7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 117 



The integral which we must evaluate is, in the limit of small /3, 









We shall now make use of a result reported by Rice (1944), namely 



V^ 1 



00 oc 







By writing 



A- = i^hny^x,, y = (i8/722)^'2A-2 



our integral becomes 



00 00 



Here 



1 1 VWhn) J J ^ ^^h 1 V(^/'22) 1 + a 



hi: 



a = 



\/{h\h22) 

 so that 



1 Vihnhi^ 



1+fl \/(^ll/»22) + /7i2 



|//|l/2 



T^ 



c 



tan 



_Jh2_ 



:|//|i/2 / ' 



Since 1¥A, ^ 2i3|//|i'^2 / ^^^/'^ 



we now have the result that, when /3 is small, 



. . y yv 2/3|//|i /2 



^^+^- ~ 4iS3/2V(/lu)[V(/^ll/^22) + /'l2]\__i 1^ 



1/2 



tan-1 



/?12 



2V(/?ii)[V(/?ii/?22) + /^i2]tan-M// 



1/2 



/^l 



2 



1 // ^d \ {knk22-kuk2ir^ 1 



ai V \2A:2i ^11/ [V(/:ii /:22) + V(^ 



12^2l)]\^^_,/^_l\ 

 \^12^21 / 



1/2 



This shows us that the mean positive amplitude of Ai in the coupled oscil- 

 lator has the same functional relationship to the talandic temperature as does 

 A+ for the single oscillator, both varying as the square root of ^. The functional 

 dependence of {A+Yx, on the coupling parameters is clearly quite complicated; 



