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Chapter 1

Beyond the Second Law: An Overview
Roderick C. Dewar, Charles H. Lineweaver, Robert K. Niven and Klaus Regenauer-Lieb

Abstract The Second Law of Thermodynamics governs the average direction of all non-equilibrium dissipative processes. However it tells us nothing about their actual rates, or the probability of fluctuations about the average behaviour. The last few decades have seen significant advances, both theoretical and applied, in understanding and predicting the behaviour of non-equilibrium systems beyond what the Second Law tells us. Novel theoretical perspectives include various extremal principles concerning entropy production or dissipation, the Fluctuation Theorem, and the Maximum Entropy formulation of non-equilibrium statistical mechanics. However, these new perspectives have largely been developed and applied independently, in isolation from each other. The key purpose of the present book is to bring together these different approaches and identify potential connections between them: specifically, to explore links between hitherto separate theoretical concepts, with entropy production playing a unifying role; and to close the gap between theory and applications. The aim of this overview chapter is to orient and guide the reader towards this end. We begin with a rapid flight over the fragmented landscape that lies beyond the Second Law. We then highlight the connections that emerge from the recent work presented in this volume. Finally we summarise these connections in a tentative road map that also highlights some directions for future research.
R. C. Dewar (&) Research School of Biology, The Australian National University, Canberra, ACT 0200, Australia e-mail: roderick.dewar@anu.edu.au C. H. Lineweaver Research School of Astronomy and Astrophysics and Research School of Earth Sciences, The Australian National University, Canberra, ACT 0200, Australia R. K. Niven School of Engineering and Information Technology, The University of New South Wales at ADFA, Canberra, ACT 2600, Australia K. Regenauer-Lieb School of Earth and Environment, The University of Western Sydney and CSIRO Earth Science and Resource Engineering, Crawley, WA 6009, Australia

R. C. Dewar et al. (eds.), Beyond the Second Law, Understanding Complex Systems, DOI: 10.1007/978-3-642-40154-1_1, ñ Springer-Verlag Berlin Heidelberg 2014

