PACE REPORT

The previously presented macroscopic framework gives an information-theoretic perspective on pattern formation in chemical systems. The continuity equation, with the corresponding information flows, connects to thermodynamic flows and thus to thermodynamic restrictions of the system. As a consequence of the second law of thermodynamics there is a destruction of information at the finest levels of resolution.

If one could keep track of single molecules and their microscopic states developing according to reversible microscopic dynamics, then there is no desctruction of information. Instead information may be spread out over increasing distances in correlations. This type of microscopic information dynamics have been investigated in the context of reversible and irreversible cellular automata [Hellvik et al 2007], exemplified above.

For simplicity let us consider a one-dimensional lattice in which a site can be occupied by at most one molecule. In order to define information quantities at this level one need a probabilistic description over sequences of lattice sites. The probability p_m(z₁,…, z_m) then denotes the probability for a sequence of sites to be in the state (z₁,…, z_m), where each z_k denotes a certain molecule or an empty site. The (microscopic) entropy s taking into account all order that may be found in correlations then takes the form

                                  (1)

where p(z_m|z₁,…, z_m–1) is the conditional probability for state z_m given that it is preceeded by (z₁,…, z_m–1) to the left. (This formalism assumes an infinite system, with a stationary stochastic process representing the spatial distribution of states. The formalism can be generalised to higher dimensions. For details on this, see [Lindgren 2008].) The microscopic quantity that corresponds to the spatial information at the macroscopic level would be a Kullback (or relative) information between an a priori distribution based on average concentrations and the conditional probability distribution,

                              (2)

In the assumption of ideal solutions in local equilibrium, the conditional probability p(z_m|z₁,…, z_m–1) transforms into a localised concentration based probability p(z|x) at position x, which means that the microscopic counterpart to spatial information takes the form

                                                                               (3)

This is slightly different from the macroscopic quantity K_spatial previously defined in the context of information dynamics, since the “concentration” p also contains the “empty state”; one can say that p is based on particles per unit volume, while the c concentration used before is based on fractions of particles of different types. This is just a difference in normalisation factor and leads to a conversion factor between the microscopic and the macrosopic quantities. This means that there is a direct relation between the macroscopic and the microscopic formalisms for quantifying information in patterns.