LINEAR THEORY OF FLUCTUATIONS 129 



In order that (5.3) give the observed S(ui) a 1/w as a result, the integral 



.iu£t reduce to scmething proportional io I k dk/(cj^ + D^k*). It is clearly 



impossible that any choice of v and any set of A,- can achieve this result. 

 Furthermore the seif-covariance i?i(/)i?i(/ + ti) corresponding to the ob- 

 served S(o)) should depend explicitly on the way S(o}) deviates from l/w 

 as CO goes to zero. The expression (5.4) is finite for all w > and does not 

 depend upon any cut-off phenomena in S(o:) at low frequencies. Therefore 

 we can exclude any physical mechanisms belonging to the class considered 

 here. However, since several mechanisms that have been proposed do fall 

 into this class, we consider them below: 



(f) Scl.iffs Mechanism. Schiff^^ considered tentatively that the fluctua- 

 tions in contact resistance may be due to the variation in concentration of 

 diffusing ions (atoms, or molecules) in a high resistance region bounded by 

 parallel planes of very small separation. Schiff arrived at a noise spectrum 

 proportional to 1/co but at the expense of disagreeing in a fundamental way 

 with thermodynamic fluctuation theory. Here we will show what the cor- 

 rect consequences of this mechanism are. 



Consider that the high resistance region is bounded on either side by planes 

 parallel to the {xi , X2)-plane and that the thickness in the Xs-direction is 

 very small. Now this is obviously a case of the general model just considered 

 in which we take v = 3 and 



Ai,2»Du,\ 



(5.5) 

 As «Du,] 



where l/u is of the order of magnitude of the frequencies of interest. It is 

 then a matter of algebraic manipulation to show that 



i2 22 72 -.00 ,2 





and 



(5.6) 



R.{l)Rr{t + u)= -^^ . .. .3;.:m.i/2 -TXF^ « > 0- (5.7) 



Thus we see that Shiff's mechanism leads to a noise spectrum proportional 

 to l/oj^'^, not l/cj. The explanation of the singularity of the self-covariance 

 (5,7) at « = lies in the inequalities (5.5). 



«L. I. Schiff, BuSldps Contract NObs-34144, "Tech. Rpt. j^3", (1946). Before the 

 publishing of this paper, Schifl informed the writer that he has discarded this mechanism. 



