Fluctuation–dissipation theorem

The fluctuation–dissipation theorem states that the random fluctuations a system shows spontaneously at equilibrium and the dissipation it exhibits when driven by an external force are not independent properties but are locked together, with the thermal energy k_BT as the constant of proportionality. The same microscopic agitation that makes a quantity fluctuate on its own is what resists an applied force, so a system cannot dissipate without fluctuating, nor fluctuate without dissipating.

Concrete instances include Einstein's relation D = μk_BT between the diffusion constant (a fluctuation) and the mobility (a dissipation) of a Brownian particle, and the Johnson–Nyquist formula V² = 4k_BTRΔf linking a resistor's voltage noise to its resistance. In each case a quantity measuring spontaneous wandering is fixed by a quantity measuring frictional response.

The theorem is a cornerstone of non-equilibrium statistical mechanics and of linear-response theory: it allows transport coefficients such as conductivity, viscosity and diffusion to be computed from equilibrium correlation functions of the corresponding fluctuations.

Anticipated by Einstein (1905) and Nyquist (1928) in special cases, proved in general by Herbert Callen and Theodore Welton in 1951, and cast in its definitive linear-response form by Ryogo Kubo in 1957.