### 9.1 Introduction

The Brownian motion of matter inside the substrates is not the only cause of noise in the optical readout. There is another cause due to temperature fluctuations in a finite volume of material. These fluctuations are called thermodynamic and can couple with strain via the thermal dilatation constant , eventually producing random motions of the surface. A good way to model this kind of noise is to start from the general thermodynamic formulas detailed by Landau and Lifshitz [24] and use the Levin approach already presented. As in the preceding Section 8, we will consider the low frequency tail of the spectral density of the effective motion of the surface (i.e., the readout noise) as depending on the energy dissipated when the body is under a virtual pressure having the same profile as the optical beam and excited at low frequency. In this case, the spectral density is still of the form (Levin’s formula)
where is the average dissipated energy. For the standard thermal noise, we had as average dissipated energy, being a global loss angle and the static strain energy. But now must be interpreted as the energy dissipated via coupling of the strain with the temperature field in the bulk. Obviously, the temperature field itself depends on the strain field. Using the same approach as used in [28], we first solve the static linear elastic problem (done in the preceding Section 8), then we compute the resulting temperature field and use it to compute the dissipated energy. For computing the dissipated energy, we use the time dependence of the entropy. The variations of the entropy density are related to the heat flux by requiring conservation of the energy in the body
where , being the thermal conductivity of the material (cf. Landau and Lifshitz [24]). Or, as well,
Therefore, total entropy variation in the body is
where the integral is extended to the whole body. This is
Neglecting the heat flow at the surface of the body, the first integral vanishes, and we have
But using the definition of , this is
so that the energy variation is
We shall say now that the temperature gradient field is caused by the small deformations of the body that we computed earlier, while is the mean temperature. This becomes
where we have replaced by a in the gradient for more clarity. On the other hand, it is well known (cf. Landau-Lifshitz) that the total entropy is the sum of two terms, one being the entropy in the reference state, and a second one proportional to the trace of the strain tensor
being the thermoelastic coefficient, so that there is, in the bulk material, a power source given by
where is the trace of . The resulting temperature field obeys the Heat (Fourier) equation
The trace of the strain tensor found in the preceding Section 8 is, in any case, a harmonic function, so that there is a trivial solution
The boundary conditions (null heat flux on the surfaces) are considered satisfied in time average ( is assumed oscillating at a few tens of Hz). In fact, they are exactly satisfied on the circular edge of the mirror. Now we reach the relevant equation for the dissipated energy
is related to the linear dilatation coefficient by
where is Young’s modulus and the Poisson ratio. Finally,
(see [28]). We have, after the preceding Section 8 on standard thermal noise, everything we need to compute .