Revising and Extending the Linear Response Theory for Statistical Mechanical Systems: Evaluating Observables as Predictors and Predictands

  title={Revising and Extending the Linear Response Theory for Statistical Mechanical Systems: Evaluating Observables as Predictors and Predictands},
  author={Valerio Lucarini},
  journal={Journal of Statistical Physics},
  • V. Lucarini
  • Published 11 June 2018
  • Mathematics
  • Journal of Statistical Physics
Linear response theory, originally formulated for studying how near-equilibrium statistical mechanical systems respond to small perturbations, has developed into a formidable set of tools for investigating the forced behaviour of a large variety of systems, including non-equilibrium ones. Mathematically rigorous derivations of linear response theory have been provided for systems obeying stochastic dynamics as well as for deterministic chaotic systems. In this paper we provide a new angle on… 
Predictors and predictands of linear response in spatially extended systems
A method for quantifying and ranking the predictive ability of observables is presented and used to investigate the response of a paradigmatic spatially extended system, the Lorenz '96 model, and it is shown that this approach can reveal insights on the way a signal propagates inside the system.
On Some Aspects of the Response to Stochastic and Deterministic Forcings
The perturbation theory of operator semigroups is used to derive response formulas for a variety of combinations of acting forcings and reference background dynamics. We decompose the response
Approximating linear response of physical chaos
Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these
Quadratic response of random and deterministic dynamical systems.
A general framework in which one can obtain rigorous convergence and formulas for linear and quadratic higher-order terms associated with the response of the statistical properties of a dynamical system to suitable small perturbations is shown.
Response theory and phase transitions for the thermodynamic limit of interacting identical systems
We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions.
Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator.
These findings support the physical basis and robustness of the EMR methodology and illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.
Space-split algorithm for sensitivity analysis of discrete chaotic systems with unstable manifolds of arbitrary dimension
This paper combines the concept of perturbation space-splitting regularizing Ruelle’s original expression together with measure-based parameterization of the expanding subspace to rigorously derive trajectory-following recursive relations that exponentially converge, and construct a memory-efficient Monte Carlo scheme for derivatives of the output statistics.
Climate change in mechanical systems: the snapshot view of parallel dynamical evolutions
We argue that typical mechanical systems subjected to a monotonous parameter drift whose timescale is comparable to that of the internal dynamics can be considered to undergo their own climate
Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory
A theory of Ruelle–Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given
Response and flux of information in extended nonequilibrium dynamics.
This work investigates spatially asymmetric extended systems and considers a simplified linear stochastic model, which can be studied analytically and includes nonlinear terms in the dynamics of the Lorenz 96 model for Earth oceanic circulation.


A statistical mechanical approach for the computation of the climatic response to general forcings
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.
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
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
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
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
On the Validity of Linear Response Theory in High-Dimensional Deterministic Dynamical Systems
This theoretical work considers the following conundrum: linear response theory is successfully used by scientists in numerous fields, but mathematicians have shown that typical low-dimensional
Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach
We consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system, which are weakly coupled to the remaining variables. In a previous paper we have
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