Wp

ED

a

ARRAA

KLMEEE

1p

tAaetseaopap1ttl

0d

Ecological Modelling 222 (2011) 1347–1353

Contents lists available at ScienceDirect

Ecological Modelling

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

hat did Lotka really say? A critical reassessment of the “maximum powerrinciple”

nrico Sciubba ∗

ept. of Mechanical & Aerospace Engineering, University of Roma 1 – La Sapienza, via Eudossiana 18, 00187 Rome, Italy

r t i c l e i n f o

rticle history:eceived 3 November 2010eceived in revised form 9 January 2011ccepted 2 February 2011vailable online 2 March 2011

eywords:otkaaximum power principle

a b s t r a c t

This paper presents a critical discussion of the so-called “maximum power principle”, often quoted instudies about the energy balance of living systems and also known in the emergy literature as “maximumem-power principle”. Several authors consider this principle highly relevant and some even proposedit as a “fourth law of thermodynamics”. A critical analysis of the original source, namely Alfred Lotka’s1921–22 papers, conducted both in an historical perspective (the connection between Lotka’s writingsand the ongoing debate at his time) and in a more modern context, leads to a more detailed and less biasedassessment. It turns out that in spite of Lotka’s very anticipatory and incredibly sharp vision of the possibleinterconnections between the second law of thermodynamics and evolutionism, doubts arise about thegeneral applicability of his “maximum power principle”. From an accurate analysis of his writings, it

xergy

ntropy generation minimizationvolution

can be concluded that: (a) Lotka explicitly and consistently addressed the “optimal use” of the flow ofexergy (available energy), and therefore the quantity defined as “em-power” is an incorrect interpretationof Lotka’s constrained maximum power principle; (b) “Lotka’s principle” can be reformulated withinZiegler’s “maximum entropy production” or Prigogine “minimum entropy generation” paradigm onlyunder two different respective sets of rather stringent additional conditions which Lotka was probably

er exp

already aware of but nev

. A brief history of the so-called “maximum empowerrinciple”

In 1921 and 1922, the 23-year old Alfred Lotka publishedwo papers (Lotka, 1922a,b) in the Proceedings of the Nationalcademy of Science to present his ideas on how the evolution offar-from-equilibrium thermodynamic system is correlated to itsnergy-absorption capacity. These papers are considered to markhe beginning of ecological thermodynamics, in that they explicitlytate that the likeliness of a living system to survive and successfullyvolve is linked to its ability to capture the energy made avail-ble by the surroundings. In the 1970s, H.T. Odum, the founderf the Emergy School, adopted and reformulated of Lotka’s princi-le, which he renamed “maximum em-power principle” (defineds the flow rate of emergy): “The time rate of change of emergy is em-ower, analogous to the time rate of change of energy, power.” (Odum,

995).: “During self organization, system designs develop and prevailhat maximize power intake, energy transformation, and those useshat reinforce production and efficiency” (Odum, 1995, p. 311). Inater writings, he omitted the second part of the above definition,

∗ Correspondence. Tel.: +39 06 44585244; fax: +39 06 44585249.E-mail address: enrico.sciubba@uniroma1.it

304-3800/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2011.02.002

licitly stated.© 2011 Elsevier B.V. All rights reserved.

and wrote: “In the self-organizational process, systems develop thoseparts, processes, and relationships that maximize useful em-power”.And he further clarified “Designs prevail that maximize em-power”(both quotes from (Odum and Odum, 1981)).

Since its rediscovery in the late 1980s, the “maximum em-power principle” (sometimes referred to as the “Lotka–Odum orDarwin–Lotka principle”) has been considered by some authorsas the fourth law of thermodynamics, mainly for its appeal as aphenomenological modelling tool, but also because of its claimedpractical validity for a very wide class of physical and biologicalsystems (Dobzhansky et al., 1977; Glansdorff and Prigogine, 1974;Sciubba and Ulgiati, 2005). One must though note that no final proofof its applicability was ever published (Valyi, 2005, p. 4).1

In analysing large complex systems, in line with Odum’s state-ments, emergists maintain that “the total available emergy flowdrives the system behaviour according to the maximum em-power

principle, determining the size of the system itself and its growthrate” (Brown and Ulgiati, 2001; Hall, 2004; Ulgiati and Brown,2001).

1 deLong’s paper on protists and metazoans (DeLong, 2008) is not considered“conclusive” here because the results shown therein may be interpreted under adifferent paradigm (Sciubba, in press).

1 dellin

ttrottaiiwtne

pdetbt

aisp

tp

2

m

tps(mi

[

opp

fmc1a

wiTn

n

ra

348 E. Sciubba / Ecological Mo

Such statements sound perplexing to exergy theorists and prac-itioners: to say that “systems prevail that maximize the integral ofheir energy absorption rate” (which is the meaning of the aboveeported quotes) does not seem to fit into the undisputable “econ-my” of evolutionary phenomena, because all available data showhat the winners in the evolutionary struggle are instead struc-ures (living or not) that use the several forms of energy they canvail themselves of (radiation, thermal, chemical. . .) not accord-ng to “maximum throughput” but in the “most efficient” way,.e., with the least dissipation compatible with their life-cycle and

ith the environment they are immersed in. Recall that “dissipa-ion” is a concept absent from the emergy theory that assumeson-dissipative “conservation laws” (Giannantoni, 2002; Jorgensent al., 2004; Odum, 1995).

