Mathematical Formulation of the Maximum Em-Power Principle

1. Introduction

The idea of giving a mathematical formulation to the Maximum Em-Power Principle originated from an explicit invitation on behalf of Prof. Howard T. Odum when we met at ENEA’s Casaccia Reserch Center (Rome) on May 25, 1995.
My interest in the Emergy Analysis had started about three years before, when Prof. Ugo Farinelli (at that time Advisor of ENEA’s President) sent to Dr. William Mebane, Director of our Energy Saving Division, a scientific paper prepared for an official publication1, from which I had a preliminary idea of the new physical quantity termed as Emergy. During the following 9 months I had several epistolary contacts with Prof. Sergio Ulgiati (first author of the paper), whom I continuously asked for additional documents to better understand the logical and physical origin of such a new concept, together with its rigorous definition. What first impressed me was the special Algebra adopted (termed as “Emergy Algebra”), which was very different from the traditional one, but at the same time so “similar” to the Differential Fractional Calculus I had begun to study six years before. In fact, the first time I met Prof. Ulgiati, I did not miss the opportunity of asking him: “Why don’t you adopt Fractional Calculus in Emergy Analysis?”. Some weeks later Prof. Ulgiati invited me to attend a special course on Emergy Analysis given by Prof. Mark T. Brown in Siena (September 1993), during which I had to the opportunity of meeting another extremely important person in my life. The same Prof. Brown, in fact, encouraged my first attempts at introducing the “mathematically Equivalent Source Terms” in Emergy Algebra, which represented the fundamental step for the successive mathematical formulation of the Maximum Em-Power Principle. However, as previously mentioned, the idea of giving a mathematical formulation to the maximum Em-Powwer Principle was born when I met Prof. Howard T. Odum. The latter, in fact, invited by our Division to give two splendid Lectures at ENEA’s Headquarters (May 24, 1995), was so kind as to spend (together with his Lady Prof. Elisabeth Odum and Prof. Ulgiati) a whole day with me, answering my questions, dissolving my doubts, and suggesting possible new lines of research (I cannot fail to mention that, among other things, he also gave me five of his most famous books as a present). At the end of that day, after experiencing such a profound willingness shown by Prof. Odum, I frankly expressed my perplexity about the Maximum Em-Power Principle when termed as “Thermodynamic” Principle. In fact, I observed, without a general mathematical formulation of such a Principle it would be difficult to decide whether the M. Em-P. Principle was to be considered a “Thermodynamic” Principle or not.Prof. Odum, with a sweet and delicate smile, promptly answered: “I agree with you, and I believe you can succeed in this task”. That hope, so openly manifested by Prof. Odum, indelibly marked the “official” birth of my attempts at giving such a mathematical formulation. I systematically worked on this subject for six years, during which his paternal look encouraged me in this difficult task. His faithful invitation sustained me especially when the major difficulties arose, during the successive stages of the formulation. In this sense, such a formulation is more Prof. Odum’s merit than mine. This is because, without his sweet smile and paternal trusting invitation, I would have never found the courage to face such a hard task.

2. Mathematical Formulation of Maximum Em-Power Principle

For the sake of simplicity, such a mathematical formulation, which was achieved through four logical steps, is synthesized in “The Rational of the Mathematical Formulation of M. Em-Power Principle” . The afore-mentioned steps, however, are fully and widely illustrated in the two annexed papers, which were presented at the First and Second Emergy Conferences, held in Gainesville (Florida, USA), in 1999 and 2001, respectively:

“Toward a Mathematical Formulation of the Maximum Em-Power Principle” (1999)

“Mathematical Formulation of the Maximum Em-Power Principle” (2001)


1 Ulgiati S., Odum H. T., Bastianoni S. Emergy use, environmental loading and sustainability. An emergy analysis of Italy. Ecological Modeling 73 (1994) 215-268 (received 9 December 1992, accepted 10 August 1993).