Fluctuation-dissipation theorem general form

The important fluctuation-dissipation theorem that expresses a balance between the dissipation and fluctuating forces to ensure the approach to equilibrium at a specified temperature is generalized to the expression that in the mass-weighted coordinates has the form... 

At the same time diffusion coefficient satisfies Einstein relation Dl = kaT/m connecting in that way two dissipative effects due to conformational changes and fractal geometry of filler surface producing that change. Our system obeys the generalized fluctuation-dissipation theorem. As a comment on the obtained distribution function we can say that from NMR experiments it is indeed possible to obtain Boltzmann distribution of energy sites (within the accuracy of the data) in the form I — Iq exp(395 + 50/F). 

The name fluctuation-dissipation theorem actually refers to another form of the relationship, namely that holding if times are replaced by frequencies. Let us switch in the explanation to general terms and choose the symbol Xy for the extensive variable and, as earlier, the notation for the field. For the linear response, we write in equivalence to Eq. (5.29)... 

An analogous relationship holds for the spectral density of the field fluctuations. The general form of the fluctuation dissipation theorem here reads... 

The power law for the time dependent modulus can be transformed into a law valid for the frequency domain, either by applying the general relations of linear response or by using the ( -dependent form of the fluctuation-dissipation theorem. The result is... 

In fact, this is another form of the fluctuation-dissipation theorem, as discussed generally in Section 6.2, now relating the resistance of the pore to the fluctuations in the pore current. 

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