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1.1 The Challenge: Understanding and Predicting Non-equilibrium Behaviour
Non-equilibrium,1 dissipative systems abound in nature. Examples span the biological and physical worlds, and cover a vast range of scales: from biomolecular motors, living cells and organisms to ecosystems and the biosphere; from turbulent fluids and plasmas to hurricanes and planetary climates; from growing crystals and avalanches to earthquakes; from cooling coffee cups to economies and societies; from stars and supernovae to clusters of galaxies and beyond. A characteristic feature of all open, non-equilibrium systems is that they import energy and matter from their surroundings in one form and re-export it in a more degraded (higher entropy) form. A sheared viscous fluid driven out of thermodynamic equilibrium by the external input of kinetic energy eventually dissipates and expels that energy to its environment as heat; the Earth absorbs short-wave radiation at solar temperatures and re-emits it to space as long-wave radiation at terrestrial temperatures; living organisms use the chemical free energy ultimately derived from photons to grow and survive, eventually dissipating it to their environment as heat and carbon dioxide. In association with these exchanges of energy and matter, spatial gradients in temperature and chemical concentration are set up and maintained, both internally and between the system and its environment. The patterns of flows and their associated gradients self-organize into intricate dynamical structures that continually transport and transform energy and mass into higher entropy forms: thus emerge plant vascular systems, food webs, river networks, and turbulent eddies such as Jupiter's Red Spot and the convective cells on the Sun's surface. Idealised systems in equilibrium with their surroundings exhibit no flows or gradients; they appear static, structureless, lifeless. In stark contrast, non-equilibrium systems, even purely physical ones, appear to be alive in a sense that perhaps even defines life itself, at least thermodynamically [1]. In view of their ubiquity in nature, understanding and predicting the behaviour of non-equilibrium systems lies at the heart of many questions of fundamental and practical importance, from the origin of a low entropy universe and the evolution of life, to the development of nanotechnology and the prediction of climate change. What is life? And what are the general requirements for its emergence on Earth and elsewhere? What determines the rate at which the universe tends towards thermodynamic equilibrium? How is the functioning of nanoscale devices affected by molecular-scale fluctuations in energy and mass flow? How will the large-scale flows of energy and mass that characterise Earth's climate respond to increased atmospheric greenhouse gas concentrations? Answering such questions has been a long-standing scientific challenge, largely because the scientific principles and tools required to understand and predict
Equilibrium is used here in the thermodynamic sense, and not in the dynamic sense of stationarity.
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non-equilibrium behaviour have been lacking. In many cases we may not know the underlying equations of motion exactly (especially the case in biology); with only the conservation laws (of energy, mass, momentum and/or charge) as guiding principles, there remains a large number of possible behaviours to choose from. Even when the underlying equations of motion are known (more or less) exactly-- for example, the Navier­Stokes equation of fluid mechanics2--computational limitations may restrict our ability to solve them. One response to this challenge is to exploit the fact that the macroscopic behaviour of complex, non-equilibrium systems represents the emergent outcome of a large number of microscopic degrees of freedom. Some of those underlying degrees of freedom may behave as `noise' that averages out at macroscopic scales. This offers the possibility of predicting the emergent macroscopic behaviour from the laws of thermodynamics. We might then hope to understand the behaviour of non-equilibrium systems statistically, in terms of the average, collective behaviour of a large number of individual degrees of freedom. Here again, however, traditional thermodynamics gives us little to go on. The First Law of Thermodynamics only gives us energy conservation, while the Second Law is qualitative--it tells us only the direction in which an isolated nonequilibrium system will evolve on the average: towards the state of equilibrium, in which the system's thermodynamic entropy adopts its largest value subject to any constraints on it. The Second Law thus implies that, on average, the total thermodynamic entropy Stot = Ssys ? Senv of an isolated system consisting of an open non-equilibrium subsystem (sys) plus its environment (env) will not decrease (i.e. dStot/dt C 0). In particular, if the open subsystem is in a steady state (i.e. dSsys/ dt = 0), then on average the entropy of the environment (Senv) will not decrease (i.e. dSenv/dt C 0). As noted above, this behaviour is evident in the observed tendency of open systems to re-export energy and matter to their environment in a higher entropy form than that in which they receive it. Crucially, however, the Second Law is mute on two counts. Firstly, it does not predict the actual value of dStot/dt (i.e. the average rate at which Stot increases). Secondly, as the mathematical physicist James Clerk Maxwell was one of the first to appreciate [2], the Second Law is statistical in character, rather than being a dynamical law. It is a statement about the average behaviour of isolated systems. However, it does not tell us the probability of statistical fluctuations in energy and mass flow for which, at least momentarily and locally, dStot/dt \ 0, as when (for example) a group of gas molecules happens to move collectively from a region of low concentration to a region of higher concentration. And yet knowing these quantities--the average rate of entropy increase, and the probability of entropydecreasing fluctuations--is central to answering some of the fundamental and practical questions mentioned above. Beyond the Second Law, the behaviour of entropy production becomes a key focus of study.

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Strictly speaking, the Navier­Stokes equation is only approximate; the (linear) expression for the stress tensor is only valid close to equilibrium.


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The aim of this introductory chapter is to orient and guide the reader of this book. We begin with a rapid flight over the landscape of non-equilibrium principles that have been proposed beyond the Second Law (Sect. 1.2). For now, that landscape is still forming; it remains a rather fragmented one and we highlight some of the challenges one encounters in trying to negotiate it (Sect. 1.3). The key challenge is to find connections within this landscape, to construct bridges between previously isolated islands. In Sect. 1.4 we highlight some of the connections suggested to us by the recent work presented in this volume. Summarizing these in Sect. 1.5, we offer a tentative road map of the current landscape, as well as possible directions for future research. Given the current state of play, we attempt no more than a partial synthesis here--partial in both perspective and scope. Thus Sects. 1.4 and 1.5 present one particular view of this exciting area of science, and where it might go next. It does not represent a consensus view of the contributing chapter authors, as will be clear from the diversity of perspectives this book brings together. And even at that, it does not pretend to paint a complete picture. Nevertheless, we hope this tentative road map will encourage the reader to develop his or her own vision of the landscape beyond the Second Law, and of the most fruitful paths to explore within it.

1.2 Beyond the Second Law: The Search for New Principles
Our main aim here is to give a brief overview of the landscape of non-equilibrium principles that have been proposed beyond the Second Law. Discussion of the key challenges in negotiating this landscape (ambiguities of meaning etc.) is deferred to Sect. 1.3.

