320 George A. Sacher 



enough to permit a calculation of the probability per unit time that fluctuations 

 will reach L. To answer these questions, we turn to a consideration of the 

 dynamic nature of the fluctuation process within the individual. The complete 

 description of a fluctuation process is given by specifying its correlation function, 

 which is in one dimension 



P(t) = {X{t)x{t + T)>av/<.v2(r)>av (2) 



where x is a deviation from the mean, 



^=^0 + ^^ (3) 



The correlation function is a measure of the degree to which a fluctuation 

 present at time t persists at a later time r -\- t, averaged over all values of t. 

 The nature of p(t) depends on the nature of the system. A process that obeys 

 the differential equation 



^ + ^x = (4) 



returns to equilibrium as 



X = :Co e-^^ (5) 



Corresponding to this, if a stationary pure random Gaussian noise source 

 f{t) is applied, 



^ + ^x =/(/) (6) 



the resulting correlation function is 



p{r) = e-P^ (7) 



It can be shown (3) that if the correlation function in one dimension is given 

 by equation (7), then the fluctuation process is Markoffian and is completely 

 described by the joint probability distribution 



W.{x^xS = 27ra2 (1 _ p2)* 



X exp 



2a\\ - p2) 



^1^ + -^2^ — 2p.YiX2 



(8) 



This gives the joint distribution of observations of x separated by time t, where 

 p is defined by equation (7). The variance, cr^, of the distribution of .y satisfies 



a2 = Dl(i (9) 



where 4D is the (constant) spectral density of the random noise (white noise) 

 source. The conditional probability distribution, which describes the distri- 

 bution of ^2 when x^ is fixed, is 



n-Yo/x, /) = [27702(1 - p2)]i 



X exp [-(.Y - xfl2a\\ - p"-)] (10) 



