Very good examples of such frameworks are given by non-equilibrium thermodynamics , statistical physics , algorithmic complexity (Kolmogorov–Chaitin complexity) and algorithmic entropy , as well as complex adaptive systems (CASs) . A general philosophical discussion of complexity might be interesting and informative, but a more quantitative scientific approach to this problem needs a more specific framework. This prediction has not come true yet, but, if it ever does, it seems most probable that the unknown laws will have something to do with studies of complexity.Īlthough complex systems can be expected to possess some common properties, complexity represents a notion that is easy to understand intuitively but difficult to define in rigorous terms . Schrödinger has also made a prediction that new laws of Nature explaining the complex working of living organisms will be discovered in the future. Erwin Schrödinger in his famous essay What is life? articulated that life forms operate in perfect agreement with the known laws of physics and must consume exergy (negative entropy) from external sources to support their existence. On the one hand, there are no violations of the first two laws of thermodynamics known to modern science on the other hand, it is not clear why Nature appears to be more complex than it has to be in order to comply with these laws. Turbulent fluid motions, the existence of life forms, the complexities of technological development and many other processes involving a substantial degree of coherent behaviour can be mentioned as phenomena that can hardly be explained by the known trend of entropy to increase in time. While thermodynamics has proved to be successful in explaining the common trend of moving towards equilibrium states, the complexity observed in many non-equilibrium phenomena may seem to be unnecessary if viewed from a thermodynamic perspective. It is not a surprise that thermodynamic principles are often invoked in relation to various evolutionary processes in which irreversibility plays a prominent role. Thermodynamics occupies a special place among other physical sciences by postulating irreversibility of the surrounding world as its principal law. Results of simulations demonstrating complex behaviour in abstract competitions are presented in the electronic supplementary material. The analogy with conventional thermodynamics weakens as competitive systems become more intransitive, while strongly intransitive competitions can display types of behaviour associated with complexity: competitive cooperation and leaping cycles. There is, however, an important difference: while conventional thermodynamics is constrained by its zeroth law and is fundamentally transitive, the transitivity of competitive thermodynamics depends on the transitivity of the competition rules. Competitive systems can thus be characterized by thermodynamic quantities (such as competitive entropy and competitive potential), which determine that the predominant direction of evolution of the system is directed towards higher competitiveness. Transitive competitions are shown to be consistent (at least qualitatively) with thermodynamic principles, which allows for introduction of special competitive thermodynamics. Two classes of competition rules, transitive and intransitive, need to be distinguished. Although the concept of abstract competition has been derived from a specific field-modelling of mixing in turbulent reacting flows-this concept is, generally, not attached to a specific phenomenon or application. This publication reviews the framework of abstract competition, which is aimed at studying complex systems with competition in their generic form.
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