LINEAR THEORY OF FLUCTUATIONS 125 



the case of a continuous distribution, a(T)dT is the weight for the relaxation 

 times in the range dr containing r. In these terms Eq. (3.17) becomes 



>S(a;) =Z-4^^,, (3.17a) 



k 1 + TftOJ 



where 



Eq. (3.19) becomes 



«* = -4, IM'- (3.17b) 



TT S 



where 



5(») = f fMlJ^ (3.19a) 



«« = -M^. I \f(i/VD-r) r d^. (3.19b) 



in which / is the unit vector in the direction of k, dUy is the differential 

 "solid" angle in the I'-dimensional /j-space, and the integration is over the 

 total solid angle (lir in 2 dimensions, or 4t, in 3). 



It is of interest to calculate the self-co variance Ri(t)Ri(t + «). In Appendix 

 I, it is shown that the self-covariance above is related to the spectral density 

 S{o}) as follows: 



Ri{i)I^i(t + u) = I S(o}) cos tico do). (3.21) 



Using S(o)) in the form (3.17), Eq. (3.21) gives 



RiiORiii + u) = xA,/^" • Z I/* P ^~^"'', (3.22) 



k 



u > 0; 

 whereas with S(oi}) in the form (3.19) we get 



Ri{t)Ri(l + u) = {ItY x/s" f I m \' r""'' dk. (3.23) 



The method of the next section yields the self-covariance directly. 



4. Smoluchowski Method or Solution 



We call the procedure employed in this section the "Smoluchowski 

 method" because it is based on an equation very closely analogous to the 

 well-known Smoluchowski equation forming the basis of the theory of 



