A fusion of concepts

TL;DR

This paper integrates concepts from climate science and statistical mechanics, linking microscopic equations and fluctuation-dissipation relations to explain the fundamental randomness in forced-dissipative systems at equilibrium.

Contribution

It advances understanding by directly connecting the fluctuation-dissipation relation to microscopic equations, explaining the origin of randomness in climate and statistical systems.

Findings

The integral fluctuation-dissipation relation resides in integrals of microscopic forcings.

Dissipation in the IFDR causes future states to be uncorrelated with initial conditions.

Randomness is a fundamental property of forced-dissipative systems in equilibrium.

Abstract

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.