Let us for the moment separate ‘information’ from ‘complexity’, in hopes of later reuniting them.

Discussions of “information theory’ to which I have paid most attention are those which relate the Boltzmann//Gibbs/Shannon (BGS) formulae designed to distinguish random from non-random states.

For one example, Bob Ulanowicz, in the application of info theory to ecosystems, in his book ‘Ascendancy’, used the idea of ‘information’ as a factor which caused, or was associated with, departures from randomness in the relationships between interacting elements in an ecosystem. He examined nutrient flows, and was able to use the BGS formulae to define degrees of organization in such systems. Bob did not develop a full description of complexity in such systems, and I did not entirely agree with every use of the word complexity in his discussion. But he did identify levels of ‘mutual information -- non random flows between nodes -- and ranked such systems in terms of ‘average mutual information’.

In his 1994 paper on ‘relational quantum mechanics’, Carlo Rovelli characterized relationships between quantum systems as the basic building block of order in the U. Those relationships were characterized in terms of correlations -- or departures from randomness --in each system’s encounter with -- or ‘measurement’ of -- any other quantum system. And Rovelli used the BGS approach to defining how information -- correlation -- would be involved in such quantum interactions.

From there you get -- via some filling-in steps -- to the question of how many distinguishable relational states are created and maintained in a definable, distinguishable system -- like a rock, or a galaxy, or a planet, or a living thing.

Here is where Chaisson enters in. He has connected the rates of energy flow through things like stars, galaxies, planets, and living things. He can point out the obvious differences in the apparent number of distinguishable states in such systems, without precisely quantifying them, and connect that with the rates of energy flow. And thus he can talk about ‘complexity’ and its connection with free energy density flow, without precisely defining complexity.

So if you follow the trajectory of Chaisson’s thoughts -- which are better than any others I have found and are extensively documented -- you are led to defining complexity as the quantity of distinguishable relationals states per unit of time per unit of mass. And this will be proportional to free energy density per unit of time and mass.

And, nicely, we are now linking relational theories of order construction in the U with non-equilibrium thermodynamics -- energy flows and structures.

So here we are. But how do we identify and count the number of distinguishable relational states in a system?

Here is where my own thoughts tend to dwell at the moment, But I am not at all satisfied with the combination of intimations, intuitions, and fragments of coherence now extant in my thinking processes. And if I am not satisfied, I could not expect you to be.

However, here are exploratory thoughts.

I am led to go from benard cells to jet engines to hierarchical levels of living creatures, to focus on some aspects of complexity -- indeed, tiered levels of complexity.

As you know, in benard cells, with a layer of fluid and a heat differential, bottom to top in a gravitational well, the fluid organizes in hexagonal columns. It creates order in dissipating the differential. Inside each column is a lot of random motion. But the column correlational structure makes a layer of complexity -- just two levels -- randomness within columnar organization -- but high energy rate density.

Take a jet engine. Whirling fans, nacelle, organized pattern of combustion and exhaust. High energy density, some layers of order containing.

Now to a cell in an organism. After a few billions of years of trial and error, a lot of differentiated structures within a membrane. The membrane imposes some order at its level, which provides for sustainable levels of high probability for the highly energetic interactions within.

Now have the cell level, over evolutionary time, get organized -- non-randomized -- into relatively simple multicellular creatures. Again, borders on the MC creature, containing first relatively simple but over evolutionary time more complex internat organization -- patterned non random energetic processes, all in a containing level of order .

Now have the MC creatures over time evolve into ‘social’ creatures, with first simple to greater interactions between them. Social structures become ‘complex’ -- they have distinguishable subsystems, and energy flows through them.

So now we have tiered complexity. That is, we have tiered levels of distinguishable relational states at each level. We have differences in complexity -- amounts of distinguishable relational states, or systems -- at different levels. And we have high energy density in the system as a whole

So now how do we sum, in complexity terms, these tiers, or nestings, of distinguishable relational sets?

Also, at the edge of the thought process, one speculates on the relationship of mass to these concentrations of tiered complexity -- tiered organized energy flows.

And of course you can take the tiering of energy density down to the atomic and molecular levels.

Welcome to this thought process. I hope it stimulates some curiosity and some thoughts or suggestions.

Jack