LINEAR THEORY OF FLUCTUATIONS 12^ 



Introducing the expansion (3.1) and the expansion 



gir, t) Z' gk{t)e'^-' (3.9) 



k 



into Eq. (3.8) we obtain the infinite set of ordinary differential equations 

 |cfe(/) = -Dk'ckd) +gfe(/), (3.10) 



I describing the time behavior of the Fourier space-amphtudes Ck(t). The 

 Fourier space-amphtudes gkO) are assumed to be random functions of / 

 possessing a white (flat) spectral density matrix Ckk > independent of fre- 

 quency. Multiplying Eq. (3.10) by 7— «"'"', integrating with respect to time 



from — ^r to -\-^t, and neglecting the transients at the end points of the 

 T-interval, we obtain 



^ / N gk(^, t) ^2Ai\ 



where Cfe(w, r) is given by Eq. (3.7) and gfe(co, r) is given by an analogous 

 equation. Forming the spectral density matrices we get for the diagonal 

 elements 



Ckk'io,) = . ^''. ^ (3.12) 



The matrix G^k' can now be evaluated by the thermodynamic theory of 

 fluctuations (See Appendix II). This theory gives 



CkiOct'iO = ^' (3.13) 



where 



(3.14) 



s and e being the entropy and energy, respectively, per unit area or volume, 

 T the average temperature, and % the Boltzmann constant. In the case 

 where c is the concentration of particles whose configurational energy is 

 constant, 5" = x/c. If c be the temperature T then s" = C/T~ where C 



