Response Operators for Markov Processes in a Finite State Space: Radius of Convergence and Link to the Response Theory for Axiom A Systems

TL;DR

This paper derives explicit response operators for finite state Markov processes, linking their convergence properties to the response theory of Axiom A systems, and demonstrates their practical application to models like Lorenz '63.

Contribution

It provides closed-form formulas for response operators at all perturbation orders, with bounds on convergence radius, applicable to empirical finite state approximations of complex systems.

Findings

Derived response operators for finite state Markov processes.

Linked convergence of response theory to system mixing rates.

Applied formulas to Lorenz '63 model with promising results.

Abstract

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 constructed - e.g. from observations or through coarse graining of model simulations - finite state approximation of statistical mechanical systems. Recent results concerning the convergence of the statistical properties of finite state Markov approximation of the full asymptotic dynamics on the SRB measure in the limit of finer and finer partitions of the phase space are suggestive of some degree of robustness of the obtained results in the case of Axiom A system. Our findings give closed formulas for the linear and nonlinear response theory at all orders of perturbation and provide matrix expressions that can be directly implemented in any coding language, plus providing bounds on the radius of convergence of the perturbative theory. In particular, we relate the convergence of the response theory to the rate of mixing of the unperturbed system. One can use the formulas obtained for finite state Markov processes to recover previous findings obtained on the response of continuous time Axiom A dynamical systems to perturbations, by considering the generator of time evolution for the measure and for the observables. A very basic, low-tech, and computationally cheap analysis of the response of the Lorenz '63 model to perturbations provides rather encouraging results regarding the possibility of using the approximate representation given by finite state Markov processes to compute the system's response.