  The internal thermal noise in LIGO's test masses is analyzed by a new
technique, a direct application of the Fluctuation-Dissipation Theorem to
LIGO's readout observable, $x(t)=$(longitudinal position of test-mass face,
weighted by laser beam's Gaussian profile). Previous analyses, which relied on
a normal-mode decomposition of the test-mass motion, were valid only if the
dissipation is uniformally distributed over the test-mass interior, and they
converged reliably to a final answer only when the beam size was a
non-negligible fraction of the test-mass cross section. This paper's direct
analysis, by contrast, can handle inhomogeneous dissipation and arbitrary beam
sizes. In the domain of validity of the previous analysis, the two methods give
the same answer for $S_x(f)$, the spectral density of thermal noise, to within
expected accuracy. The new analysis predicts that thermal noise due to
dissipation concentrated in the test mass's front face (e.g. due to mirror
coating) scales as $1/r_0^2$, by contrast with homogeneous dissipation, which
scales as $1/r_0$ ($r_0$ is the beam radius); so surface dissipation could
become significant for small beam sizes.