On the other hand, if Lotka’s principle were a “maximum exergyower principle”, that is, if it could translate in a form like “In nature,esigns prevail that are capable of capturing exergy at the high-st possible rate compatibly with the environmental conditions”,his would be perplexing as well from the exergist point of view,ecause also in such a form the statement would not account forhe “conversion efficiency” of the system, however measured.

Thus, even leaving aside the ongoing debate between emergistsnd exergists (Sciubba, 2008, 2010; Sciubba and Ulgiati, 2005), thessue of the maximum em-power is a troubling one, and to gainome enlightenment a closer analysis of the original formulationroposed by Lotka (Lotka, 1922a,b) is in order.

Scope of this paper is to critically reexamine Lotka’s statements,o reassess them and to re-evaluate their validity in light of ourresent thermodynamic concepts.

. Relevant thermodynamic definitions

This section provides a list of definitions for some of the ther-odynamic quantities used in the paper.Energy is a prime concept in Thermodynamics. It is defined as

he ability of a system to cause change and, more specifically, toerform mechanical work. In living systems, “energy” is never con-idered as per its “content” but as per its “flowing”: thermal energyheat) flows from a body to another, and so do chemical energy,

echanical energy, electrical energy, and so on. Energy is measuredn Joules [J].

Power is the rate of flow of energy, and is measured in WattsW].

Emergy is the integral over space and time, in a life-cycle sense,f the solar radiation energy embodied in a natural or man-maderoduct. It is measured in equivalent solar J (seJ) per unit ofroduct.2

Em-power is the rate of flow of emergy, measured in seJ/s.Exergy is the maximum amount of energy that can be extracted

rom a system by bringing it into equilibrium with its environ-ent, under the condition that the set system∪environment is a

losed one (no interactions with the rest of the universe). Until the990s, following Gibbs’ definition, the name of available energy wasdopted in the anglo-saxon literature. Measured in J.

A system is said to be in equilibrium when there are no gradientsithin it. This is clearly a macroscopic approach, and when speak-

ng of equilibrium, the scale of the analysis ought to be mentioned.

wo systems are in equilibrium if when connected to each othereither one undergoes macroscopically detectable changes.

A system is said to be in a steady (or stationary) state whenone of its macroscopic measurables varies with time. Equilib-

2 For an exact definition, for a rather extensive list of references, and for a vividepresentation of the emergy/exergy debate which represents both the backgroundnd the motivation for the present paper, readers are referred to Sciubba (2008).

g 222 (2011) 1347–1353

rium states are by definition steady, while non-equilibrium statesmay be steady (the inner gradients in the system are continuouslymaintained by an external source of energy) or unsteady (the innergradients vary themselves with time).

3. Lotka’s statements

To begin with, it is important to understand that Lotka was con-sistently talking of available energy (i.e., exergy): in his footnote n.1in (Lotka, 1922a) he makes explicit reference to Boltzmann’s 1886paper (Boltzmann, 1886) in which not the energy, but the avail-able energy content of a flux was considered. This is an essentialpoint, and in the remaining of this paper Lotka’s postulate shall bereferred to as the “maximum exergy rate principle”.

A second important point is the following: Lotka correctly inter-preted the evolutionary principle in the “random” sense in which ithad been formulated by Darwin. He writes: “If sources are presented,capable of supplying available energy in excess of that actually beingtapped by the entire system of living organisms, then an opportunity isfurnished for suitably constituted organisms to enlarge the total energyflux through the system” (Lotka, 1922a, p. 147). Here again he addsa footnote to stress that the “energy flux” he is referring to must beintended as exergy flowrate.

Thus it appears that Lotka had in mind a sort of “theory of con-strained growth”: assume a system S absorbs a total exergy influxEin, which its N subsystems are using to survive: if – under what-ever circumstances – an additional exergy source �Ein becomesavailable, and as long as it remains available, some of these N sub-systems will modify their configuration in such a way to be able toabsorb this additional resource. It is interesting to remark, in view ofthe discussion in Section 4 herebelow, that he founds his postulateon what he presents as “empirical observation”: “In every instanceconsidered, natural selection will so operate as to increase the totalmass of the organic system, to increase the rate of circulation of mat-ter through the system, and to increase the total energy flux throughthe system, so long as there is presented a unutilized residue of matterand available energy” (Lotka, 1922a, p. 148).

Lotka also correctly understood that a minimum thresholddegree of complexity is necessary to allow for the emergence ofsystems capable of tapping an “above average” exergy flux: “It isnot lawful to infer immediately that evolution tends thus to make thisenergy flux a maximum. . .. If the material furnished for selection isstrictly limited. . .the range of operation of the selective influences isequally limited” (Lotka, 1922a, p. 148). He notes that in living organ-isms, intrinsically more complex than non-living systems, the poolof “types presented for selection” (the genes, in modern terms) is solarge that it appears highly probable that “sooner or later” a suit-able combination thereof emerges that allows evolution to enforceMERP rule.

But Lotka makes also another important distinction: he states(Lotka, 1922a, p. 149) that his maximum exergy rate principleapplies “compatibly with the constraints”. In modern terms, thisamounts to saying that of the N random mutations in a gene pool,only those that are capable of adjusting to the boundary conditionsand of tapping a higher-than-average exergy flux will be favouredin the struggle for existence. This is exactly the “random emerging,environment-dependent opportunism” that defines our modernvision of evolution (Dobzhansky et al., 1977).

