LINEAR TUEORY OF FLUCTUATIONS 137 



Thus we have illustrated how one modification of the model has removed 

 the divergence at co = 0. 



It appears from the treatment here and in part b that roughness and 

 diffusion in two dimensions are essential (at least in a linear treatment) 

 features in obtaining S(ui)a 1/co. In the case of a non-linear coupling (to be 

 considered in a later paper) a "self-induced" roughness effect may occur 

 without introducing roughness ab initio as an intrinsic feature of the model. 



6. Summary 



(a) If the resistance deviation i?i(/) is related to the concentration devia- 

 tion Ci(r, t) of a diffusing medium (particles or heat) by the linear functional 



RxU) = f firMr, dr, (6.1) 



where r is a vector and dr a volume element in a j^-dimensional space of 

 volume A^ , then the spectral density S{co) of Ri(t) is 



, . 2xA.D k'lfkf ,... 



where D is the diffusion constant, s" is defined by Eq. (3.14), x is the Boltz- 

 mann constant, w is the frequency (in radians per sec), k is the wave number 

 vector in v-dimensional /j-space limited to a discrete lattice of points (defined 

 by Eq. (3.2)) over which the summation is taken, and/^ is the /jth Fourier 

 component of /(r) (Eq. {2>2)). 



(a) If the important terms in (6.2) vary slowly between lattice points in 

 /j-space (true if co > coo given by Eq. (3.18)), then (6.1) can be replaced by 

 the integral 



where the integration extends over the entire /j-space and where f{k) is 

 given by (Eq. 3.20) 



(b) Let oj' be a frequency in the middle of a wide range. Suppose [ f{k) \^ 

 averaged over ths total solid angle in j'-dimensional /j-space is proportional 

 to k" ', where n is an integer, in a wide range of k with k = \/o:'/D in its 

 middle. It follows then that S(ui) a D'"""'^ ^-i+«+W2 ^^ ^^^^^ as — 1 < 2w 

 + j'+l<3. Asa consequence, we see that with n an integer (as is true for 

 the simple cases considered in Section 5) v must be 2 — ^the only even di- 



