5. THE STATISTICAL MECHANICS OF THE EPIGENETIC SYSTEM 61 



surface G^ = constant but can move freely in phase space, varying its G by 

 exchange with the rest of the system. We may ask how the point representing 

 this part will be distributed in phase. According to a basic proposition in statis- 

 tical mechanics, the distribution law is 



p^ = e^4:-G.)ie (29) 



where G, = G(.Yi, Xj, . . ., x^; jVi, >'2, • • •» >'v)- This law defines the Gibbs canonical 

 ensemble. Since the distribution must be normalized, we have 



J Pvdv = 1 



where dv = dxi . . . dx^ dyi... dy^,, and the integral is taken over the space of 2i^ 

 dimensions. pX-Xu ■ ■ -, x^; yi, . . ., y^) represents the probability that a member 

 of the ensemble (which is in stationary equilibrium) chosen at random will 

 be found in the volume element dv around (xi, . . ., x^; yi, . . ., y^). 

 From the above normalization we have the relation 



e-'P,l9 = z^= { e~^-'^dv 



because 0^ is independent of the variables x^, . . ., x^, y^, . . ., y^,- The quantity 

 Z^, is known as the Gibbs phase integral, and it will be much used in the statis- 

 tical mechanics. The range of integration for the variables x,- is from -pi to 

 00, and for y; from -r, to oo. The upper limits of infinity may seem rather 

 unexpected insofar as our variables are population numbers of different 

 macromolecular species, and obviously no species reaches an infinite size in a 

 cell. However, since we cannot set upper limits for the variables, because we 

 have no information which tells us exactly what these limiting sizes are (unlike 

 the lower limits, which are fixed by the fact that the original variables, Xi and Yi, 

 have zero as their minima, negative concentrations having no meaning), we 

 use a common device in statistical mechanics which in effect makes very large 

 population sizes extremely improbable, while not entirely excluding their 

 possibility. Thus the Gibbs phase integral, written in extenso for v compo- 

 nents is 



00 CO oo 00 00 



— pi — P2 — /'v — Tl — Ti, 



dxi ... dXydyi ...dy^ 



Since the variables in the case of the simple system without strong coupling 

 are separated, the integrals can be evaluated separately and the expression 

 reduces to a product which we write in the form 



^v — [ [ '^Pi^C/i 



(=1 



