Complexity as the Rate of Entropy Growth

We propose that the complexity of an open system is the rate at which it increases entropy. We place this claim in a specific setting: a focal subsystem sits inside a quasi-isolated host system. The host’s internal free-energy store drives entropy redistribution among its subsystems. We formalise and compare five candidate definitions: (1) the entropy growth rate of the subsystem together with its environment; (2) the logarithmic growth rate of the subsystem’s own entropy; (3) the partial rate of change of host entropy attributable to the subsystem, with all other subsystems held fixed; (4) definition 3 normalised by the subsystem’s entropy; and (5) definition 3 normalised by the host’s entropy. Definition 1 cannot tell subsystems apart once the environment is the rest of the host. Definition 2 gives zero for any stationary dissipative structure, a living cell included. Defini- tion 3, which we call absolute entropic complexity, avoids both failures. It is non-negative, it is additive over subsystems, and it equals the exergy destruction rate divided by the reference temperature (Gouy–Stodola). Definitions 4 and 5 then give an intensive specific complexity, theinverseofanentropy-turnovertime, anda relative complexity whosesumoversubsystems equals the logarithmic entropy growth rate of the host. This last identity links the levels of a hierarchy. Definition 2 applied to the host is assembled from definition 5 applied to its parts. Quasi-isolation is exactly the condition under which the same rules can be applied one level up. We discuss how subsystems compete withinasharedfree-energybudget,howthemeasuresrelatetoChaisson’senergyratedensity, maximum entropy production and Kolmogorov–Sinai entropy, and where the approach is weak. Its main weakness is that it cannot separate organised from merely fast dissipation.