  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 characterised 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 the ideas and methods that are originated in analysis of
turbulent combustion but reviews a much broader scope of issues linked to
mixing and competition, including thermodynamic characterisation of complex
competitive systems with self-organisation. The approach presented here is
interdisciplinary and is addressed to a general educated reader, while the
mathematical details can be found in the Appendices.
