Entropic Contributions to Mortality and Aging 321 



where a~ and p are deiincd as above and 



X = x^e'^'^ (11) 



As /— > 00, this becomes the stationary distribution of fluctuations, 



This distribution is Gaussian because of the linearity of mechanism specified 

 by equation (4). Fluctuation processes are not in general Gaussian if the 

 dynamical equations are non-linear. 



Equation (12) gives the stationary frequency distribution of fluctuations over 

 the entire :>c-axis. We now introduce the condition that there is at Xg^ an 

 absorbing barrier at a distance X from the mean 



X— X^ — Xq 



An individual remains in the distribution only as long as the path described 

 by his fluctuation process remains in the region .v < ).. If this situation prevails 

 for a time sufficiently longer than the relaxation time of fluctuations, 1//?, a 

 stationary distribution is again established and there will be a stationary 

 probability q per unit time that the path will intersect x = /I. This 'absorption 

 rate' is the mortality rate for the model fluctuation process. The stationary 

 frequency distribution in the presence of an absorbing barrier may be obtained 

 from equation (12) by the following argument. 



In the steady state there is a stationary diffusion current, j, into the barrier. 

 The desired frequency distribution Q{x) must satisfy the steady state difl'erential 

 equation for diffusion in the presence of a force field (4). This equation is, in 

 the notation of equations (6) and (9), 



j = KQ{x)-(^a''^^^Q{x) (13) 



where K = —fix is the restoring force. 



A solution of equation (13) satisfying the boundary condition 



Q(x) = (x^ A) 

 is 



3:2 (x-2;)2-l 



Q(x) 



1 



(27702) 



2\i 



2(72 . ^ 2(T2 



(14) 



The mortality rate, q, is equal to the diffusion current, j, normalized by the 

 area under the distribution, Q{x), so that from equation (13) we find the 

 constant mortality rate to be 



A rigorous discussion of this class of stochastic processes (4) indicates that the 

 validity of equation (15) is subject to the limitation 



