Although it has roots in the study of Brownian motion by Einstein in the
early years of the 20th century,[3] the fundamental theoretical
underpinning of our understanding of thermal noise is the
Fluctuation-Dissipation Theorem. The first clear statement of the theorem in
its general form is usually credited to Callen and his collaborators.[4] This powerful theorem describes the fluctuations of any linear
system in thermal equilibrium at a temperature T. For our purposes, the
most useful statement the theorem is the expression for the displacement
power spectrum 
at the point of interest
where Y(f) is the admittance of the system, v/F. The real part of the
admittance, 
, is a measure of the amount of
dissipation in the system.
In an interferometric gravitational wave detector, the most important point at which thermal noise will appear is at the front surface of a test mass. The thermal noise power spectrum at that point is dominated by motion in three kinds of modes: the fundamental pendulum mode, modes of the pendulum wires, and internal modes of the test mass itself. For any single normal mode, the thermal noise power spectrum can be found by solving for the admittance Y. It is given by
where 
is Boltzmann's constant, T is the temperature, k is the
spring constant, m is the mass, and 
is the loss angle,
representing the fractional part of the spring constant associated with a
dissipative (out-of-phase) response as opposed to the elastic (in-phase)
response of the spring.
The integral of the thermal noise over all frequencies corresponds to an
energy per degree of freedom of 
, as one would expect from the
Equipartition Theorem. When the level of dissipation is very low, then
almost all of that noise power is concentrated at the resonant frequency. A
fundamental part of the design strategy of gravitational wave
interferometers is to place as many of the mechanical resonances as possible
outside of the signal band of interest. The pendulum mode will have a
frequency of about 1 Hz, while the test mass internal modes will all be in
excess of 10 kHz. In this circumstance, only a tiny fraction of the noise
remains at frequencies far from the resonance. The smaller the dissipation,
the lower the spectrum of off-resonance thermal noise.
(The one exception to the design strategy is the ``violin modes'' of the pendulum wires. But, since the vibrating parts have small mass compared to the test masses themselves, the net motion of the test masses is very small, and will only be visible in very narrow bands right at the resonant frequencies.)
The Fluctuation-Dissipation Theorem itself is an established part of the
field of non-equilibrium statistical mechanics. But the linkage of the
theorem with the phenomenology of internal friction in materials, as
described in the P.I.'s 1990 paper,[5] has some surprising
features. Most striking is the common case in which damping has the
so-called ``structural'' form, in which the loss angle 
is a
constant. When the frequencies of interest are low compared to the resonant
frequency of an oscillator (such as for a test mass mode) the thermal noise
power spectrum has a frequency dependence proportional to 1/f. This
spectrum is formally divergent if extended to arbitrarily low frequency.
Damping with this form would be troubling from a practical as well as
theoretical point of view if it applied to the fused silica test masses of
LIGO; it could lead to thermal noise being the dominant (and nearly white)
noise term in the 100 Hz band in any advanced receiver, unless dissipation
levels were reduced substantially from today's 
.