LINEAR THEORY OF FLUCTUATIONS 141 



where 







when 5 and e are the entropy and energy, respectively, per unit volume (of 

 jz-dimensional space). In calculating (II-6) it was assumed that 



/ ci{r, /) dr = 0, 



J A, 



expressing the fact that the system is closed. In order to put (II-6) into a 

 form strictly analogous to (II-2) we write it 



AS = -I- f f s"6{r - r')ci{r, t)ci{r', l) dr dr'. (II-7) 



2. J A, J A, 



We see that the equation analogous to Eq. (II-5) must be 



I [ y{r')s"d{r - r")ci(r", t)c,{r, t) dr dr' = x7(r) (II-8) 



JAy J Ay 



where 7(r) is an arbitrary function. Integrating (II-8) with respect to f 

 and using the fact that the delta function is defined by 



fy{r')8(r' -r) dr' = y{r) 



we readily arrive at the result 



ci(r, l)cy{r', t) = ^, b{r - r'). {11-9) 



Using the Fourier space-expansions of Ci and 6(r) 



k 



Kr) =~T.e''\ 

 A, k 



in the region A;, = LiX • • • XL„ with ki = lirni/Li , we can write (II-9) 

 over into the equivalent expression: 



^ u s 



where 



fl if ft = k', 

 bkk' = { 



otherwise. 



