A relation valid near thermodynamic equilibrium that connects the linear response of a system to small external perturbations with the time-correlation functions of its spontaneous equilibrium fluctuations.

Principle

Microscopic reversibility and linear-response: the same microscopic degrees of freedom that produce equilibrium fluctuations determine how the system dissipates energy when gently driven, so response kernels are proportional to fluctuation correlators under equilibrium conditions.

Demonstration

In a prototypical overdamped stochastic system, the mobility coefficient that measures response to a small applied force is proportional to the integral of the velocity autocorrelation function at equilibrium; equivalently, the spectrum of the response function equals a symmetrized fluctuation spectrum times temperature in appropriate units.

Misapplication

Applying the theorem far from equilibrium, to strongly nonlinear perturbations, or to systems with active drives or non-reciprocal interactions; doing so leads to incorrect predictions of response magnitudes and signs.

Consequence

Allows experimental determination of response properties from passive fluctuation measurements and provides constraints on allowable forms of linear transport coefficients; it grounds many fluctuation-based parameter estimation and noise-to-signal conversion methods.

Reversal

When time-reversal symmetry is broken or driving is large, fluctuation statistics and response decouple: spontaneous fluctuations no longer predict dissipation, and one must use extended, non-equilibrium response formalisms.

Boundary

Requires near-equilibrium conditions, linear perturbations, stationarity, and typically time-reversal invariance; it does not hold universally for active matter, driven steady states with entropy production, or strongly nonlinear regimes.

Semantic Tension

Competes with non-equilibrium response theories that introduce additional terms (housekeeping fluxes, frenetic contributions) absent in the equilibrium FDT; the tension is between simplicity under equilibrium and complexity away from it.

Synthesis

The Fluctuation–Dissipation Theorem states that, near equilibrium and under linear response, measurable spontaneous fluctuations encode the system's dissipative response: correlation functions and response kernels are proportional, enabling prediction of dissipation from equilibrium noise.