1.2.1 Paltridge's MaxEP, the Fluctuation Theorem ...
Within the last few decades, significant progress has been made towards developing and applying new principles of thermodynamics for non-equilibrium systems that go beyond the Second Law. With regard to the average value of dStot/dt and the probability of entropy-decreasing fluctuations, two key concepts have emerged: respectively, the principle of Maximum Entropy Production (MaxEP) and the Fluctuation Theorem (FT). MaxEP is often stated verbally as a sort of codicil to the Second Law, according to which it is asserted that an open system adopts the stationary state (dSsys/dt = 0) in which dStot/dt = dSenv/dt attains its largest value possible within the constraints acting on the system. That is, a stationary open subsystem plus its environment not only tends to equilibrium (dStot/dt = dSenv/dt C 0) but, it is claimed, does so as fast as possible (maximum dSenv/dt) subject to any constraints.


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In the seminal work of Garth Paltridge [3­5] in the 1970s and 1980s, MaxEP was applied to simple steady-state energy balance models of Earth's climate. Maximizing the entropy production associated with material heat transport in the atmosphere and oceans produced realistic predictions of the stationary latitudinal profiles of surface temperature, cloud fraction and equator-to-pole material heat transport. Somewhat surprisingly, this success was achieved when the maximization was subject to the sole constraint3 of global energy balance, in the absence of any dynamical information such as planetary rotation rate. Paltridge's MaxEP principle selects one among several climate states compatible with global energy balance [3­6]. It is the archetype for analogous MaxEP principles constrained only by global mass balance that have been applied with similar success to other non-equilibrium selection problems (e.g. crystal growth morphology, macromolecular evolution, plant growth strategies) [7­13]. For brevity, in the following we will refer to these collectively as `Paltridge's MaxEP'--i.e. MaxEP principles in which the key constraints are global energy and/or mass balance. Despite these successes, the theoretical basis for Paltridge's MaxEP has remained elusive and this has hampered its acceptance by the wider scientific community. The Fluctuation Theorem (FT) [14­16] concerns the probabilities of trajectories and their time reverse in microscopic phase space. Roughly speaking, the FT states that the probability of observing an entropy change ­d relative to that of an entropy change +d over a given time period is exponentially small in d. Since d is an extensive quantity in both space and time (i.e. the entropy change increases with both the size of the system and the time period), the FT implies that macroscopic decreases in entropy, although possible, are extremely rare. In contrast, we expect to see frequent entropy-decreasing fluctuations in small (e.g. nanoscale) systems observed over short periods. Significantly, the FT also implies the Second Law inequality, i.e. the ensemble average4 of d is non-negative.

1.2.2 ... and other Principles
Prior to Paltridge's MaxEP principle, several earlier non-equilibrium principles had also been proposed, involving entropy production or dissipation in one guise or another (see e.g. the excellent review in [13]). A selection of these are summarised in Table 1.1: they include Onsager's MaxEP principle [17, 18], Prigogine's minimum entropy production (MinEP) theorem [19], Kohler's MaxEP principle in statistical transport theory [20, 21], and Ziegler's MaxEP principle for

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However, Paltridge's energy balance model still contained a number of ad hoc assumptions and parameterizations (see Herbert and Paillard Chap. 9). 4 The ensemble average is over the probability distribution of microscopic trajectories in phase space (see Sect. 1.4.1).


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Table 1.1 A fragmented landscape. A selection of different dissipation- and entropy-related variational principles H(y|C), defined in terms of the function that is maximised (H), the variables being optimised (y), the constraints (C), and the key prediction. The table entries above and below the dashed line describe principles that are conjectured to apply, respectively, close to and far from equilibrium (linear and non-linear regimes). Ri denotes a sum over i = 1, .... n (this may be generalised to continuous systems). Abbreviations: EP entropy production; GCM general circulation model; KE kinetic energy; LBE linearized Boltzmann equation; RTE radiative transport equation; UBT upper bound theory of fluid turbulence Variational Maximised function, H Variables, y Constraints, C Key prediction principle Fluxes1 Ji Ïi ¼ 1; .. .n÷ Fixed forces: Xi ¼ X ö Ïi ¼ 1; .. . n÷ i Linear flux-force relations: Ji ¼ Rj Rþ1 X ö j ij Ji ¼ 0 Ïi ¼ k × 1; .. .n÷ Stationary f(v) near equilibrium

Onsager MaxEP [17, 18]

