activity and interactions in the departure from equilibrium linear response.

2019

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