122 BELL SYSTEM TECHNICAL JOURNAL 



Before proceeding to the solution itself let us consider what it is that we 

 wish to know about Ck{t). Expanding the function /(r) of Eq. (2.4) in a 

 Fourier space-series in the region A^ , 



f(r) = Zfke'"-', (3.3) 



k 



we can write Eq. (2.4) in the form 



RiU) = A.Z'ftck(i) (3.4) 



k 



where /fe is the conjugate oifk. 



The spectral density 5(w) of Ri(t) is then 



Sic) = A;Z'Ckk'(o:)ftfk', (3.5) 



kk' 



where Ckk' («) is the spectral density matrix for the set Ck{i) given by 

 C*fe'(co) = 2ir Lim - [ckico, T)ck'{o3, r) + Ck( — c», T)ct'( — o}, r)] (3.6) 



7-»W T 



in which 



1 /.+T/2 



Cki<^, r) = ^ Ckide-''"' dl. (3.7) 



ZTT J-t/2 



For a full discussion of spectral densities and spectral density matrices see 

 Appendix I. Consequently our objective in this section is the calculation 

 of the maxtix Ckk' {<^) defined by Eq. (3.6). 



Now we assume that Ci(r, /) satisfies the diffusion equation 



^ c,{r, I) = Z)v'ci(r, /) + g{r, i) (3.8) 



ot 



where Z) is a constant, V^ is the Laplacian operator in two or three dimen- 

 sions, and where g{r, t) is a random source function, whose Fourier space- 

 amplitudes gk{l) possess statistical properties to be discussed presently. 

 The random source function g is required for exciting a sufficiently to main- 

 tain (he fluctuations given by equilibrium theory. In the case of material 

 diffusion the random source function g may be discarded in favor of a ran- 

 dom force term of the form —D/x T-^-[f{c + Ci)], where V/ = Jn/, x 

 is the Boltzmann constant, T is the temperature, and / is the random force; 

 however, in the linear approximation these two procedures will give identical 

 final results. In the case of heat flow it is understood that the diffusion 

 constant \% D = K/pC where A' is the thermal conductivity, p the density, 

 and C the specific heat. Eq. (3.8) as written is valid only for D a constant 

 and Ci small. 



