Integral fluctuation-dissipation relation and turbulence as equilibrium fluctuations

Fluctuation-dissipation relation (FDR)—a well-known theorem in statistical mechanics—comes in various versions. In an early version  (Nyquist 1928, Callen and Welton, 1951), a FDR is thought to be responsible for  the emergence of dynamical equilibrium, characterized by well-defined statistics such as variances and spectra.  A later version, proposed by Kubo (1957) and  introduced to climate research by Leith (1975) and further extended by Lucarini et al. (2017),  focuses on the response of a system to an external forcing perturbation and relates this response to the system’s restoring behavior found in the absence of perturbation.  Geophysical turbulence—generated by   dissipative systems under constant external forcing  and characterized by variances and spectra conform with the given external forcing—represents fluctuations in a dynamical equilibrium. As such, it should be governed by Nyquist’s FDR. 

 

However, it is not clear how such a FDR  is related to the differential equations that govern the evolution of  turbulent flows, not mentioning the way dissipation operates and controls the statistics of turbulent flows.  The integral fluctuation-dissipation relation (IFDR) (von Storch 2026) generalizes and extends Nyquist’s FDR.  It postulates that the IFDR resides in integrals of  differential forcings that define the governing differential equations, and represents a principle that is complementary to but  distinct from these differential equations. It is complementary in the sense that turbulent flows are described not only by solutions of the differential  equations but also by statistics, such as variance and spectra, which only emerge due to  the IFDR. It is distinct in the sense that IFDR does not exist as a time rate of change and hence cannot be included in the governing differential equations. This situation is a manifestation of the fact that in a dynamical equilibrium, the differential forcing of a component x of the full state vector is effectively non-dissipative and acts as a driver of x, while dissipation of x arises from dissipative processes implemented in equations of all components that interact with x. Such a dissipation only unfolds  when the system is integrated forward in time and reaches its maximum strength for sufficiently long integration period. The IFDR is exemplified using the Lorenz 1963 model. The identification of IFDR opens a new perspective for understanding the macroscopic behaviors of turbulent flows characterized by well-defined variances and spectra.

 

von Storch 2026: https://doi.org/10.1016/j.physa.2025.131218