5. THE STATISTICAL MECHANICS OF THE EPIGENETIC SYSTEM 69 



Furthermore, in view of the theorem mentioned in Chapter 1 showing that 

 for an arbitrary class of complex dynamic system the probability of any trajec- 

 tory ending in a continuing oscillation approaches 1 as the complexity of the 

 system increases, it may be found that a parameter such as 6 is of fundamental 

 signiticance in a "thermodynamic" description of general dynamic systems. 

 However, one parameter does not make a thermodynamics. Other macroscopic 

 parameters analogous to pressure, volume, entropy, specific heat, etc., must 

 first be found and then relationships analogous to the gas laws and the heat 

 theorems of classical thermodynamics. This is an ambitious programme, and 

 in the present work we can do no more than indicate the direction in which such 

 a study might develop in the particular context ofcellular control systems. The 

 first step is to introduce the conventional thermodynamic functions which arise 

 in connection with the quantity G, the talandic energy. 



The "Thermodynamic" Functions of the Epigenetic System 



In obtaining thermodynamic variables for the epigenetic system, we will 

 make continuous use of Gibbs phase integral, which was defined as 



where 



•' (=1 



00 



Pi 



CO 



Z . = r e-(6./^)f>'i-iog(i+>',)]^^. 



It will be convenient to write ^=1/6 in the following. These phase integrals 

 can be reduced to familiar functions in the following manner. Writing 



i = 



(W^' 



the first integral becomes 



iwf / -'■*=(l,)"-" 



-PiViPciiD 



where Erfc{z) is the error function, defined by 



(34) 