In his closing remarks to his first 1922 paper, Lotka makesanother novel point, extremely important for the present analysis:

quoting a contemporary work by Johnstone (Johnstone, 1921), hecomments that it cannot be simply inferred, as Johnstone did, that“in living processes the increase of entropy is retarded”, but that theissue must be addressed from a global and systemic point of view.Lotka’s idea is that it is the system as a whole that may (although

dellin

tms(stt

benrL(eop

fimwecedttp(toon1ttipe(mofiainosao

hsbcewsepi

c(

E. Sciubba / Ecological Mo

hrough oscillations) evolve towards a lower dissipation rate (inodern terms, lower exergy destruction): subsystems may occa-

ionally prevail that display a higher degree of entropy productionhigher exergy destruction), but then again, sooner or later theseubsystems will find themselves challenged by competing subsys-ems operating within the same niche that are “more efficient”, i.e.,hat destroy a lower percentage of the incoming exergy flux.3

Lotka also indicates (Lotka, 1922a, p. 150) that it remains toe defined what the expression “compatible with the constraints”xactly means: today, we know from non-equilibrium thermody-amics that a complex system sufficiently far from equilibrium mayespond to even a small perturbation in a non-predictable way, butotka did not have such knowledge at his time. He had, thoughprobably through his work on systems of non-linear differentialquations of “evolutionary” type), an incredibly sharp perceptionf the possible implications of imposing weak constraints on a com-lex system.

Alfred Lotka’s 1921–1922 papers must be considered within therame of a debate that was ongoing at that time about “the phys-cal meaning of the evolutionary theory”. In addition to the above

entioned work by Johnstone, he quotes – among others – anotherork by Lodge (Lodge, 1906), who was trying to analyze the influ-

nce of life and mind upon the external world under the strongonstraint of the second law of thermodynamics (dissipation andntropy generation). In a different form, this problem is still beingebated today: are we living in a universe ruled by an extremiza-ion (minimum or maximum) of the entropy generation or not? Inhe first two decades of the twentieth century, the question wasosed differently: do the first and second law of thermodynamicsand the third, which Lotka quotes by making an explicit referenceo Nernst) constitute a sufficient axiomatic basis for the descriptionf our universe? Lotka thinks they do not: “Whether life is presentr not, something more than the first and second law of thermody-amics is required to predict the course of events” (Lotka, 1922b, p.52). “Real phenomena are irreversible” he notes, and in particularhreshold phenomena (which he calls “trigger action phenomena”)hat “release available energy from a false equilibrium”, require thentroduction of new principles. Lotka thinks that evolution (morerecisely, natural selection driven by some Lagrangian) is the nec-ssary “fourth principle” missing hitherto. He also explicitly statesLotka, 1922b, p. 153) that he “has long realized” that his constrained

aximum principle may well be considered the fourth principlef thermodynamics, but that he is inclined to wait until the neweld of the “dynamics, or energetics of evolution” has been system-tically explored: he writes that he has enough material on thisssue, and that it “will be brought forward” at a later time, which itever was. We can only conjecture that he was referring to his workn non-linear systems of equations (Lotka–Volterra predator–prey

ystems, which in his formulation are though non-dissipative4). Inlater, more formal work (Lotka, 1925), Lotka treats the subject

nly in passing.

3 In Lotka (1921), an extraordinarily anticipatory paper published in 1921 in whiche tried to link economics to second law principles, Lotka had been even bolder: aftertating that “the body of a living organism in not in equilibrium with its surroundings”ut that, within the frame of his analysis, it is in an approximately steady state, heoncludes that “every organism. . .must be provided with devices for capturing availablenergy”, and that, due to the competition for survival, “the advantage must go to thosehose energy-capturing devices are most effective in directing available energy into

uch channels as are favorable to the preservation of the species”. This statement isntirely equivalent to saying that organisms prevail that destroy the least possibleortion of the incoming exergy (i.e., that operate with the least irreversibility), and

s somewhat at odd with the “maximum power principle”.4 Recently, it has been shown that a set of equations of the Lotka–Volterra type

an lead to “emerging behaviours” if dissipation (i.e., exergy destruction) is addedSciubba, in press).

g 222 (2011) 1347–1353 1349

4. Can Lotka’s principle be regarded as “The Fourth Law ofThermodynamics”?

As we have seen, long before emergy analysts elected their re-formulation of Lotka’s principle as the perspective fourth law ofthermodynamics, Lotka himself had regarded his constrained max-imum exergy rate principle as worthy of being added to the threeexisting thermodynamic laws. On the basis of the critical analysisdeveloped in Section 2 above, is it correct to conclude that besidesthe assumptions of existence of an a priori tendency towards equi-librium (0th law), the energy conservation (1st law) and the entropygeneration (2nd law), a fourth postulate is necessary and sufficientto produce a complete axiomatic set of thermodynamic principles?The question is obviously important because its answer has pro-found implications on our concept of thermodynamics as a science.Let us separately consider some of the many facets of the problem.

4.1. Systems at equilibrium (classical thermodynamics)

In this field, Lotka’s principle is clearly of limited physical andlogical relevance. The analysis of systems of whatever degree ofcomplexity, as long as they remain within the bounds of equi-librium thermodynamics, can be performed on the basis of thethree classical laws, and does not need any additional assump-tion. All equilibrium processes and cycles we have studied to datecan be perfectly described by means of the three existing postu-lates and of proper material property laws. There is no “maximumpower” optimum: the conversion performance of a system is com-pletely identified, both in a qualitative and quantitative sense, byits structure (connectivity), by the performance of its elementarycomponent processes, by the physical properties of the workingmedia and materials, and by the applied boundary conditions (i.e.,by the ways the system interacts with its environment).

