For a system in thermal equilibrium and within the linear-response regime: the linear response of an observable to a small external perturbation is directly related to the equilibrium correlation (fluctuation) function of that observable; equivalently, the spectrum of spontaneous fluctuations determines the dissipative (imaginary) part of the response function.

Principle

At thermal equilibrium, dissipation (the system's irreversible response to perturbations) and spontaneous fluctuations share the same underlying physical origins so that measurement of one constrains the other via temperature-dependent proportionality factors.

Demonstration

Illustrative scenario — A micron-sized particle in a fluid at temperature T experiences random thermal forces (Brownian motion). Its velocity autocorrelation function, measured without applied force, determines its mobility (response to a small applied force) through the fluctuation–dissipation relation (Einstein relation in this context).

Misapplication

Applying the theorem to systems far from thermal equilibrium (e.g., driven steady states, active matter) or to strong, nonlinear perturbations yields incorrect predictions because the required equilibrium and linear-response assumptions fail.

Consequence

Provides a method to obtain response coefficients (viscosity, electrical conductivity, mobility) from passive fluctuation measurements and sets consistency checks on noise levels in physical and electronic systems at a given temperature.

Reversal

In non-equilibrium steady states or systems with active energy input, modified fluctuation–dissipation relations with extra terms or entirely different relations are required; microscopic detailed balance breakdown invalidates the standard form.

Boundary

Valid for systems at thermal equilibrium, in the linear-response regime, and for observables whose correlation functions are well defined; it excludes strongly nonlinear responses, explicitly time-dependent driving, and non-thermal noise sources unless generalized forms are used.

Semantic Tension

The theorem ties equilibrium statistical mechanics to measurable transport properties, creating tension with approaches that treat fluctuations and dissipation as independent in driven or biological systems.

Synthesis

FDT operationally links passive observation and active perturbation: under equilibrium and linearity, one can deduce how a system will respond to small forces solely from its spontaneous fluctuations.