Principles of equilibrium fluctuations

Fluctuations in systems maintained in dynamical equilibrium – spanning phenomena from Brownian motion to internal climate variability – are commonly analyzed through fluctuation–dissipation relations (FDRs) derived from the underlying microscopic dynamics. Such a derivation often relies on specific approximations or coarse-graining techniques, leaving the precise origin of FDRs and their connection to the governing differential equations conceptually unsettled. Using the Lorenz–63 model as a representative forced dissipative system, this paper identifies an integral fluctuation–dissipation relation (IFDR)—a FDR that constitutes (apart from a constant) the integrals of the system’s differential forcing without any approximation. The IFDR does not exist as a time rate of change and can hence not be embedded in the microscopic differential dynamics. It only emerges when the considered system is integrated forward in time. Macroscopic quantities such as variances and spectra result from the joint effect of the dissipation and fluctuation terms of the IFDR, and cannot be determined by the system’s differential forcing itself. Thus, equilibrium fluctuations of a system are governed by two principles that are complementary but not reducible to one another: the microscopic differential equations that govern individual trajectories and the IFDR that determine the macroscopic quantities. The identification of IFDR provides a deterministic foundation for equilibrium fluctuations – random solution arises internally from deterministic forward integration – and clarifies how macroscopic quantities arise intrinsically from time-integrated dynamics.