4.2. Systems “near” equilibrium (linear non-equilibriumthermodynamics)

Somewhat different considerations apply to systems operatingin perturbed conditions but whose state parameters (all the rel-evant variables influencing the system performance) remain in asufficiently small region of the solution space around their steadyor even dynamic equilibrium state.5 Notice that oscillating and“threshold” behaviour (as displayed by the Benard cells, by thechemical oscillators studied by Prigogine, etc.) also fall under thiscategory, as long as the participating structures tend to a dynamicequilibrium state (i.e., until the system remains in a state in whichthe topology of its internal dynamics does not change in time). Theabove rather general statements have been exactly quantified byOnsager (Onsager, 1931a,b), and later reformulated in a mathe-matically exact fashion by Prigogine and co-worker (Glansdorff andPrigogine, 1964; Nicolis and Prigogine, 1989): given a complete set

x of state parameters xj (where “complete” means that the xj are allrelevant and that x completely identifies the state of the system),and a control parameter ε that identifies the possible transitionsof the system from one to another region in its phase space,6 the

5 The undetermined quantifier “sufficiently” makes this definition somewhatambiguous: how “actually small” the region of (linear or non-linear) quasi-equilibrium is depends on the amplitude of the perturbation, on the system itself andon its interaction with the environment. But the fact that the n-dimensional radiusof this region can be determined in all known cases (possibly by trial and error)provides sufficient generality to the statement. A more precise definition providedby Zupanovic et al. (2010) is based on a Taylor expansion of the fluxes about theequilibrium state, but it is really not necessary in the present general discussion.

6 These transitions are real physical phenomena, like the onset of turbulence(ε = Re), the appearance of the Benard cells (ε = �T/T), the excitation of a diode

1 dellin

g

x

f

i

f

s1s(sst

yet

E

tepiuofi

�

rLfiaafb

(

(

(

wjatc

(c

rmt(

(

(

350 E. Sciubba / Ecological Mo

eneral evolution equations are of the form:

˙ j = f (xj, ε), j = 1, 2 . . . N (1)

At equilibrium the time derivative vanishes:

(xj,eq, εeq) = 0, j = 1, 2 . . . N (2)

But also at a non-equilibrium steady-state, with obvious mean-ng of the suffixes:

(xj,neq,ss, ε) = 0, j = 1, 2 . . . N (3)

For all cases in which Eq. (3) applies, and under a set of rathertringent conditions,7 Nicolis and Prigogine (Nicolis and Prigogine,977) derived their minimum entropy generation theorem, thattates that – as long as there is at least one thermodynamic forcei.e., an externally imposed gradient) of sufficient (but not too large)trength, the system maintains itself at a steady non-equilibriumtate by generating the least amount of entropy compatible withhe boundary conditions.

In the context of this paper, it is interesting to perform the anal-sis using exergy as the relevant global variable: in a system S inquilibrium or in a non-equilibrium steady-state, the time evolu-ion of the system’s exergy is governed by the exergy rate equation:

˙ in[J/s] = Eout + E� (4)

For an open system like those of interest here, Eq. (4) tells us thathe incoming exergy flux Ein (light, power, chemical- or materialxergy, heat. . .) is partially destroyed within S by internal dissi-ative processes (E�) while the remaining exergy flux is released

nto the environment: notice that Eout may include useful “prod-cts” (shaft- or electrical power in energy conversion devices, heatr motion in living cells) and “waste flows” (like hot gases, spentuels, and detritus): Eout = Eu + Ew . The “conversion efficiency” of Ss defined as:

= Eu

Ein

= 1 − Ew + E�

Ein

(5)

The response of such systems to an increase in the input exergyate from Ein to Ein + �Ein (the “excess available energy flux” ofotka’s principle) depends essentially on their structure (i.e., con-guration, or “connectivity”) and on the type of equilibrium theyre in (the curvature of the entropy, ı2s, at equilibrium (Glansdorffnd Prigogine, 1974)). If �Ein is “small enough”, the displacementrom equilibrium will be non-disruptive, and the system can adjusty three mechanisms:

a) by increasing its useful output Eu, usually with an increase ofE�;

b) by decreasing Eu, usually with a corresponding increase inE�/Ein;

c) by generating the same Eu.

In man-made systems, all three responses are observed: if S is aell-designed system operating at nominal conditions, and is sub-

ected to an increase of its exergy input, it will usually generate

higher useful output at reduced efficiency (case “a” above); but

here are some engineering systems, like wind turbines or axialompressors for example, that will display a lower output at a lower

ε = �V/V), etc. The critical value of ε at or beyond which a transition appears isalled in modern terms the threshold value.

7 These assumptions are: (a) local equilibrium; (b) validity of the linear Onsagerelations; (c) time constancy of the Onsager coefficients; (d) isotropic and isothermaledium in mechanical and thermal equilibrium with the environment. It is obvious

hat conditions c and d scarcely apply to biological systems. See the critique by Kay1984).

g 222 (2011) 1347–1353

efficiency (case “b”) and yet others (cogenerating plants in electri-cal load-following mode) that can be controlled to provide the sameuseful output with an increased waste exergy flux (case c). In anycase, such systems cannot “grow”, and Lotka’s principle does notapply to them.

