LINEAR TUEORY OF FLUCTUATIONS 131 



Suppose that the contact in an idealized form consists of two rough sur- 

 faces close together. Let positions on the surfaces be measured with respect 

 to a plane between the surfaces, which we will call the mid-plane. Let the 

 coordinate system be oriented so that the .Ti and X2 axes lie in the mid- 

 plane. Furthermore let the region in the mid-plane corresponding to close 

 proximity of the rough surfaces be a rectangular area A2 = LyX Lo. Now, 

 for convenience, we describe positions on the mid-plane by a two dimensional 

 vector r = (.ti , X2), and henceforth it will be understood that all vector 

 expressions refer to this two-dimensional space. Let the distance between 

 the upper and lower surfaces at r be denoted by h{r). The geometry of the 

 above model is illustrated in Fig. I. 



Now suppose that each surface is covered by a diffusing absorbed layer, 

 such that the sum of the concentrations on both surfaces is c(r, t) at the 

 time / in the neighborhood of r. Now consider the conduction of current 

 between the surfaces. Let us assume that the conductance per unit area 

 (of mid-plane) is a function of the separation // of the surfaces and the total 

 concentration c of absorbate near the point in question, i.e., F{h, c). The 

 total conductance will be the sum of the conductances through each element 

 of area: hence, the instantaneous resistance R{l) at time / will be given by 



\/R{l) = f F{h{r), c{r, /)) dr (5.9) 



''A 2 



where dr is the differential area on the mid-plane and the integration extends 

 over the rectangle A2 = Li X L2 . Behind the above statements lies the 

 tacit assumption that the radii of curvature of the rough surfaces are gen- 

 erally considerably larger than the values of //. However, we will not explic- 

 itly concern ourselves with this implied restriction. 



At this point it is expedient to imagine that we have an ensemble of con- 

 tacts identical in all respects except for different variations of the separa- 

 tion h{r). If we have any function of h, \f/{h) say, which we wish to average 

 with respect to the variations of h, we simply average the function over the 



(e) 



above ensemble giving a result which we denote by \l/(h) . 

 Now let us write 



and, as before, 



h(r) = h''' + //i(r), (5.10) 



R{t) = R + R,0), ] 



(5.11) 

 c(r,l) = c-}-cj(r, /). 



We assume that the ensemble average h''" and the time averages R and c 

 are constants independent of r and /. Let us also assume that the integrals 



