2 - Fluctuation–dissipation relations

Summary

In any physical system the dependence of the fluctuations' correlation function on time or, equivalently, the frequency dependence of the spectral density, on one hand, and the response of the same system to external perturbation, on the other hand, are governed by the same kinetic processes, and one can expect that there is some relationship between the two kinetic characteristics of the system. For example, the velocity correlation function of a Brownian particle t ψ(t1 – t2) decays exponentially with t1–t2 (Sec. 1.9). The corresponding relaxation time τ depends on the viscosity of the liquid, on the mass and linear dimensions of the particle. If the Brownian particle is brought into motion by an external perturbation (e.g., by an electric field if the particle is charged) the particle's stationary velocity and the time of its acceleration and deceleration after switching off the force are determined by the same parameters and, consequently, by the same relaxation time τ.

Such qualitative considerations are usually true for any physical system. However, for equilibrium systems an exact relationship holds between the spectral density of fluctuations at any given frequency f and that part of the linear response of the same system to an external perturbation of the same frequency f, which corresponds to the dissipation of the power of the perturbation. This fundamental relation is called the fluctuation–dissipation relation (FDR), or theorem (FDT).