If, however, S is a (very simple) living system capable of growth(a cell) or containing sub-systems that can grow within it (a pond),different behaviours are possible. Let us consider the simple caseof an ideal unicellular being S that uses light and oxygen (Ein) tosynthesize a material useful output Eu that helps it grow (for acell, by adding protoplasmic material, for a pond, by increasingthe numerosity of one of its algae colonies). If an additional �Ein

becomes available to S, the response prescribed by Lotka’s princi-ple would be the following: S generates more Eu, grows, and thusthe circulation of matter and energy within S is increased. Here wemust make the additional stipulation that the changes are so slowthat the system is not subjected to disruptive changes (fires, floods,epidemics, or similar catastrophes) in the entire period of interest.The question is then, is the growth mechanism guided by a “max-imum exergy rate capture” principle the only possible behaviour?The answer is clearly negative, as there are three instances in whichS would not show a tendency to grow:

a) the increment �Ein is so small that the additional useful out-put �Eu is destroyed by the increased dissipation rate �E�

(efficiency limit);b) the increment �Ein generates an additional useful output �Eu

that is below a certain threshold required by the system to grow(activation limit);

(c) the increment �Eu is impossible under the prevailing boundaryconditions (saturation limit).

We must conclude that Lotka’s maximum exergy rate princi-ple is not general enough to describe the behaviour of systemsthat remain “near” equilibrium or in a non-equilibrium (possiblydynamic) steady-state. In other words, the maximum power prin-ciple does not apply to systems that fall within the domain of linearnon-equilibrium thermodynamics.

4.3. Systems “far from” equilibrium (non-linear non-equilibriumthermodynamics)

Here, the situation is quite different: systems “far enough” fromequilibrium are known to behave in a way that is no longer pre-dictable solely on the basis of the 1st, 2nd and 3rd Law8 (Glansdorffand Prigogine, 1964; Martyushev and Seleznev, 2006; Nicolis andPrigogine, 1977). A system far enough from equilibrium may goback to its previous equilibrium state following different relax-ation paths, may move to other equilibrium states or may remainin the non-equilibrium region forever. Such a behaviour is oftendescribed by saying that the system acquires or displays “emerg-ing properties”: this expression is misleading though, because itonly means that the system behaves in a way that we could notpredict in advance (nothing really “emerges”). The problem wasclear to Lotka, who writes “If these laws do not fully define the courseof events, this does not necessarily mean that this course, in nature,

is actually indeterminate, and requires, or even allows, some extra-physical influence to decide happenings. It merely means that the laws,as formulated, take account of certain factors only, leaving others outof consideration; and that the data thus furnished are insufficient to

8 This does not mean that such systems contradict the three general laws, whichcontinue to be applicable a posteriori (i.e., once the state of the system is determined).It means though that the time-evolution of the system (its future states) cannot bepredicted solely by means of the three laws (as Lotka explicitly stated).

dellin

yeotci

pta“gpe

ejteaepw

(

etS1elnabtsenmt

to re-process Cai’s results to see if they were compatible with aminimum exergy destruction paradigm (which is essentially thesame as EGM).

E. Sciubba / Ecological Mo

ield an unambiguous answer to our enquiry regarding the course ofvents in a physical system” (Lotka, 1922b, p. 152). In this sense, thenly “emerging” fact is that we need additional principles (possiblyaking other hitherto neglected state parameters into account) toorrectly describe (i.e., predict) the time evolution of these systemsn their state space.

The prevailing opinion today is that for all relatively sim-le auto-catalytic systems that fall under this category, of whichhe Glansdorff and Prigogine chemical oscillators (Glansdorffnd Prigogine, 1964) are perhaps the best known example, theadditional principle” is a properly generalized form of entropyeneration extremization rather than the “maximum exergy rate”roposed by Lotka (Martyushev and Seleznev, 2006; Zupanovict al., 2010).

Ninety years after Lotka’s writings, we still lack though a gen-ral theory for the description of such phenomena, and thus theudgement ought to remain suspended. My personal feeling ishat a consistent application of a properly constrained minimumntropy generation (EGM) principle (Bejan, 1996; Gyftopoulosnd Beretta, 1995) may be successful even in non-linear non-quilibrium thermodynamics, but the prevailing opinion seems torefer the maximum entropy production principle (MEP): muchork is needed in this direction, but we should consider that:

(a) if EGM could be proven to be applicable to real systems, itwould be a suitable complement for Lotka’s maximum exergyrate postulate (reshaping it into a form of “constrained growth”assumption). We would, in practice, see an embodiment ofPrigogine’s intuition: that periodic, quasi-periodic and chaoticbehaviour of far from equilibrium systems may be consistentlydescribed by the three laws of classical thermodynamics withthe addition of an “economic” principle that guides the evolu-tion of these systems in such a way as to minimize the exergydestruction rate under the given boundary conditions (exergyinput, a prescribed target output and the environment condi-tions);

b) if MEP could be proven to be applicable to real systems, thiswould be a rewording of Lotka’s maximum exergy rate postu-late: the system evolves in such a way as to deplete at the fastestpossible rate the externally imposed gradients (Beretta, 2009;Kay, 2002; Martyushev and Seleznev, 2006), so that the rateof entropy generation would be a maximum. This is somewhatdifferent from Lotka’s idea of “maximum exergy rate capture”:in exergy terms, MEP can be rephrased as “for a constant exergyinput rate, a far-from-equilibrium system evolves in such a wayas to maximize the exergy destruction”, while Lotka meant “thefar-from-equilibrium system evolves in such a way as to maximizeits exergy input rate, no matter at what cost in terms of exergydestruction”.