Ri Ji Xi þ 1=2Rij Rij Ji J

j

Fixed forces: Xi ¼ X ö Ïi ¼ 1; .. .k÷; Prigogine MinEP þRij Rþ1 Xi Xj Free forces i ij [19] Xi Ïi ¼ k × 1; ...n÷ Linear flux-force: Xi ¼ Rj Rij Jj Kohler MaxEP EP of molecular collisions rcoll Ï f ÷ (source One-particle velocity Fixed temperature and concentration (solution to term inRcontinuity equation for entropy (v) distribution fields; rcoll Ï f ÷ ¼ entropy export LBE) [20, 21] (steady-state condition) function f(v) s ¼ þ f ln fdv)
m

-------------------------------------------------------------------Various moment constraints on Im Radiation transport coefficients

Radiation intensity, I Radiative MinEP þrtot ÏIm ÷, where rtot ÏIm ÷ = total (radiation ? matter) EP (solution to RTE) (Chap. 12) Ziegler MaxEP Dissipation r(J) (assumed to be a known Fluxes1 Ji Ïi ¼ 1; .. .n÷ [22] function of the fluxes Ji )

Fixed forces: Xi ¼ X ö Ïi ¼ 1; .. .n÷; i rÏJ ÷ ¼ Ri Ji X ö i

Paltridge MaxEP Various EP-related functions of the form Ji and Xi Ïi ¼ 1; .. .n÷ Steady-state energy/mass balance (neither Xi nor Ji need be fixed) [3­13] Ri Ji Xi

Non-linear flux-force relations: X ö / i orÏJ ÷=oJi (orthogonality) Stationary Ji and Xi (non-linear regime) (continued)

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Table 1.1 (continued) Variational Maximised function, H principle Variables, y Mean velocity field A restricted set of spatial integrals of the Navier­Stokes equation Steady-state GCM dynamics Constraints, C Key prediction

Malkus UBT [23­28] GCM parameters

Various dissipation-related functions of turbulent flow (including KE dissipation) Dissipation of KE in a GCM

Mean stationary velocity field (non-linear regime) Dynamical climate features Most likely sampling distribution pi

Max KE dissipation [29] BoltzmannGibbs-Jaynes MaxEnt [30­34] Posterior probability of outcome i, pi Available or relevant information: Ri pi fik ¼ Fk Ïk ¼ 1; .. .m÷; Prior probability qi of outcome i

Relative entropy ­Ri pi lnÏpi =qi ÷

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The Onsager and Ziegler MaxEP principles can also be formulated in the space of forces Ïy ¼ X ÷ subject to fixed fluxes ÏJ ¼ J ö ÷

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dissipative materials [22]. Also, starting in the 1950s, several variational principles for fluid turbulence were developed by Malkus and others, based on maximising various dissipation-like functions of the flow [23­28]--an approach known as the Upper Bound Theory (UBT) of fluid turbulence. Table 1.1 also includes three other variational principles: a variant of Kohler's principle applied to radiative transport, in which entropy production is minimized rather than maximized (Christen and Kassubek Chap. 12); a principle of maximum kinetic energy (KE) dissipation, suggested by recent climate simulations using a General Circulation Model (GCM) [29], which is also one of the principles emerging from UBT; and the Boltzmann-Gibbs-Jaynes Maximum Entropy (MaxEnt) algorithm [30­34]. Anticipating the discussion in Sect. 1.3, the landscape presented by these principles is a fragmented one. In order to compare and contrast the elements of this landscape, Table 1.1 describes each principle in terms of the dissipation- or entropy-related function H that is maximized, the variables (y) being optimised, and the constraints (C). Key predictions of each principle are given in the last column.

1.2.2.1 Onsager's MaxEP, Prigogine's MinEP Onsager's original motivation was to establish a theoretical framework for the development of near-equilibrium thermodynamics [17, 18]. Specifically, Onsager's MaxEP principle may be used to derive the near-equilibrium, linear `constitutive relations' between generalised thermodynamic fluxes Ji and forces Xi, i.e. Ji = RjLijXj (generalisations of the laws of Fick and Ohm, for example), where the matrix of coupling coefficients5 is symmetric (i.e. Lij = Lji, also known as reciprocity). Prigogine's principle [19] assumes linear flux-force relations as a starting point, and some of the forces are then relaxed: it describes the behaviour of the entropy production, given by RijLijXiXj, `when we let go of some of the leads' [21].

1.2.2.2 Kohler's MaxEP, Radiative MinEP In a separate context, Kohler [20] established a mathematical variational principle to solve the linearised Boltzmann equation (LBE) describing the statistical transport properties of a rarified gas. Subsequently, Ziman [21] recast the Boltzmann equation in the language of thermodynamic fluxes and forces and showed Kohler's principle to be mathematically equivalent to Onsager's MaxEP principle. This suggested to Ziman that Kohler's principle was not just a convenient mathematical trick but had the following physical interpretation: the entropy production of molecular collisions is maximized subject to fixed thermodynamic forces (e.g. temperature and concentration fields), and to the steady-state condition

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Here Lij is the inverse of the matrix Rij in Table 1.1.


