Fluctuation–dissipation theorem

A family of relations connecting a system’s linear response to external perturbations with the spontaneous fluctuations of its observables at thermal equilibrium.

Linear response theory: A framework for calculating how a system changes under small external perturbations. The theorem supplies equilibrium formulas for the response functions studied in this framework.
Statistical mechanics: The framework connecting microscopic states and probabilities to macroscopic physical properties. Its equilibrium ensembles define the fluctuations used in the theorem.
Brownian motion: The irregular motion of particles suspended in a fluid, driven by collisions with surrounding molecules. The Einstein relation connects particle diffusion to mobility through thermal fluctuations.
Nonequilibrium statistical mechanics: The study of systems whose macroscopic states are not in thermodynamic equilibrium. Away from equilibrium, the standard theorem generally needs modification or loses its simple form.
Kubo formula: A formula expressing a system’s linear response through equilibrium correlations of the perturbation-coupled observable. It is a central quantum and statistical-mechanical form of the theorem.
Thermal equilibrium: A state in which macroscopic properties remain steady and no net heat flows between coupled parts. The theorem’s standard relations require equilibrium before perturbation.
Einstein relation: A relation connecting a diffusing particle’s diffusion coefficient and mobility to temperature. It is a classic low-frequency fluctuation–dissipation result for Brownian particles.
Detailed balance: A condition in which transitions between states balance their reverse transitions at equilibrium. Its equilibrium constraints underpin familiar fluctuation–dissipation relations.
Green–Kubo relations: Transport coefficients expressed as time integrals of equilibrium current-correlation functions. These relations turn the theorem into formulas for viscosity, diffusion, and thermal conductivity.
Correlation function: A function measuring statistical dependence between values of observables at different points or times. Equilibrium time correlations encode the spontaneous fluctuations entering the theorem.

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