Response Formulae for $n$-point Correlations in Statistical Mechanical Systems and Application to a Problem of Coarse Graining

@article{Lucarini2017ResponseFF,
  title={Response Formulae for \$n\$-point Correlations in Statistical Mechanical Systems and Application to a Problem of Coarse Graining},
  author={Valerio Lucarini and Jeroen Wouters},
  journal={arXiv: Statistical Mechanics},
  year={2017}
}
Predicting the response of a system to perturbations is a key challenge in mathematical and natural sciences. Under suitable conditions on the nature of the system, of the perturbation, and of the observables of interest, response theories allow to construct operators describing the smooth change of the invariant measure of the system of interest as a function of the small parameter controlling the intensity of the perturbation. In particular, response theories can be developed both for… 
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References

SHOWING 1-10 OF 80 REFERENCES
A statistical mechanical approach for the computation of the climatic response to general forcings
TLDR
This paper shows for the first time how the Ruelle linear response theory, developed for studying rigorously the impact of perturbations on general observables of non-equilibrium statistical mechanical systems, can be applied with great success to analyze the climatic response to general forcings.
Response Operators for Markov Processes in a Finite State Space: Radius of Convergence and Link to the Response Theory for Axiom A Systems
Using straightforward linear algebra we derive response operators describing the impact of small perturbations to finite state Markov processes. The results can be used for studying empirically
Stochastic Perturbations to Dynamical Systems: A Response Theory Approach
Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation
Fluctuations, Response, and Resonances in a Simple Atmospheric Model
Ruelle's linear response formula, ensemble adjoint schemes and Lévy flights
A traditional subject in statistical physics is the linear response of a molecular dynamical system to changes in an external forcing agency, e.g. the Ohmic response of an electrical conductor to an
Beyond the linear fluctuation-dissipation theorem: the role of causality
In this paper we tackle the traditional problem of relating the fluctuations of a system to its response to external forcings and extend the classical theory in order to be able to encompass also
Fluctuation-dissipation: Response theory in statistical physics
Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig Relations
We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We
A review of linear response theory for general differentiable dynamical systems
The classical theory of linear response applies to statistical mechanics close to equilibrium. Away from equilibrium, one may describe the microscopic time evolution by a general differentiable
Evidence of Dispersion Relations for the Nonlinear Response of the Lorenz 63 System
Along the lines of the nonlinear response theory developed by Ruelle, in a previous paper we have proved under rather general conditions that Kramers-Kronig dispersion relations and sum rules apply
...
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