In either case, in spite of some very authoritative doubtsxpressed in (Nicolis and Prigogine, 1977), the prevailing opinionoday (Beretta, 2009; Gyarmati, 1970; Lucia, 2007; Martyushev andeleznev, 2006; Nicolis and Prigogine, 1977; Ziegler and Wehrli,987; Zupanovic et al., 2010) is that the behaviour of far-from-quilibrium systems may be consistently described by the threeaws of classical thermodynamics with the addition of an “eco-omic” principle that guides the evolution of these systems in suchway as to extremize the exergy destruction rate under the givenoundary conditions (exergy input, a prescribed target output andhe environment conditions). Fath et al. (2001) have analysed a

eries of “indicators” based on different metrics (energy, emergy,xergy, and dissipation), concluding that – in the framework of aetwork analysis – all the corresponding extremal principles areutually consistent. There is, indeed (perhaps?), a “lagrangian”,

hough of a different type of those hitherto devised.

g 222 (2011) 1347–1353 1351

4.4. Thermodynamics of living organisms

To consider “life” as an “emerging property”, thus drawing arigid separation line between “brute matter” and “living organ-isms”, introduces a sort of quantum jump in the “quality” of thesystems we wish to include in our study, and represents a logicallyunnecessary complication. Scientists who choose such an approachto describe a living system must somehow account for this “jump”,and by doing so are forced to introduce a plethora of philosophical,ethical and metaphysical considerations that end up blurring thescientist’s judgement.9 But if we recall that “natura non facit saltus”(nature does not proceed by discontinuities (von Linné, 1751)), wemay eliminate this qualitative jump, and see that we do not needan abstract definition of “life” to quantify the “emerging” attributesof the structure: certain systems may evolve to a complex enoughstate to display additional state properties that cannot be describedby the laws of classical thermodynamics, and the threshold valueof complexity can be – as of today – only fuzzily described. Thebehaviour of a living organism falls thus in the same category asthat of “very complex, auto-catalytic systems”, and to describe itsbehaviour we need, as in that case, the three laws and some addi-tional postulate. Here, too, Lotka had an incredibly sharp foresightwhen he wrote “. . .whether life is present or not, something definitedoes happen, the course of events is determinate, though not in terms ofthe first and second laws alone. The freedom of which living organismsavail themselves under the laws of thermodynamics is not a freedomin fact, but a spurious freedom arising out of an incomplete state-ment of the physical laws applicable to the case” (Lotka, 1922b, p.152). Notice that this statement is very far from a purely mecha-nistic Lagrange-type approach, and Lotka’s position is very modernindeed, especially because he uses the above consideration to intro-duce the need for an “evolutionary principle” that, complementingthe classical laws, may lead to a correct description of systems thatare and remain far from equilibrium for their entire existence.

No proof exists to-date that Lotka’s maximum exergy rate prin-ciple is the required postulate. From a theoretical point of view,evolution seems to be more apt to include some combination ofan “optimal efficiency” with a “maximum dissipation” principle(Martyushev and Seleznev, 2006). An attractive assumption is that afar-from-equilibrium system “tries” to go back to equilibrium (or atleast to a steady-state non-equilibrium) by first reducing the exces-sive external driving forces (maximum exergy destruction, MEP)and then, when most forces have become small enough, adjustingto minimum dissipation (EGM). It is though difficult to extrapo-late such general assumptions to practical cases, because “life” is asequel of interactions with the environment in the course of whichthe externally imposed gradients (the forces) vary in an unpre-dictable way. The idea is of course the same as Schrödinger’s andPrigogine’s: life is generated and maintained solely by the exergyinput at the boundary that balances its internal dissipation: theargument is whether it is the dissipation or the mass throughputthat “adapts” to the exergy flux.

From an empirical point of view, no experimental verificationexists of the fact that colonies of living organisms evolve under amaximum power principle. To my knowledge, the only experimentconducted in this area is that reported in (Cai et al., 2006), and itsconclusions may be interpreted equally well in terms of an “optimalconversion efficiency” paradigm. It would be therefore interesting

9 In this perspective, it is very instructive to read the famous paper by Schrodinger(1946) as well as the related reflections of Schneider and Kay (1994).

1 dellin

5

pd

ecbm(amwpbattei

dhehbwovimtdLmbtidbispbttas

edt(

i“opbaomcee

352 E. Sciubba / Ecological Mo

. Discussion: Lotka’s view in a modern perspective

Three points can be extracted from the analysis presented in thisaper, that are here figuratively “posed on the table” for furtheriscussion:

(a) Lotka thought (and wrote!) in terms of exergy (availablenergy) and not energy. It is somewhat unfortunate that his “prin-iple” has been taken up by emergy – rather than exergy analysts,ecause the latter would have immediately compared the maxi-um exergy rate with the minimum exergy destruction paradigm

which was the real matter of debate between Johnstone and Lotka)nd might perhaps have found possible intersections. A “maxi-um emergy power” principle cannot be extracted from Lotka’sritings. Thus, a first conclusion is that in modern terms Lotka’srinciple ought be restated as follows: “Given N systems interactingoth among themselves and with a common environment at time t0,nd given that for t > t0 the environment supplies a surplus of exergy,he M < N systems that shall prevail for very large values of t arehose that are capable of tapping the highest possible amount of suchxergy surplus under the boundary conditions prevailing for t > t0 andn accordance with the three Laws of Thermodynamics”.

