Entropic Contributions to Mortality and Aging 325 



plus higher order partials. At equilibrium the first partial is equal to zero, so 



^S^lljf^x,x, (31) 



= -i2 S,/^x,x, (32) 



where 5*,/ is a positive-definite matrix. 



There is a formal equivalence between the S^j^ and the /■; defined by equation 



(24), 



SO = ^A (33) 



where k is Boltzmann's constant. Thus the X^j, which we may term the partial 

 coefficients of the frequency distribution of fluctuations, are proportional to 

 the coefficients 5",/ of the quadratic form for the mean entropy decrease due to 

 fluctuation in the system. The physiological systems that are under consideration 

 and are not completely described by a small number of variables, and accord- 

 ingly the complete fluctuation distribution and fluctuation entropy would 

 not be estimated. However, the S^j, or the A,^, are additive, so an initially 

 incomplete description can be completed as knowledge of the system increases. 

 From the definition of entropy by Boltzmann 



S = k J p{x) log p(x) dx (34) 



it follows that the fluctuation entropy coefficient S^^^ in equation (32) can be 

 written 



where /J = p{x^, • • • , x,). 



In one dimension this reduces to 



S» = -J^%./.v (36) 



This is identical with the definition of information given by R.,A. Fisher (11). 

 The equivalence continues to hold in the n-dimensional case. It should be noted, 

 however, that the Fisher infoiTnation is a defined quantity, whereas the 5',_," 

 are terms in an approximation formula. 



There is a close relationship between information theory and the analysis 

 of fluctuation processes as can be made evident in terms of the equivalences 

 brought out above. Where there are distinct classes the information is 



H=-Y.p,\ogp, (37) 



In the one-dimensional continuous case we write 



H{x) = -J p{x) log p{x) dx (38) 



In a large number of cases, the representation of log/?(.T) in terms of three 

 terms of a Taylor series is a good or even an exact description. The expression 

 for the information then becomes 



^W = - J /?(a-) log /X'Yo) + \x^ p{x) ^^ log /j(.Vn) 



dx (39) 



22 



