LINEAR THEORY OF FLUCTUATIONS 133 



face concentration of the absorbate, be measured in molecules (atoms, or 

 ions) per unit area, then, for the ideal system above, it follows that s = —x 

 c log c and finally that s" = xli- However, in the mechanism discussed in 

 this part we have a compound system consisting of two separate layers on 

 the upper and lower surfaces respectively. Nevertheless, a detailed analysis 

 reveals that with c equal to the sum of the concentrations of both layers 

 we still have ^ = x/^ even though s itself is no longer given by an expres- 

 sion the same as that above. In conclusion let us consider the factor xA" 

 in Eq. (5.14). This factor is under the above idealization simply equal to 

 c. That is, the spectral density 5(co) is directly proportional to the average 

 concentration of absorbed molecules, meaning simply that each molecule 

 makes its contribution to the resistance fluctuations independently of the 

 others. Of course, in any real system this will not be quite true; however, 

 the existence of interactions will be manifested only by making xh" not 

 equal to c in Eq. (5.14). 



The results quoted thus far apply to a system with a given hiy). Now we 

 shall average the right-hand side of Eq. (5.14) over the ensemble of varia- 

 tions of //(r), it being supposed that S{<ji) itself on the left-hand side will 

 be negligibly affected by this operation. This amounts to replacing |/fe |2 



(e) 



by I /ftp . We then have 



I 

 where /^^ are the Fourier space-amplitudes of hxif) 



We now consider more closely the problem of calculating | hk |^ . We want 

 to assume that h\{y) is a more or less random function of r. If h\{y) were a 

 random function of r in the same way that the thermal noise voltage is a 



(e) 



random function of the time /, then | hk ^ would be a constant independent 



(e) 



of k and the self-co variance hi{r)/h(r') would vanish for r 9^ r'. This 

 clearly cannot be so, since the function hi(r) with such statistical properties 

 would represent a highly discontinuous type of surface incapable of physical 

 existence. We then fall back upon the more reasonable assumption that the 

 gradient of hi possesses statistical properties of the above type. This notion 

 is precisely formulated by means of the following equations: 



V//i(r) =/>(r) (5.16) 



where 



fkf = a^R' Ihkl''"' (5.15) 



-.M 



and 



f p{r) dr = 0, (5.17) 



p{r)p{r'f^ = /3U(r - r'). (5.18) 



