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Reference: Niven, R.K. and Ozawa H. (2016), Entropy production extremum principles, Chapter 32 in Singh, V. (ed.) Handbook of Applied Hydrology, 2nd ed., McGraw-Hill, NY, ISBN 978-0071835091.
Chapter
32
Entropy Production Extremum Principles BY
ROBERT K. NIVEN AND HISASHI OZAWA
ABSTRACT
We review a number of extremum principles of nonequilibrium thermodynamics based on dissipation, power consumption or entropy production, and their significance to hydrology. The connections between these and various allied methods are reviewed and mapped using the rigorous framework supplied by maximum entropy analysis. Two conjugate maximum and minimum entropy production principles, originating from Paltridge, Gaggioli, and others, are found to be of considerable importance and warrant further investigations in hydrology. 32.1 INTRODUCTION
For several decades, the maximum entropy production (MaxEP) principle of Paltridge (1975, 1978) has been applied as a heuristic tool to infer the stationary state of a variety of far-from-equilibrium dissipative systems, including many fluid and heat flow systems of interest to hydrologists. Examples include the atmospheric, oceanic, and mantle circulation systems on the Earth and other planets (e.g., Paltridge, 1975, 1978; Vanyo and Paltridge, 1981; Ozawa and Ohmura, 1997; Lorenz et al., 2001; Shimokawa and Ozawa, 2001, 2002; Ozawa et al., 2003; Kleidon and Lorenz, 2005); turbulent shear and heat convection systems (Ozawa et al., 2001, 2003); global planetary cycles and the biosphere (Kleidon, 2004, 2009a-b, 2010a-b; Kleidon and Lorenz, 2005); vegetation spatial distribution (Kleidon et al., 2007; del Jesus et al., 2012); ecosystem selection (Meysman and Bruers, 2007); photosynthesis and plant optimization (Dewar et al., 2006; Dewar, 2010); fluvial geomorphology (Paik and Kumar, 2010; Beven, 2015); particle sedimentation (Chung and Vaidya, 2008, 2011); jumpwise colloidal processes (Kozvon et al., 2002); crystal growth (Martyushev and Axelrod, 2003); tropical cyclones (Ozawa and Shimokawa, 2015); plasma dynamics (Yoshida and Mahajan, 2008; Kawazura and Yoshida, 2010, 2012); and earthquake dynamics (Main and Naylor, 2008). Several reviews and monographs have been published (Ozawa et al., 2003; Kleidon and Lorenz, 2005; Martyushev and Seleznev, 2006; Kleidon, 2010a; Dewar et al., 2013). Numerous studies also reveal the existence of a “conjugate Paltridge” minimum entropy production (MinEP) principle (e.g., Paulus and Gaggioli, 2004; Martyushev, 2007; Niven, 2010b; Kawazura and Yoshida, 2010, 2012), quite different to the MinEP principle of Prigogine (1967). Of course, the application of variational methods to nonequilibrium systems has a long pedigree, with a variety of minimum or maximum dissipation, power or entropy production principles proposed by Helmholtz, Rayleigh; Onsager, and Machlup (1953); Prigogine (1967), and in turbulent fluid mechanics (upper-bound theory) (e.g., Malkus, 1956, 2003; Busse, 1970; Howard, 1972), finite-time thermodynamics (e.g., Salamon and Berry, 1983; Nulton et al., 1985), and engineering design (Bejan, 1996). One apparent advantage of the
