Chemical and spatial information

We shall use Eq. (1) as a starting point for our combined information-theoretic and thermodynamic analysis of the system. Since homogeneity is assumed to be broken, we introduce spatially varying concentrations ci(x), expressed in the probabilistic form pi(x) = Vci(x)/N. (We assume that the number of molecules per unit volume, N/V, does not depend on position.) This results in probability distributions pi(x) over the different molecules that are normalised at each position. Then, Eq. (1) is replaced by

 

         clip_image003 ,                                                                                      (2)

 

where the information K is now an integral over Kullback information quantities for each position in the system,

 

         clip_image006 .                                                 (3)

 

We shall use the average concentration within the system, defined by

 

         clip_image009,                                                                                  (4)

 

in order to decompose the total information K in Eq. (2) into two terms, one that quantifies the deviation of the average concentrations from equilibrium, Kchem, and one that quantifies the deviation from homogeneity, i.e., the presence of spatial structure, Kspatial,

 

         clip_image012                                                      (5)

 

Therefore we define the spatial information, Kspatial,

 

         clip_image015,                                                               (6)

 

and the chemical information, Kchem,

 

         clip_image018,                                                                          (7)

 

and we can write the total information K as

 

         clip_image021 .                                                                                 (8)

 

Thermodynamically, the chemical information is related to the presence of a chemical non-equilibrium even when spatial variations are not taken into account. This means that there is an abundance of “fuel” and a low level of “waste” products in the system. The spatial information reflects the presence of a non-equilibrium for diffusion processes, i.e., that there is some spatial pattern in the system.