120 BELL SYSTEM TECHNICAL JOURNAL 



general way a large class of models involving resistance fluctuations arising 

 from diffusional mechanisms. In the next section we propose a general 

 mathematical model embracing a class of linear diffusional mechanisms. 

 In Sections 3 and 4 the consequences of the general mathematical model 

 are obtained by the "Fourier" and "Smoluchowski" methods, respectively, 

 these alternative methods leading to identical results. In Section 5, the 

 general results are speciahzed to several physical cases, some of which are 

 introduced only for the purpose of providing some insight into the relations 

 between the possible physical mechanisms and the resultant resistance 

 fluctuations, and one of which along with its refinement is a successful'* 

 attempt to provide a theory of Eq. (1.2). Section 6. is a summary. 



2. The General Mathematical Model 



The physical models which we consider in this paper are concerned with 

 the fluctuations of contact resistance arising from a diffusional process. We 

 are consequently led to consider the following general mathematical model 

 embracing a rather extensive class of the physical models as special cases: 

 Let us consider the instantaneous contact resistance R{l) to be related to 

 the intensity cir, i) of some diffusing quantity as follows'^: 



G{R{t)) = j F{r, c{r, /)) dr, (2.1) 



where r is a vector in two or three dimensional space depending on whether 

 the diffusion takes place on a surface or in a volume, and dr is correspond- 

 ingly a differential area or volume. The intensity c{r, t) may be either a 

 concentration (in the case of diffusion of material in two or three dimensions) 

 or a temperature (in the case of heat flow in three dimensions). In writing 

 Eq. (2.1) we have evidently assumed that the contact resistance R{t) is 

 independent of the applied voltage. Eq. (2.1) may of course allow a de- 

 pendence on voltage through the quantity c; however, we will consider no 

 processes involving a dependence of c on the voltage. These restrictions, 

 strictly speaking, make the model applicable only in the limit of low applied 

 voltages. 



Before proceeding further let us limit the treatment to the case in which 

 the deviations of R and c from their average values are sufficiently small 

 for higher powers of these deviations to be neglected. Let 



R{1) = R-\- R,{t), (2.2) 



c{r, t) = c + c,{r, 0, (2.3) 



'* That is, successful in so far as agreement with the form of Eq. (1.2) is concerned. 



"A relation more general than R{t) — SF{t, c{t, /)) dr is retiuired as one can see from 

 considering the special case of a total resistance composed of a parallel array of resistive 

 elements. 