Paltridge MaxEP heuristic (and its conjugate)—as applied by many researchers—is that the user can omit most details of the dynamics. If correct, this would suggest the operation of a new variational principle of science, consistent with conservation laws but which acts outside the framework of the four laws of equilibrium thermodynamics. The aim of this chapter is to impart a more rigorous theoretical foundation and understanding of minimum and maximum entropy production (MinEP and MaxEP) principles, and to review their current and possible future applications in hydrology. In Sec. 32.2, we examine the entropy production concept for both macroscopic and infinitesimal dissipative systems, and then review and attempt to classify the many variational methods based on dissipation, power consumption, or entropy production. From this review, we draw out the significance of the MaxEP principle of Paltridge for flow systems subject to particular constraints, as well as its conjugate MinEP principle for flow systems subject to different constraints. In Sec. 32.3, we apply the maximum entropy method formulated in Chap. 31 to the analysis of flow systems, both locally and globally, in each case leading to an extremum principle based on a thermodynamic-like “flux potential.” Under different conditions, this furnishes subsidiary MinEP and MaxEP principles, in the same fashion that the minimum free energy principle of thermodynamics gives rise to subsidiary minimum and maximum internal energy (or enthalpy) principles. This analysis provides a theoretical justification of the two conjugate Paltridge principles. In Sec. 32.4, we review the existing (relatively sparse) literature on applications in hydrology, leading in Sec. 32.5 to our conclusions and recommendations for further research. 32.2 BACKGROUND AND REVIEW 32.2.1 Entropy Production
From the second law of thermodynamics, the thermodynamic entropy is not conserved but rather is preserved; that is, once created, it cannot be destroyed. For a macroscopic open system subject to external flows and internal entropy-generating processes, the rate of increase in entropy is measured by the total (thermodynamic) entropy production, as defined by (Jaumann, 1911; de Groot and Mazur, 1984; Niven and Noack, 2014):
σ =
∂S in + FSout ,tot − FS ,tot ≥ 0 ∂t
(32.1)
where S is the thermodynamic entropy within the system, t is time, and FSin,tot and FSin,tot are, respectively, the flow rates of thermodynamic entropy out of and into the system. For a fluid flow system represented by a control volume (CV) enclosed by its control surface (CS), this becomes (de Groot and Mazur, 1984; Niven and Noack, 2013):
32-1
32_Singh_ch32_p32.1-32.8.indd 1
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32-2 Entropy Production Extremum Principles
σ = ∫∫∫
∂ ρs dV + ∫∫ ( jS + ρsv ) ⋅ n dA ≥ 0 ∂t CS
32.2.2 Entropy Production Extremum Principles
(32.2)
where Q is the volumetric flow rate, PL is the loss in piezometric pressure, H L is the head loss, and g is the gravitational acceleration. The sign of (32.7) indicates spontaneous flow in the direction of decreasing pressure or head. For open channel flow, H L is the negative change in total head H L = −∆H = −∆( z + y + U 2 /2 g ) , where z is the channel base elevation, y is the water surface elevation, and U is the mean cross-sectional velocity; furthermore, for uniform flow H L = SL ≈ Sx, where S is the bed slope, L is the downstream length, and x is the downstream horizontal distance. Whether microscopic or macroscopic, for nonradiative systems (32.4) and (32.5) or (32.6) we can refer to pairs of conjugate fluxes and thermodynamic forces, which combine to give individual components of the entropy production. If a pair of extremum principles are identical in all respects but lead to opposing extrema (minimum or maximum), depending on whether they are constrained by flux(es) or thermodynamic force(s), they can be termed conjugate extremum principles.
