7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 1 1 1 



the range of ^2 is from 



I'^i'K/,!,) 



I '2 

 Pi 



M , ^'12 



to infinity, while the range of |i is from 

 to infinity. When I2 is at its lower bound, 



- '(B 



1 2 



(hnPi+hnPz) = -pi, say 



In the (^1,12) plane, then, the phase integral is taken over the area between the 

 the line 



^2= -|//|i/2|Aj'"^2(callthis-p2) 

 and the line 



ii-|^2i2= -iihi^r'Pi 



the area extending out to infinity in the positive direction of the axis. This 

 transformation puts the integral into the form 



^-- = «L^ J ^^2 J ^-^'^-'^V^, 





When jS is very large, the double integral is clearly approximated by 



00 CO 



— 00 — 00 



In this case we have simply a product of two integrals, each of which is a/"^ 

 so that the result is (60) Z^,^^ ~ 7t/^\H\^'2 for very large /S. This is the limit 

 d->0, and it is interesting to note that the decomposition of the double integral 

 into two single integrals reflects the " uncoupling" of the components when the 

 talandic temperature is very small. This means that there is very little inter- 

 action between strongly coupled components when 6 is small, and the motion 

 in the system is effectively linear. 



In the other limit, for ^ very small (d very large), the phase integral becomes 

 approximately 



00 00 



Z X — \ d(^ I e-^^'"+^'"^d^i 



P'P* R\H\i^2 j "^2 J *^ «?i 



(Ai.|//|'/2)f, 



