128 BELL SYSTEM TECHNICAL JOURNAL 



which is identical with Eq. (3.23) (provided that we let ^^ -^ <» in the lat- 

 ter). Thus the methods of approach used in Section 3 and in this Section 

 are completely equivalent. 



5. Special Physical Models 



In the previous two Sections we have developed by two different methods 

 the consequences of the general mathematical model discussed in Section 2. 

 Here we apply the general results to some special physical cases. In this 

 task we will be principally concerned with finding the form of the function 

 /(r) and establishing the number of dimensions v of the diffusion field. The 

 main objective here is to provide some orientation on what mechanisms are 

 or are not reasonable and to find at least one mechanism leading to the 

 observed spectral density (inversely proportional to the frequency). 



a. A General Class of Models. Here we consider all at once mechanisms 

 which can be adequately represented by having /(r) a j/-dimensional Gaussian 

 function of the form 



where aJ'^ is the "width" of the function measured along the t-th coordinate 

 Xi. This form of /(r) can represent approximately several types of localiza- 

 tion of the coupling between R\ and Ci , as will be seen in the special examples 

 later. Now if we work with Av= oo , we will then have to consider the Fourier 

 space-transform of /(r), which is readily shown to be 



f^'^-Tk^l^^^^^-^'"'' 



(5.2) 



Inserting this result into Eq. (3.19) we obtain immediately 



exp ( — 2^ ^i ^i ) k~ dk 



^w^^rnss/ 



2xD _ 



7r5" VfJi 27ry J «2 + I>2^* {S.2,) 



1=1 

 Inserting this expression (5.2) into Eq. (3.23) gives 