A schematic diagram is given in Fig. 32.1 of several variational methods proposed in equilibrium and nonequilibrium thermodynamics. In equilibrium thermodynamics, the purpose of an extremum principle is to predict the equilibrium position of the system. Analogously, for a nonequilibrium system, a useful extremum principle should facilitate the prediction of the stationary state, usually termed the steady-state flow2. Some principles have also been proposed for transient flows. As evident from Fig. 32.1, a tremendous assortment of variational methods—each denoted by a small roman letter— have been proposed. The validity of and borderlands between many of these methods have not been adequately charted; indeed, the authors do not even proclaim that the present representation is correct or complete. Furthermore, some methods are of relatively narrow applicability (indeed, some may not be valid at all). Briefly summarizing the essential points: 1. All of equilibrium thermodynamics and many branches of nonequilibrium thermodynamics are underpinned by the maximum entropy (MaxEnt) method (a) developed by Boltzmann (1877) and Planck (1901), rewritten in generic form by Jaynes (1957, 2003). This method is discussed in detail in Chap. 31. For example, applying MaxEnt to the contents of a thermodynamic system (b) gives the four laws of thermodynamics (c) and, for an isolated system, can be applied directly (max S ) to infer its equilibrium state (d). For an open thermodynamic system, MaxEnt leads to a thermodynamic potential Φeq (analogous to a Planck potential or free energy/temperature)—representing the (negative) entropy of the universe—which is minimized (min Φeq ) at equilibrium (e). Both isolated and open equilibrium formulations have strong connections to empirical thermodynamics (e.g., Clausius 1876; Gyftopoulos and Beretta, 2005) (f), finite-time thermodynamics (e.g., Salamon and Berry, 1983; Nulton et al., 1985) (g), and probabilistic dynamics based on the Liouville, Hamiltonian, Fokker-Planck, or Master equations (e.g., Lanczos, 1970; Risken, 1996) (h). However, while the framework of equilibrium thermodynamics indicates the direction of spontaneous change of a system—from which the total global or local entropy production (32.1)–(32.3) must be positive—it makes no statement concerning the rate of change. Many prominent researchers have been confused on this point. For this reason, any justification of an entropy production extremum principle must lie outside the established framework of equilibrium thermodynamics. 2. The conservation of mass, momentum, energy, and charge gives rise to the conservation laws of fluid mechanics, thermodynamics, and chemical reactions (i), including the continuity, Navier-Stokes, energy and mass action equations (e.g., de Groot and Mazur, 1984; White, 2006). These can be used to directly model steady-state and time-variant flows (j). However, these analyses can involve considerable complexity, due to the nonlinear Reynolds stress terms and chemical reaction kinetics, motivating the present search for direct variational methods. 3. A spectral representation of turbulent flow, in conjunction with dimensional arguments, yields the Kolmogorov (1941) equation and related analyses of the turbulent energy cascade (k). Furthermore, the MaxEnt analysis of spectral decompositions of flow systems (l), e.g., by Galerkin proper orthogonal decomposition, can also be used to infer spectral coefficients (modal amplitudes) in certain classes of flows (e.g., Noack and Niven, 2012, 2013). 4. As will be discussed, the application of MaxEnt to an open, flow-controlled system (m) gives a thermodynamic-like potential Φst which is minimized at steady state (min Φst ; Niven, 2009, 2010a) (n). This behavior is analogous to that of the thermodynamic potential of an open equilibrium system (e). Related analyses give steady-state analogs of the four laws of thermodynamics (Niven, 2009) (o), as well as the Fluctuation Theorem (p) for the ratio of probabilities of forward and backward fluxes of the same magnitude (e.g., Evans et al., 1993). The MaxEnt analysis of hybrid systems constrained by both thermodynamic contents and flow rates gives the field of Extended Irreversible Thermodynamics (e.g., Jou et al., 1993) (x). 5. An important pair of variational principles consist of the MaxEP principle of Paltridge (1975, 1978) (q) and its conjugate MinEP principle (e.g., Paulus and Gaggioli, 2004; Martyushev, 2007; Niven, 2010b; Kawazura and Yoshida, 2010, 2012) (r). The former is commonly allied with a MaxEP orthogonality principle developed by Ziegler (1977) (u), often applied to mechanical systems. These methods select the observed steady state from a set of possible steady-state flows; they are therefore of considerable utility and, as discussed, have been widely applied. Examples include the state of a heat convection system based on Rayleigh
Individual components of the entropy production can be negative, for example heat transfer and chemical reaction, provided they are coupled to be non-negative in total.
