A continuity equation for information

Closed system

We can view the information density k as a generalised form of exergy (or free energy). By the flow across length scales jr we have accounted for the destruction of information from entropy production in chemical reactions and diffusion within the system. We will now derive a continuity equation for information which will connect the previously defined flow across length scales, jr(r, x, t), with the change of local information k(r, x, t) and a flow of information in space j(r, x, t). For a closed chemical system such a continuity equation takes the form

 

         clip_image003.                                                            (24)

 

The equation states that, for a closed system, the local information density k changes due to accumulation of of the information flows across length scales (in the downward direction) and across space. This continuity equation then implies a definition of the clip_image006 term,

 

         clip_image009                                                (25)

 

If we require that j(rxt) = 0 when clip_image012 is uniform and that j is rotation-free, the spatial flow is defined

 

         clip_image015                                        (26)

 

Note that the spatial flow depends on the presence of reactions. This means that when reactions are not present, the only flow is the one across length scales. This is a direct consequence of the fact that the resolution operator, the Gaussian blur, is equivalent to a diffusion process. The flow across length scales, though, depends on both reactions and diffusion. This should be expected since it is a generalisation of the entropy production, and entropy is produced in both these processes.


Open system

Finally, we open the system for inflow and outflow of molecules, given by the term B in the reaction-diffusion dynamics, Eq. (16). In addition to the terms in the continuity equation for the closed system, we need to introduce a local source/sink term J(r, x, t) which will capture the effects from the system being open. The continuity equation for an open system then takes the following form,

 

         clip_image003 .                                               (27)

 

For an open system, there are two ways in which the information characteristics of the system is directly affected from the flow of molecules across the system border. First, an inflow and an outflow of components changes the average concentrations of the transported components and in that way the chemical information is changed. For a driven chemical system, with an inflow of a fuel component and an outflow of a product, the flow across the system boundary typically keeps the chemical information at a level sufficient for driving the information flows in the system. But, there is also a direct influence that the inflow may have on spatial patterns in the system. For example, diffusion into or out of the system of a component that has a spatial variation, directly leads to that spatial pattern being less accentuated, and in that way the flow destroys the local information density clip_image006.

 

The negative effect on informaiton density k from the diffusion over the system boundary is captured by the sink term J in the continuity equation. For a diffusion controlled flow, as in Eq. (17),  we get (after some calculations) the following expression for J,

 

         clip_image009,                                                        (28)

 

which shows that J is a sink for information. In conclusion we have

 

         clip_image012

         clip_image015                      (29)

         clip_image018,

         clip_image020.

 

These quantities can be integrated over either position space x or resolution length scale r in order to derive aggregated information quantities like k(r, t), i.e., the resolution-dependent information, or k(xt) and j(xt) which are the normal information density and the corresponding information flow. If we integrate the continuity equation (27) over resolution lengths r, we get the following balance equation for the information density k(x, t),

 

         clip_image023 .                                           (30)

 

Here the entropy production and the change in chemical information come from the upper and lower limits of the flow jr.