### 8.1 The Fluctuation-Dissipation theorem and Levin’s generalized coordinate method

We are interested in the spectral density of thermal noise. There is a general derivation of this spectral
density, based on the Fluctuation-Dissipation (FD) theorem of Callen and Welton [9]. For
an elementary dynamic system described by a degree of freedom and any driving force
, one can consider the resulting velocity , and compute a mechanical impedance
as . Then, the power spectral density of displacement is (this is the FD theorem)
We can now address the problem of internal degrees of freedom in the mirrors. Internal elastic waves
eventually distort the reflecting surface, causing a phase noise. We have already discussed how to obtain the
information on the surface relevant to the beam. Let be the component of
the displacement vector of matter at the surface of the mirror. The equivalent displacement
(generalized coordinate ) is, as usual, the axial displacement, averaged by the intensity profile,
where is the normalized light-intensity distribution in the readout beam. We now follow the
method proposed by Levin [26]. Let be the corresponding driving force. The interaction energy is
or
where the displacement may be thought of as being caused by the pressure distribution . We
address now the case of low frequencies. This case is very relevant, because resonances of mirrors
are at relatively high frequencies (several kHz) and the region where internal thermal noise is
disturbing lies long before the first resonance, in the low frequency regime. Thus, although a general
knowledge of internal thermal noise is useful, it is nevertheless extremely interesting to have
the low frequency tail. This can be obtained as follows. If we consider a force
oscillating at very low frequency, the frequency will be lower than the cutoff for any standing
waves. The pressure will produce an oscillating stationary displacement , of the form
This is equivalent to neglecting inertial forces in the motion of matter. The phase represents a
retardation effect that dissipation may cause; in the case of thermoelastic dissipation, we know that can
be considered very small and independent of the frequency (at least in the GW detection band). In the
Fourier domain, this is
The impedance is
so that
, where the numerator of the fraction appears as the elastic energy stored in the solid stressed by
the pressure distribution . The strain energy is defined in classical elasticity theory by
where is the pressure distribution causing the displacement at the surface on which it is
applied. Thus, we can write for the spectral density of displacement,
In fact, is proportional to , so that is the strain energy for a static pressure
normalized to 1 N. The spectral density of displacement takes the general (low frequency) form
The problem is reduced to the computation of and this is the main point of Levin’s approach. This can
be difficult in the general case of an arbitrary solid, but numerical finite-element codes are able to give
more-or-less accurate estimates. However, it is possible to obtain analytic solutions in the case of
axisymmetry.