A fusion of concepts
This essay fuses concepts and approaches used to describe fluctuating phenomena in climate systems and statistical mechanics, and explores new ideas essential for understanding such phenomena. Its starting points are the Langevin equation (LE) and the fluctuation-dissipation theorem (FDT). The former was introduced to climate research by Klaus Hasselmann through his stochastic climate models. While a version of the latter, formulated within the framework of linear response theory, has found wide application, the deeper origin of the relation between fluctuations and dissipation has remained inconclusive. This essay goes one step further by seeking the cause of the apparent randomness, rather than merely describing it as in the LE, and by directly linking a fluctuation-dissipation relation to the governing microscopic equations. It postulates that such a relation, also referred to as the integral fluctuation-dissipation relation (IFDR), resides in integrals of the forcings that determine microscopic evolutions (individual trajectories) of the considered system. The IFDR ensures the emergence of well-defined macroscopic quantities, such as variance and spectra, quantities that characterize the system's fluctuations, provided the system is in dynamical equilibrium with constant external forcing. It is the dissipation embodied in IFDR that renders future states uncorrelated with initial conditions, thereby generating apparent randomness. Randomness is therefore not an unphysical artifact; on the contrary, it is a fundamental property of forced-dissipative systems in dynamical equilibrium. Fluctuation phenomena in such systems must be described by two principles: the governing microscopic equations and the IFDR. The two principles are complementary but not reducible to one another.