140 BELL SYSTEM TECHNICAL JOURNAL 



APPENDIX II 

 Thermodynamic Theory of Fluctuations 



The value of the quantity fi(r, i)ci{r^, t) or {ck{l)ck^{l)) is determined from 

 equilibrium considerations. Before going into the above continuum problem 

 let us first consider the problem for the case of a system described by a finite 

 set of variables. More specifically let us suppose that the state of the system 

 subject to certain restraints (i.e. fixed total mass and energy) is described 

 by the set of variables Xi , • • ■ , Xn . Let the equilibrium state be given by the 

 values Xi , ■ • • , x„ , and let 



Xi = Xi -{- ai . (ITl) 



If the system is constrained to constant average energy E, the entropy of the 

 non-equilibrium state S = S^ -{- AS will be less than vS", the entropy of the 

 equilibrium state, by an amount 



AS = -liHSijaiUi, (II-2) 



where 



d'^S 



I ( d'E\ 



r^\dxidxj. 



Obviously, AS must be the negative of a positive definite quadratic form, 

 otherwise the equilibrium state would not be a state of maximum entropy. 

 The probability distribution''*' for the a's is given by 



P{a,,--- ,«„) = Ne''"'' (II-3) 



where TV is a normalization factor and x is the Boltzmann constant. Averag- 

 ing the products cci a; we find that 



2_/^ij«j«/fc = X^ik- (II-4) 



i 



Multiplying (II-4) by the arbitrary set ji and summing over i we get 



22 yiSijOijaK- = XTi- (II-5) 



The generalization to a system described by a continuous set of variables 

 is not difficult on the basis of (II-5). Now suppose that, in a i^-dimensional 

 space ^^ , we have a system whose state at time / is defined by the continuous 

 set of values of the variable c{r, l) = c -{- Ci{r, t) ; we have 



AS = -\l s"c\(r,/)dr (II-6) 



«« H. B. G. Casimir, Rev. Mod. Pliys. 17, Nos. 1 and 3, 343-4 (1945). 



