Linear Response and Fluctuation-Dissipation

Linear-response theory expresses how a system reacts to weak perturbations in terms of its equilibrium fluctuations, with the fluctuation-dissipation theorem tying dissipation to spontaneous noise.

Definition

Linear-response theory describes a system's response to a weak external perturbation to first order through a response function, and the fluctuation-dissipation theorem states that this dissipative response is determined by the correlation function of the system's spontaneous equilibrium fluctuations.

Scope

This topic covers the linear-response function and generalized susceptibility, the Kubo formula expressing transport coefficients as equilibrium time-correlation functions, the fluctuation-dissipation theorem relating the dissipative response to the spectrum of fluctuations, the Green-Kubo relations for viscosity and conductivity, the Johnson-Nyquist noise as a concrete instance, and Onsager's reciprocal relations as a near-equilibrium constraint.

Core questions

• How does a weak perturbation elicit a response governed by equilibrium correlation functions?
• What does the fluctuation-dissipation theorem say about the link between noise and dissipation?
• How do the Green-Kubo relations express transport coefficients as time-correlation integrals?
• Why do Onsager's reciprocal relations follow from microscopic time-reversal symmetry?

Key concepts

• Response function and generalized susceptibility
• Kubo formula and time-correlation functions
• Fluctuation-dissipation theorem
• Green-Kubo relations
• Johnson-Nyquist noise

Key theories

Fluctuation-dissipation theorem

The dissipative part of a system's linear response to a perturbation is proportional to the power spectrum of its equilibrium fluctuations of the conjugate variable, so the same microscopic processes cause both spontaneous noise and energy dissipation.

Onsager reciprocal relations

Near equilibrium the kinetic coefficients linking thermodynamic forces to fluxes form a symmetric matrix, a constraint on coupled transport phenomena derived from the time-reversal symmetry of the microscopic dynamics.

Clinical relevance

Linear-response theory provides the standard route to computing transport coefficients in simulations and connects measured noise to dissipation in resistors and detectors, while Onsager's relations govern coupled effects such as thermoelectricity used in sensors and energy harvesting.

History

Onsager's 1931 reciprocal relations and the Johnson-Nyquist analysis of thermal noise foreshadowed a general principle that Callen and Welton stated as the fluctuation-dissipation theorem in 1951, and that Kubo cast into the linear-response formalism in the 1950s.

Key figures

• Lars Onsager
• Ryogo Kubo
• Herbert Callen
• Harry Nyquist

Seminal works

• onsager1931
• reif1965

Frequently asked questions

What is the intuition behind the fluctuation-dissipation theorem?

The same random molecular kicks that make a quantity fluctuate at equilibrium are responsible for the friction that dissipates energy when the system is driven, so measuring spontaneous fluctuations tells you how strongly the system will dissipate under a perturbation.