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Abstract:The algorithmic entropy of a system, the length of the shortest algorithm that specifies the system's exact state adds some missing pieces to the entropy jigsaw. Because the approach embodies the traditional entropies as a special case, problematic issues such as the coarse graining framework of the Gibbs' entropy manifest themselves in a different and more manageable form, appearing as the description of the system and the choice of the universal computing machine. The provisional algorithmic entropy combines the best information about the state of the system together with any underlying uncertainty; the latter represents the Shannon entropy. The algorithmic approach also specifies structure that the traditional entropies take as given. Furthermore, algorithmic entropy provides insights into how a system can maintain itself off equilibrium, leading to Ashby's law of requisite variety. This review shows how the algorithmic approach can provide insights into real world systems, by outlining recent work on how replicating structures that generate order can evolve to maintain a system far from equilibrium.

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Replication can be envisaged as a computational process that is able to generate and maintain order far-from-equilibrium. Replication processes, can self-regulate, as the drive to replicate can counter degradation processes that impact on a system. The capability of replicated structures to access high quality energy and eject disorder allows Landauer's principle, in conjunction with Algorithmic Information Theory, to quantify the entropy requirements to maintain a system far-from-equilibrium. Using Landauer's principle, where destabilising processes, operating under the second law of thermodynamics, change the information content or the algorithmic entropy of a system by ΔH bits, replication processes can access order, eject disorder, and counter the change without outside interventions. Both diversity in replicated structures, and the coupling of different replicated systems, increase the ability of the system (or systems) to self-regulate in a changing environment as adaptation processes select those structures that use resources more efficiently. At the level of the structure, as selection processes minimise the information loss, the irreversibility is minimised. While each structure that emerges can be said to be more entropically efficient, as such replicating structures proliferate, the dissipation of the system as a whole is higher than would be the case for inert or simpler structures. While a detailed application to most real systems would be difficult, the approach may well be useful in understanding incremental changes to real systems and provide broad descriptions of system behaviour.

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