The Fluctuation-Dissipation Relations: Growth, Diffusion, and Beyond

Abstract

In this review, we scrutinize historical and modern results on the linear response of dynamical systems to external perturbations with a particular emphasis on the celebrated relationship between fluctuations and dissipation expressed by the fluctuation-dissipation theorem (FDT). The conceptual foundation of FDT originates from the definition of the equilibrium state and Onsager's regression hypothesis. Over time, the fluctuation-dissipation relation has been vividly investigated also in systems far from equilibrium, which often exhibit wild fluctuations in measured parameters. In this review, we recall the major formulations of the FDT, including those proposed by Langevin, Onsager and Kubo. We discuss the role of fluctuations in a broad class of growth and diffusion phenomena and examine the violation of the FDT resulting from a transition from Euclidean to fractal geometry. Finally, we highlight possible generalizations of the FDT formalism and discuss situations where the relation breaks down and is no longer applicable.