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that the internal entropy production is balanced by dissipation of heat into the environment. For the problem of radiative transfer in gases or plasmas, the relevant principle appears to be one of MinEP rather than MaxEP (Christen and Kassubek, Chap. 12 and references therein; see also Niven and Noack, Chap. 7). Moreover, when the radiative transfer equation is considered as a LBE, the linearisation is exact because photons do not interact with each other, so that the solution is valid for radiation that is arbitrarily far from thermal equilibrium.

1.2.2.3 Ziegler's MaxEP The original motivation behind Ziegler's MaxEP principle [22] was to derive the non-linear constitutive relation between generalised forces Xi and fluxes Ji (e.g. stress­strain relations) in dissipative materials far from equilibrium. What are the fluxes given the forces (and vice versa)? As Table 1.1 indicates, the key prediction of Ziegler's MaxEP (subject to fixed generalised forces Xi = Xi* and the constraint r(J) = RiJiXi*) is a constitutive relation that satisfies an orthogonality condition (OC), according to which the generalised force X* (considered as a vector with components Xi*) lies in the direction normal to the contours of r(J) in flux space.6 Ziegler originally derived the OC using a geometrical argument, based on the assumption that the vector X can be derived solely from properties of the scalar dissipation function r(J); the existence and nature of the function r(J) were also assumptions (Houlsby, Chap. 4). Ziegler noted the equivalence of the OC to a variational principle (Ziegler's MaxEP)--i.e. maximizing r(J) with respect to J under the constraints in Table 1.1--as a possibly more general thermodynamic basis for the OC. And yet a fundamental basis for the assumptions underlying either derivation of Ziegler's OC (geometrical or variational) has yet to be established; moreover, a direct experimental test of the OC has yet to be derived [35]. In practice, therefore, the OC has been adopted as a working hypothesis for classifying different theoretical behaviours of dissipative materials.

1.2.2.4 Upper Bound Theory of Fluid Turbulence, Maximum KE Dissipation The UBT of fluid turbulence was developed by Malkus and others [23­28] to predict the mean turbulent velocity field, by maximizing various dissipationrelated functionals of the flow. Initially these took the form of maximum transport principles (maximum heat flow, maximum momentum transport) [23­25, 27].

An equivalent orthogonality condition for the direction of the generalised fluxes J in force space can be stated in terms of the contours of r(X).

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Later, Kerswell [26] analysed a more general family of functionals related to KE dissipation by the mean and fluctuating components of the flow. Maximum KE dissipation by the mean flow was also proposed more recently by Malkus [28]. Crucially, the maximization was subject to a restricted number of dynamical constraints, obtained as integral properties of the Navier-Stokes equation (e.g. global power balance and horizontal mean momentum balance within a horizontally sheared fluid layer) rather than the full dynamics. The UBT is thus analogous in spirit to Paltridge's MaxEP: i.e. select one of many possible stationary states compatible with a restricted number of constraints representing the relevant physics on macroscopic scales, the rest being treated as `noise'. However, one key difference is that UBT includes some dynamical information (momentum balance) in addition to global energy balance; another key difference is in the nature of the extremized function (e.g. viscous dissipation of KE [28] rather than thermal dissipation [3­6]). Intriguingly, simulations using the FAMOUS GCM [29] also showed that key dynamical features of Earth's climate were close to a maximum of KE dissipation (Table 1.1).

