Introduction to Focus Issue: Linear response theory: Potentials and limits

Abstract

Linear response theory (LRT) constitutes a cornerstone of statistical mechanics. Developed in the 1960s for thermostatted Hamiltonian systems, applications now include modern areas of research such as neurodynamics and climate science. If a system has linear response, one can estimate the change of expectation values caused by a perturbation using only information of the unperturbed system. LRT has been successfully applied for many dynamical systems in a predictive mode to determine their response to prescribed perturbations in the realistic case when the equilibrium density is not known. LRT also found its way into science as a tool to design and calibrate model reductions of high-dimensional complex systems. Almost independently from these success stories in applying LRT to understanding and controlling the natural world, mathematicians studied the dynamical ingredients necessary in a system to assure its linear response behavior, and found that many simple dynamical systems actually fail to obey LRT. Understanding, applying, and developing LRT remains an exciting and important endeavor. This Focus Issue brings together physicists and mathematicians from several areas to provide a state-of-the-art perspective.