Title: Self-organization and the Maximum Empower Principle in the Framework of max-plus Algebra,
Abstract: Self-organization is a process where order of a whole system arises out of
local interactions between small components of a system.
Emergy, defined as the amount of (solar) energy used to make a product or a
service, is becoming an important ecological indicator. To explain observed
self-organization of systems by emergy the Maximum Empower Principle (MEP) was
proposed initially without a mathematical formulation.
Emergy analysis is based on four rules called emergy algebra. Most of emergy
computations in steady state are in fact approximate results, which rely on
linear algebra. In such a context, a mathematical formulation of the MEP has
been proposed by Giannantoni (2002).
In 2012 Le Corre and the second author of this paper have proposed a rigorous
mathematical framework for emergy analysis. They established that the exact
computation of emergy is based on the so-called max-plus algebra and seven
coherent axioms that replace the emergy algebra. In this paper the MEP in
steady state is formalized in the context of the max-plus algebra and graph
theory. The main concepts of the paper are (a) a particular graph called
'emergy graph', (b) the notion of compatible paths of the emergy graph, and (c)
sets of compatible paths, which are called 'emergy states'. The main results of
the paper are as follows:
(1) Emergy is mathematically expressed as a maximum over all possible emergy
states. (2) The maximum is always reached by an emergy state. (3) Only prevail
emergy states for which the maximum is reached.