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Self-organization is a process where order of a whole system arises out oflocal interactions between small components of a system.  Emergy, spelled with an 'm', defined as the amount of (solar) energy used tomake a product or service, is becoming an important ecological indicator. TheMaximum Empower Principle (MEP) was proposed as the fourth law ofthermodynamics by the ecologist Odum in the 90's to explain observedself-organization of energy driven systems. But this principle suffers a lackof mathematical formulation due to an insufficiency of details about theunderlying computation of empower (i.e. emergy per time).  For empower computation in steady-state an axiomatic basis has been developedrecently by Le Corre and the second author of this paper. In this axiomaticbasis emergy is defined as a recursive max-plus linear function.  Using this axiomatic basis and a correspondance between ecological theory anddynamic systems theory, we prove the MEP. In particular, we show that theempower computation in steady-state is equivalent to a combinatorialoptimization problem.

Self-organization and the Maximum Empower Principle in the Framework of max-plus Algebra