Detailed Balance, Fluctuations and Dissipation

Abstract

Relations between spontaneous and forced deviations from equilibrium occur over and over again in statistical mechanics. It starts already in equilibrium situations with the well-known relations between fluctuations and static suscep-tibilities [Ref. 5.1, Sects. 14,49,50]. In qualitative terms: the larger the response to external perturbations, the larger the spontaneous fluctuations. Another example from equilibrium theory is Planck's formula (8.1) for the amplitude fluctuations of a harmonic oscillator. Keeping in mind the relation mW5 = f between the force constant f and eigenfrequency wo of the oscillator one has again: small force constant-large susceptibility-large amplitude fluctuations. Note in addition that this holds not only for the (classical) thermal fluctuations at large temperatUres kT ~ liw, but also for the (quantum mechanical) zero-point fluctuations at T = O. Einstein's relation D = BkT between the diffusion constant D and mobility B is the first one connecting a fluctuation quantity D with a dissipative one B. We have discussed the connections between diffusion and the position fluctuations of Brownian particles in Chap.4, in particular (4.14). Einstein's relation was extended to a relation between noise and dissipation in electric circuits by Schottky and NyquisL Callen and Welton have put it into its general form, which we are now going to derive. The starting point of this celebrated fluctuation-dissipation theorem, as it is now called, is an innocent looking symmetry relation between product correlations that we have come across several times already [compare (3.26,27) and (6.17), which is essentially the Fourier transform of (3.26)]. We write down (3.26) for the product correlation Skl(t), making use of time translation invari-ance, together with its Fourier transform (6.17): (8.2) 42 W. Brenig, Statistical Theory of Heat