Self-organization and the maximum empower principle in the framework of max-plus algebra
Introduction
It has been observed since a long time (see e.g. Boltzmann, Podolinsky, 1880) that energy, as the ability to do work, plays an important role in our civilization. Nowadays, more and more people realize that complex systems such as ecological networks, social organizations, economic systems are energy driven systems.
Self-organization, or spontaneous order principle, states that any living or non-living disordered system evolves towards an “equilibrium state” or coherent state, also called attractor. Self-organization is observed e.g. in physical systems (Bar-Yam, 1997, Haken, 1978, Nicolis, Prigogine, 1977), in biological systems (Camazine et al., 2003), in social systems (Anzola et al., 2017), in mathematical systems/models, in economics, in information theory and informatics (see e.g. Kaufman, 1993 and references therein).
To explain self-organization of energy driven systems, the maximum power principle has been proposed in e.g. Lotka (1922) and Odum and Pinkerton, 1955. This principle states that:
“system designs develop and prevail that maximize power intake, energy transformation, and those uses that reinforce production and efficiency”.
The major drawback of such approach is that complex energy systems can use energies of different kinds, e.g. renewable energies (solar, wind, ...) fossiles energies (fuel, gaz, coal), nuclear energy. Moreover, different energies do not have the same time scale. In Odum (1996, Chap. 2) the concept of energy hierarchy is introduced. It means that if the Sun is the reference point and is considered to be instantaneously available, then e.g. the fuel requires thousands of years to be used by human beings. And these two energies do not have the same calorific power.
In order to address this problem the ecologist Odum proposed the concept of emergy (spelled with an ‘m’ which is a neologism for energy memory). This term was coined in the mid-80’s in e.g. Scienceman (1987). In Odum (1996, p. 7) it is defined as follows: “Emergy is defined as the available energy of one kind previously used up directly or indirectly to make a service or a product”. It is a cumulative function of available energy and its unit is the emjoule. Recalling as above mentioned that different kinds of energies do not have the same ability to do work, Odum proposed to take the Sun as the reference point and defined the solar emergy as the available solar energy used directly or indirectly to make a service or product. Its unit is the solar emjoule, abbreviated sej (Odum, 1996, p. 8). Thus, solar emergy can be considered as a metric for environmental assessment which allows to compare different energy systems doing the same functions on the same basis: the Sun.
The major contributions of Odum are:
Transformity. To take into account the different time scale of energies, Odum introduced the dimensionless number he called transformity. The transformity is defined as the emergy required to make one Joule of a service or product (Odum, 1996, p.10, p. 288), so that we have:Process path function. Emergy of a product or service is a function of solar energy and its value depends on the scenario followed by the solar energy to generate the product or service under examination.
Maximum Empower Principle (MEP). Defining the empower as the emergy per time Odum proposed the maximum empower principle (MEP) to explain self-organization of energy networks as a Universal principle (fourth law of thermodynamics).
MEP: “In the competition among self-organizing processes, network designs that maximize empower will prevail” (Odum, 1996, p. 16). A network design that maximizes empower is named a sustainable design (Odum, 1996, p. 279).
The concept of emergy as an holistic paradigm which allows to compare two energy systems on the same basis (i.e. solar emergy) has generated a lot of literature on the subject and has been successfully applied on many domains (see e.g. Chen et al., 2017 and references therein).
But the concept of emergy has also generated debates and criticisms (see e.g. Hau and Bakshi, 2004 and references therein). As mentioned in Hau and Bakshi (2004, Section 3.2): “it is important to note that many criticisms are also valid for other methods [...] including Life Cycle Assessment, Cumulative Exergy analysis,...”.
However, the major drawback of the empower computation was a lack of mathematical formalism. Assuming the following hypothesis (see Section 2 for details):
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(A0) Steady-state analysis.
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(A1) No creation of emergy.
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(A2) Emergy in feedbacks cannot be added more than once,
The maximum empower principle as the fourth principle of thermodynamics has received criticisms (see e.g. Mansson and McGlade, 1993) and rebuttals (see e.g. Li, Lu, Tilley, Ren, Shen, 2013, Odum, 1995, Odum, 2002) since it was stated.
Under:
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assumptions (A0)–(A2)
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axiomatic basis developed in Le Corre and Truffet (2012a)
It is important to notice that our result does not depend on the exact definition of available energy. It is just implicitly assumed that it is a nonnegative quantity linked to energy concepts.
To the best knowledge of the authors only one pioneering work concerning the mathematical formulation of the MEP was developed in Giannantoni (2002). This work is based on:
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assumptions (A1)–(A2),
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linear algebra,
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fractional calculus,
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available energy defined as exergy.
The framework of Giannantoni (2002) is more general than the one of this paper however we can make the following two remarks:
Linear algebra is not the appropriate framework for emergy computation. Indeed, it has been noticed in e.g. Patterson (2014) that this approach can lead to absurd results such as negative transformities.
The emergy is defined as the space and time integral of the exergy but, in fact, the Gibb’s free energy is used (see Sciubba, 2010, p. 3700, footnote 4).
First, we introduce in Section 2 two important notions, which are emergy graph and emergy path, then we recall the axiomatic basis (developed in Le Corre and Truffet, 2012a) on which the MEP is proved.
Then, in Section 3, we present the correspondence between ecological theory and dynamic systems theory (see Table 2). Using this correspondence we establish the MEP (see Theorem 2).
Section 4 is devoted to a numerical example which illustrates all the concepts developed in the paper.
Finally, in Section 5, we reformulate the MEP using our settings and suggest a new line of approach for empower computation.
Section snippets
Emergy calculus reminder
In this section we recall basic materials to compute empower or emergy through networks. A network is modelled by a particular valued directed graph named in the sequel emergy graph which is a multiple inputs multiple outputs system. Let us recall and detail our main three assumptions.
(A0) Steady-state analysis. It means that the characteristics of the emergy graph (topology, valuation) do not depend on the time.
(A1) No creation of emergy. The emergy received by an output cannot be greater than
A mathematical formulation of Odum's maximum empower principle
Let us consider an emergy graph G, where (see Section 2.1). We recall that the set of emergy paths (see Definition 1) is assumed to be given. We mainly use Notations 1, so denotes the set of all emergy paths ending by arc [l; l′]. The emergy function θ and the weight function ω of G (see Section 2.3) are assumed to be known. The emergy flowing on arc [l; l′] of G is defined as (see Definition 3) where φ is the auxiliary function of Definition 2.
Numerical example
We consider the example taken from Li et al. (2010) (whose emergy graph is given in Fig. 1), where the emergy of the sources are and . Let us compute the emergy flowing on arc [9; 13], i.e. Em([9; 13]).
By Theorem 1, there exists an attractor such that and, by Theorem 2, .
Here, contains at most 26 emergy states because there are 6 emergy paths that end by arc [9; 13] (see Table 1). However, it is possible to
Conclusion
Recall that the maximum empower principle (MEP) was expressed by Odum as maximization of ecological network designs:
“In the competition among self-organizing processes, network designs that maximize empower will prevail” (Odum, 1996, p. 16).
In this work, we have proposed and used a correspondence between ecological theory and dynamic systems theory (see Table 2), so that the MEP can be stated as in Theorem 2:
“In the competition among self-organizing processes, emergy states that maximize
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
CRediT authorship contribution statement
Chams Lahlou: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. Laurent Truffet: Conceptualization, Methodology, Writing - original draft, Writing - review & editing.
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