Fluctuation-dissipation Theorem - Derivation

Derivation

We derive the fluctuation-dissipation theorem in the form given above, using the same notation. Consider the following test case: The field f has been on for infinite time and is switched off at t=0

We can express the expectation value of x by the probability distribution W(x,0) and the transition probability

The probability distribution function W(x,0) is an equilibrium distribution and hence given by the Boltzmann distribution for the Hamiltonian

For a weak field, we can expand the right-hand side

here is the equilibrium distribution in the absence of a field. Plugging this approximation in the formula for yields

(*)

where A(t) is the auto-correlation function of x in the absence of a field.

Note that in the absence of a field the system is invariant under time-shifts. We can rewrite using the susceptibility of the system and hence find with the above equation (*)

Consequently,

(**)

For stationary processes, the Wiener-Khinchin theorem states that the power spectrum equals twice the Fourier transform of the auto-correlation function

The last step is to Fourier transform equation (**) and to take the imaginary part. For this it is useful to recall that the Fourier transform of a real symmetric function is real, while the Fourier transform of a real antisymmetric function is purely imaginary. We can split into a symmetric and an anti-symmetric part

Now the fluctuation-dissipation theorem follows.