(b) Keeping in mind his other fields of interest (he wrote onemographics, biology, statistics, and is today mostly known foris mathematical competition model, the so-called Lotka–Volterraquations), it is probably legitimate to say that when Lotka wroteis 1922 papers he was devising a strategy to explore the possi-ility of a mathematics of the evolution of dissipative systems (forhich an energy-conserving Hamiltonian does not suffice). He was

bviously a strong supporter of evolutionism and, in line with theiews of his time, thought that “growth” was a bonus in nature ast was for the society of the 1920s. Therefore, his idea was to use

ass-, energy and entropy balances to quantitatively describe theime-dependent behaviour of real systems. Since such systems areissipative and complex, he realised that a non-energy-conservingagrangian was needed to drive evolution, and he proposed hisaximum exergy rate principle. The idea is indeed interesting,

ecause any amount of “untapped excess exergy” is, from the sys-em point of view, wasted exergy. Lotka probably discarded thedea of introducing a maximum conversion efficiency or minimumissipation principle because he was convinced, on the basis of theiological notions of the time, that “the advent of animals organisms

n a world peopled with a purely vegetable population would certainlyeem to be an acceleration of the process of dissipation” (Lotka, 1922a,. 149). But he does not find the minimum dissipation principle toe in contrast with his maximum exergy rate, provided, he says,hat it can be proven that “not merely the animal organism evolves inhat direction, but that the system of coupled transformers, plant andnimal, as a whole has so evolved” (Lotka, 1922a, p. 150): a veritableystemic view!

A second conclusion is that Lotka was correct in thinking that thevolution of complex systems far enough from equilibrium may beescribed by an additional Lagrangian. Today we are more inclinedo think that such a Lagrangian ought to be based on dissipationexergy destruction) rather than on growth (input power).

(c) One of the most modern aspects of Lotka’s 1922 paperss their emphasis on “boundary conditions”. The evolution of thetypes of organisms” (the genes) is driven not only by the three lawsf thermodynamics, not only by the maximum exergy rate princi-le (or by whatever additional Lagrangian may be found), but alsoy the boundary conditions imposed on each part of the systemt every instant of time. This adds to the mathematical complexity

f the problem position, but avoids determinism: since both theutations and the boundary conditions are random (and scarcely

orrelated with each other), the existence of Lagrange’s “perfectquation” is negated. The conclusion is that Lotka’s paradigm isxactly the same as that of modern physics and biology!

g 222 (2011) 1347–1353

Summarizing the above reflections, it appears that while Lotka’smethod was correct and his fundamental intuition right, his con-strained maximum exergy rate principle represents the weak linkof his theory. A possible solution, proposed here only tentatively,is to augment the principle by combining it with EGM, so that thefinal result is a statement like “. . .for very large values of t, the M sys-tems that prevail are those capable of tapping the maximum exergyrate with the minimum exergy destruction (entropy generation), foreach given conversion task (process) and under the boundary con-ditions prevailing between t0 and t”. This definition is perhaps notvery rigorous, but it captures Lotka’s intent while adapting it tomodern non-equilibrium thermodynamics. Notice that the expres-sion “for very large values of t” allows for an initial “adjustmentphase” in which the trajectory of the system would be described bya maximum entropy production principle. Notice also that Lotkathought that after an initial phase of “maximum power capture”, amature ecosystem would evolve towards a better exergy efficiency,in that the initial “exergy predators” would be displaced by moreefficient subsystems each covering its own environmental niche(Lotka, 1922a, p. 150).

6. Conclusions

This paper was intended to present a critical analysis of the con-tents and significance of Alfred Lotka’s three seminal 1921–1922papers, and the fact that it ended up being almost a eulogy for Lotkareflects the profound respect that his intuition, his way of argu-menting and his rigor demand of the reader. The analysis showsthough that, while his reasoning was correct and his argumenta-tions accurate, the very weakness of Lotka’s theory seems indeed tobe his insistence on the absolute validity of a “maximum principle”.Today we are inclined to think that the experimentally measur-able difference in the way systems “near” or “far” from equilibriumbehave depends on the different Lagrangian they may be follow-ing: a system near equilibrium tends to “conserve exergy” (evolvewith maximum efficiency), while one far from equilibrium tendsto dissipate the existing gradients (both internal and external) inthe fastest possible way (therefore evolving with the maximumpossible absorption – and destruction – of the available exergyrate). Lotka’s principle is for sure not “the Fourth Law of Thermody-namics”, but it is relevant in all applications to systems very muchremoted from equilibrium, including obviously living systems thatwere Lotka’s original concern.

References

Bejan, A., 1996. Entropy Generation Minimization. CRC Press.Beretta, G.P., 2009. Nonlinear quantum evolution equations to model irreversible

adiabatic relaxation with maximal entropy production and other non-unitaryprocesses. Rep. Math. Phys. 64, 139–168.

Boltzmann L., 1886. Der zweite Hauptsatz der mechanischen Waermetheorie (TheSecond Principle of the Mechanical Theory of Heat), Lecture to the ImperialAcademy of Science, Berlin, May 29, 1886. (in German).

Brown, M.T., Ulgiati, S., 2001. Emergy measures of carrying capacity to evaluateeconomic investments. Popul. Environ. 22 (5), 471–501.

