5. THE STATISTICAL MECHANICS OF THE EPIGENETIC SYSTEM 57 



having all variety of initial values of .v, and ;,. In the Cartesian space of the 

 variables .v, and j, of dimension 2n, known as phase space, the configuration 

 of each copy is represented by a point, the ensemble of copies by an ensemble of 

 points. These points move through phase space in a manner governed by the 

 differential equations. Taking these points to be sufficiently numerous (i.e. 

 for a large enough number of copies, hence a great enough variety of initial 

 conditions) they constitute a fluid of density which we represent by p{xi, . . ., x„; 

 >'ii • • •» >'n) at a point (.Vi, . . ., .v„; yi, .. ., j,,). The velocity of the fluid at this 

 point is K = (.Vi, . . ., x„; y^, . . ., x„). Since fluid is neither created not destroyed, 

 we must have the hydrodynamical equation of continuity, 



; 1 L 



= 



Expanding the partial derivatives, we get 



The expressions in the second summation vanish because we have 



dxi _ _ d^G_ dji _ ^G_ 



dxi ~ dxidy/ dy^ dy^dXi 



these terms being equal and opposite. We are left with Liouville's theorem on 

 the conservation of density in phase: 



This means that as one follows the motion of one point the density in its neigh- 

 bourhood remains invariable. The significance of this theorem is that there is 

 no tendency of the motional equations to cause an accumulation of points in 

 one part of phase space, so that all parts receive an equal distribution of points. 

 This also implies that any element of volume of phase space, though changing 

 its shape, maintains a uniform size or measure as the motions of its points 

 unfold, so long as it consists always of the same set of points. 



The importance of this theorem is that we can now define for phase space 

 a probability density which is stationary in time — i.e. we can introduce pro- 

 bability arguments which will allow us to calculate, instead of exact quantities 

 associated with the motion of the control system, mean or expected values of 

 these quantities. It is our ignorance of initial conditions and the dynamics of 

 the biochemical space in which the control units function which forces us to 

 take this probabilistic or statistical point of view. The probability densities 

 which are most often used in statistical mechanics are functions of G alone: 

 p = p(G). These are stationary and satisfy Liouville's equation (28). The mean 



