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.
Highlights
1.1 A Brief Summary of Response TheoryThe development of methods for computing the response of a complex system to small perturbations affecting its dynamics is the subject of very active investigation in many fields of science and of technology
We want to rephrase the previous results in the context of continuous time dynamical systems and derive some formulas previously presented in the literature concerning Axiom A systems
While previous approaches focus on the constructing a theory able to account for the effect of adding small perturbations to the baseline flow, we focus on computing the change in the invariant measure and for the change in the expectation values of general observables occurring when the Markov transition matrix
Summary
The development of methods for computing the response of a complex system to small perturbations affecting its dynamics is the subject of very active investigation in many fields of science and of technology. Non-trivial implications of the nonequilibrium/equilibrium dichotomy regarding the validity of the fluctuation-dissipation relations are discussed in [2,5,7], while the a physical interpretation of the first and second order terms occurring in Ruelle’s response formalism is provided in [8] At this stage one needs to bridge the gap between mathematical formalism and physical meaningfulness, One manages to bring Ruellle’s formalism into the realm of applicability by adopting the chaotic hypothesis [9,10], which basically says that a high-dimensional chaotic physical system can be treated at all practical purposes as if it were Axiom A if we focus on macroscopic observables.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