Cai, T.T., Montague, C.L., Davis, J.S., 2006. The maximum power principle: an empir-ical investigation. Ecol. Model. 190 (3–4), 317–335.

DeLong, J.P., 2008. The maximum power principle predicts the outcomes of two-species competition experiments. Oikos 117, 1329–1336.

Dobzhansky, Th., Ayala, F.J., Stebbins, G.L., Valentine, J.W., 1977. Evolution. W.H.Freeman, San Francisco.

Fath, B.D., Patten, B.C., Choi, J.S., 2001. Complementarity of Ecological Goal Functions.J. Theor. Biol. 208, 493–506.

Giannantoni, C., 2002. The Maximum Em-power Principle. S.G.E., Padova.Glansdorff, P., Prigogine, I., 1964. On a general evolution criterion in macroscopic

physics. Physica 30, 351–374.

Glansdorff, P., Prigogine, I., 1974. Thermodynamic Theory of Structure, Stability and

Fluctuations. Wiley Interscience.Gyarmati, I., 1970. Non-equilibrium Thermodynamics – Field theory and variational

principles. Springer, Berlin.Gyftopoulos, E.P., Beretta, G.P., 1995. Thermodynamics: Foundations and Applica-

tions. Dover Pub.

dellin

H

J

J

K

K

L

L

L

L

LL

M

N

NO

von Linné, C., 1751. Philosophia Botanica, Stockholm.Ziegler, H., Wehrli, C., 1987. On a principle of maximal rate of entropy production.

E. Sciubba / Ecological Mo

all, C.A.S., 2004. The continuing importance of maximum power. Ecol. Model. 178(1–2), 107–113.

ohnstone, J., 1921. The Mechanism of Life in Relation to Modern Physical Theory. E.Arnold & Co, London.

orgensen, S.E., Brown, M.T., Odum, H.T., 2004. Energy Hierarchy and Transformityin the Universe. Ecol. Model. 178, 17–28.

ay J.J., 1984–2002. About some Common Slipups in Applying Prigogine’sMinimum Entropy Production Principle to Living Systems, www.jameskay.ca/musings.

ay, J.J., 2002. Some Observations About the Second Law and Life,www.jameskay.ca/musings.

odge, O, 1906. Life and Matter, a Criticism of Professor Haeckel’s ‘Riddle of theUniverse’, G.P. Putnam’s Sons. Knickerbocker Press, New York/London.

otka, A.J., 1921. Note on the economic conversion factors of energy. Proc. Natl. Acad.Sci. 7, 192–197.

otka, A.J., 1922a. Contribution to the energetics of evolution. Proc. Natl. Acad. Sci.8, 147–151.

otka, A.J., 1922b. Natural selection as a physical principle. Proc. Natl. Acad. Sci. 8,151–154.

otka, A.J., 1925. Elements of Physical Biology. Williams & Wilkins, Baltimore.ucia, U., 2007. Irreversible entropy variation and the problem of the trend to equi-

librium. Physica A 376, 289–292.artyushev, L.M., Seleznev, V.D., 2006. Maximum entropy production principle in

physics, chemistry and biology. Phys. Rep. 406, 1–45.

icolis, G., Prigogine, I., 1977. Self-organization in Non-equilibrium Systems. J. Wiley

& Sons, NY.icolis, G., Prigogine, I., 1989. Exploring Complexity. Freeman & Co, NY.dum, H.T., 1995. Self-organization and maximum empower. In: Hall, C.A.S. (Ed.),

Maximum Power: The Ideas and Applications of H.T. Odum. Colorado Univ. Press,Colorado.

g 222 (2011) 1347–1353 1353

Odum, H.T., Odum, E.C., 1981. Energy Basis for Man and Nature. McGraw-Hill, NY.Onsager, L., 1931a. Reciprocity relations in irreversible processes, part 1. Phys. Rev.

37 (February).Onsager, L., 1931b. Reciprocity relations in irreversible processes, part 2. Phys Rev.

38 (December).Schneider, E.D., Kay, J.J., 1994. Life as a manifestation of the second law of thermo-

dynamics. Math. Comp. Model. 19 (6–8), 25–48.Schrödinger, E., 1946. What is Life? Macmillan Pub (based on lectures delivered at

Trinity College, Dublin, in 1943).Sciubba, E., 2008. Why emergy- and exergy analyses are non-commensurable meth-

ods of system analysis. Int. J. Exergy 6 (4).Sciubba, E., 2010. On the second law inconsistency of emergy analysis. Energy 35,

3696–3706.Sciubba E., Zullo F., in press. Exergy-based population dynamics: a thermody-

namic view of the sustainability concept. J. Ind. Ecol., doi:10.1111/j.1530-9290.2010.00309.x.

Sciubba, E., Ulgiati, S., 2005. Emergy and exergy analyses: complementary methodsor irreducible ideological options? Energy 30. (10).

Ulgiati, S., Brown, M.T., 2001. Emergy accounting of human-dominated, large-scaleecosystems. In: Jorgensen, S.E. (Ed.), Thermodynamics and Ecological Modelling.CRC Press, pp. 63–113.

Valyi R., 2005. About the Emergy Concept, Thesis, Ecole Centrale de Lyon.

J. Non-Eq. Thermodyn. 12, 229–243.Zupanovic, P., Kuic, D., Bonacic-Losic, Z., Petrov, D., Juretic, D., Brumen, M., 2010.

The maximum entropy production principle and linear irreversible processes.Entropy 12, 996–1005.