Mixing, entropy and competition

Abstract

Non-traditional thermodynamics, applied to random behaviour associated with turbulence, mixing and competition, is reviewed and analysed. Competitive mixing represents a general framework for the study of generic properties of competitive systems and can be used to model a wide class of non-equilibrium phenomena ranging from turbulent premixed flames and invasion waves to complex competitive systems. We demonstrate consistency of the general principles of competition with thermodynamic description, review and analyse the related entropy concepts and introduce the corresponding competitive H-theorem. A competitive system can be characterized by a thermodynamic quantity-competitive potential-which determines the likely direction of evolution of the system. Contested resources tend to move between systems from lower to higher values of the competitive potential. There is, however, an important difference between conventional thermodynamics and competitive thermodynamics. 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. Intransitivities are common in the real world and are responsible for complex behaviour in competitive systems. This work follows ideas and methods that have originated from the analysis of turbulent combustion, but reviews a much broader scope of issues linked to mixing and competition, including thermodynamic characterization of complex competitive systems with self-organization. The approach presented here is interdisciplinary and is addressed to the general educated readers, whereas the mathematical details can be found in the appendices.

Complexity and intransitivity in technological development

“…Existence of technological, economic and historic cycles is well documented in the literature (as reviewed by Klimenko, 2008Klimenko, , 2010Klimenko, , 2012 Figure 4.…”

“…Canonical (Gibbs) distributions represent a system property, which in addition to conventional thermodynamics can be encountered in different physical circumstances: two-dimensional turbulence, probability measures on graphs, and competitive systems (see Klimenko, 2012). We understand these distributions quite well and do not see them as complex.…”

“…Intransitivity is also known to pose a problem of choice in democratic elections (Arrow, 1951). Intransitive choices are common or even ubiquitous in nature (especially when there exist multiple selection criteria) and can be identified as causes of complexity in competitive systems (Klimenko, 2012(Klimenko, , 2013.…”

A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

“…Those additional axioms lean on additivity properties, and the aim here is finding entropy functions that are unique up to linear transformations. In a similar way, a battery of papers by Klimenko (see [38,[73][74][75][76][77]) deals with concepts as mixing (see also [78]) and competition of systems. It is not our intention here to make a criticism on those papers.…”

“…Coming back to the thermodynamical approach, we pay attention to the reference [73] as well as to Section 3 in [1].…”