What can we learn from a master plot of energy rate versus mass for a very wide variety of (complex) systems?
Authors: Van Duin Martin 
Published: January 2024

Mass and energy rate (ER) data have been collected for a wide variety of (complex) systems from the biological, cultural, and cosmological realms. They range from the cytochrome oxidase protein (10-22 kg and 6x10-19 W) to the observable universe (1.5x1053 kg and 1048 W) and, thus, span 75 mass and 66 ER orders of magnitude. Many of these systems are relevant for the big history (BH) narrative, i.e., the development of complexity over “big time” from the Big Bang up to the human society on Earth of today. The purpose of this paper is not per se to describe their history though, but to explore a master plot of ER vs. mass. Notably, the development of systems over big time has followed a rather tortuous path criss-crossing over this ER vs. mass master plot. The true mass of the system as a whole is used (for example trees including the non-living wood, living organisms including their intrinsic water, and social systems including the built constructs), because these inactive parts are essential for the performance of the system and facilitate its ER. A double logarithmic master plot of all ER vs. mass data shows clusters of data points. To some extent, this provides quantitative support for the distinction between the (sub-)realms, which is based on a qualitative description of their material structure and energy processing. In the master plot, small systems with low mass and ER converge into larger systems with larger mass and ER, which is typically accompanied by a decrease of the energy rate density (ERD = ER/mass). Correlation of ER with mass for various groups of systems demonstrates both sub- and supra-linear scaling with the power law β constant varying between 0.5 and 4.0, showing that the mechanisms of self-organisation are quite different for the corresponding system groups. The combination of convergence and scaling with β always larger than zero explains why the ER & mass data points fall in a diagonal band with a width of 17 orders of magnitude.

ER and mass have changed over wide ranges during the evolution of groups of systems, suggesting that evolution can be viewed as a process of systems exploring a larger ER vs. mass area until they run into ER and/or mass limitations. Indeed, there is a diagonal ER vs. mass limit for stable systems in all realms, corresponding to an ERD value of around 105 W/kg. Systems with ER & mass combinations above this limit, such as bombs, super-novae and cosmological transients, are unstable and “explosive”. This raises the interesting question of whether such an ERD maximum puts a limit on the development of complexity over big time. It seems that the low, right side of the master plot is empty. However, it is argued here that it is full of systems with low ER, such as dormant, living organisms, technological systems with their power adjusted or even switched off, as well as cooling, cosmological objects. Such systems are typically considered of less interest in a BH context, but they are viewed here as simple, complex systems which are out of equilibrium with matter, energy and information stored in their structure. While ERD appears to increase with the ‘advancement’ of systems over big time [5,51,52], there are quite some confounding factors regarding the efficacy of ERD as a metric for complexity in BH. For example, ERD decreases during the lifetime of a human and the human society (the mass of human-made constructs has grown faster than the global energy consumption), as well as during the evolution of living organisms and stars, whereas complexity is considered to increase. High ERD values of system parts may be illustrative for the complexity of the larger system, but are not representative for ERD of the system itself. Machines with an increased efficiency of energy conversion have a lower ERD, but could be considered more complex. The smallest and largest ERD values observed for the various realms appear to correlate with activity level and reciprocally with size, which do not per se reflect complexity. It is hoped that the raw data collected and the major trends observed in this paper will offer new insights into various aspects of the evolution of the universe over big time, and serve as an important resource for other related studies.