1.2.2.5 Boltzmann-Gibbs-Jaynes Maximum Entropy Finally we have the Boltzmann-Gibbs-Jaynes principle of Maximum Entropy (MaxEnt) [30­34]. This principle stands somewhat apart from the others in Table 1.1, both conceptually and in practice (see Dewar and Maritan, Chap. 3; Niven and Noack, Chap. 7). MaxEnt predicts a probability distribution pi over microscopic outomes i, from which macroscopic quantities may be predicted as averages over pi. The maximized function H is the relative entropy (or negative Kullback­Leibler divergence) of pi and a prior distribution qi; H reduces to the Shannon entropy when qi is uniform. The maximization is subject to constraints on certain moments of pi (representing available or relevant physical information), as well as the specified prior probabilities qi. MaxEnt has several interpretations (Chaps. 3 and 7, and references therein). One fairly concrete interpretation of the MaxEnt distribution is that it corresponds to the most likely frequency distribution of outcomes that would be observed in a long sequence of independent observations of a system that is subject to the given constraints; MaxEnt also has an information-theoretical interpretation as the `least-informative' pi [32, 33]. MaxEnt has a long history, starting with Boltzmann's discovery that MaxEnt expresses the asymptotic behaviour of multinomial probabilities [30], and the early development of equilibrium statistical mechanics by Gibbs [31]. The later reappearance of ­Ripilnpi (Shannon entropy) in the development of information theory [36, 37], as a measure of missing information, led Jaynes to see MaxEnt as a general method of statistical inference from incomplete information [32, 33]. In view of its general nature, Jaynes promoted MaxEnt as a theoretical framework for non-equilibrium as well as equilibrium statistical mechanics. When applied to non-equilibrium systems, MaxEnt leads to non-linear flux-force relationships that


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automatically satisfy Onsager reciprocity and reduce to linear form in the nearequilibrium limit [38­40]. Although MaxEnt provides a foundation for equilibrium and non-equilibrium thermodynamics, its physical interpretation remains a subject of debate (e.g. [41]) that has, like Paltridge's MaxEP, hampered its wider acceptance.

1.3 A Fragmented Landscape
After this rapid tour, the student would be forgiven for being confused by the sheer number and variety of entropy production-related principles, as well as by the diverse ways in which they have been applied to non-equilibrium systems. One sees a fragmented landscape of principles and applications, and faces three key difficulties in negotiating it.

1.3.1 Different Histories
One difficulty is historical: the above theoretical principles (Table 1.1) were developed at different times, more or less independently of one other. Some theoretical links between the earlier variational principles (Onsager, Prigogine, Ziegler) have been identified [13]. For example, Ziegler's MaxEP principle reduces to Onsager's in the near-equilibrium limit, while Prigogine's is a corollary of Onsager's that involves additional constraints. However, the links (if any) between Paltridge's MaxEP, the Fluctuation Theorem, Kohler's MaxEP, radiative MinEP, UBT, maximum KE dissipation and MaxEnt (and between these and Ziegler's principle) have remained obscure.

1.3.2 Different Meanings
A second difficulty, and one that compounds the first, is semantic. The terms entropy production or dissipation are defined and used by different workers in different ways, creating ample room for confusion. Some approaches take entropy production as a given function of thermodynamic fluxes and forces [3­9, 11, 12, 17­19, 22­28], while others define entropy production from an underlying microscopic picture [10, 14­16, 21, 42, 43]. Moreover it does not help that Onsager called his MaxEP principle `least dissipation of energy', or that Paltridge originally called his principle `minimum entropy exchange' before resorting to MaxEP! Yet further scope for confusion arises in the context of extremal principles. For example, from the name alone one might conclude that Paltridge's MaxEP


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principle contradicts Prigogine's principle of MinEP, whereas these two extremal principles refer to quite different situations. Paltridge's MaxEP principle (e.g. as applied to climate systems [3­6]) is a selection principle between different farfrom-equilibrium stationary states. In contrast, Prigogine's principle describes the non-stationary behaviour of the entropy production of near-equilibrium systems7 when a subset of the thermodynamic force constraints is relaxed; it says only that the unique stationary state has lower entropy production than any non-stationary state, and does not provide a selection principle in situations where there are multiple stationary states (for a further critique of Prigogine's MinEP, see [44]). To have any chance of making sense of the landscape, one must look beyond the semantics and identify three key aspects (Table 1.1) of each extremal principle, which we denote by H(y|C): (1) which entropy production or dissipation function (H) is being maximised? (2) with respect to which variable(s) (y)? and (3) subject to which constraint(s) (C)? Unless extremal principles are clearly stated in this way, the potential for confusing apples with pears is essentially infinite.

1.3.3 Lack of Foundations
A third related difficulty in negotiating the current landscape lies in the somewhat ad hoc way in which, for example, Paltridge's MaxEP has been applied in practice, with many aspects open to ambiguity. Which entropy production function (H) is to be maximised? R some discussions of the physical interpretation of H, the system In entropy Ssys ¼ ssys dV (V = system volume) is treated as a physical quantity obeying a local continuity equation (ossys =ot ¼ þr à js × r with entropy flux js and local entropy production rate r). In a stationary state all the entropy production R rdV