2 The term “steady-state” is somewhat misleading, since it refers only to the mean flow and not its fluctuations. A steady-state flow need not be steady in time, only in the mean.
CV
where ρ is the fluid density, s is the specific thermodynamic entropy (per unit mass of fluid), jS is the nonfluid entropy flux (per unit area of the control surface), v is the fluid velocity (whence ρsv is the fluid-borne entropy flux), n is the unit normal pointing out of the control surface, “⋅” is the vector scalar product, and dV and dA are infinitesimal volume and area elements, respectively. Using σ = ∫∫∫ σˆ dV to define the local entropy production σˆ , expressed CV per unit volume of fluid, we obtain from (32.2) using Gauss’ theorem: Ds ∂ ρs + ∇ ⋅ ( jS + ρ sv ) = ρ + ∇ ⋅ jS ≥ 0 σˆ = (32.3) Dt ∂t where D/Dt is the substantial derivative. Whether local or global, from the second law of thermodynamics the total entropy production must be nonnegative.1 By a standard analysis of a nonradiative system at local thermodynamic equilibrium, subject to flows of heat, fluid, chemical species, and with chemical reactions, it can be shown that the non-fluid entropy flux and entropy production are given by (de Groot and Mazur, 1984):
1 µ jS = jQ − ∑ c jc T T c
(32.4)
τ : ∇v 1 µ ˆ G − ∑ ξ d ∆ d (32.5) σˆ = jQ ⋅∇ − ∑ jc ⋅∇ c − T T c T T d
where T is the absolute temperature, jQ is the heat flux, jc and µc are respectively the flux and chemical potential of the cth chemical species, τ is the stress tensor (here positive in compression), ξˆd and G d are, respectively, the rate and molar Gibbs free energy of the dth chemical reaction, and “:” is the tensor scalar product.
Equations (32.4) and (32.5) can be summarized respectively by jS = ∑ jr λr and r σˆ = ∑ jr ⋅ fr , where jr ∈ jQ , jc , τ , ξd are generalized fluxes, λr ∈{1/ T , − µc / T } r −1 are intensive variables, and fr ∈{∇T , −∇( µc / T ), −∇v / T , −∆(G d / T )} are
{
}
thermodynamic forces conjugate to jr. Note that the fluid velocity (volumetric flux) v does not appear as a flux in (32.5), but is connected to these relations via (32.3) or by the specific Gibbs-Duhem equality (de Groot and Mazur, 1984). For systems with a potential field or interactions with electromagnetic radiation, more complicated (not necessarily bilinear) relations are required (de Groot and Mazur, 1984; Niven and Noack, 2014). Relations (32.4) and (32.5) are often extended to larger (macroscopic) systems with uniform flow rates and gradients:
σ = ∫∫∫ σˆ dV = ∫∫∫ ∑ jr ⋅ fr dV
CV
CV
r
=
uniform flows
− ∑ Fr ∆λr (32.6) r
where Fr ∈{ FQ , Fc } are generalized flow rates and ∆λr are their conjugate intensive variable differences. For example, the total steady-state entropy production in an incompressible fluid flow with only turbulent dissipation—typical of hydrological flows—can be calculated by (Adeyinka and Naterer, 2004):
σ bulk = − ∫∫∫
CV
1
32_Singh_ch32_p32.1-32.8.indd 2
τ : ∇v P ρ gH L dV = Q L = Q T T T
(32.7)
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BACKGROUND AND REVIEW 32-3
(ab) Steepest entropy ascent (Beretta)
Extended (x) irreversible thermodynamics (b) MaxEnt on contents (a) MaxEnt (Boltzmann, Jaynes)
(c) Zeroth,first, second and third laws of thermodynamics
?
(d)
?
(f) Empirical thermodynamics
Isolated systems max Ssys
Fluctuation (p) theorem (steady state)
(n)
Turbulence cascade (Kolmogorov)
?
(y)
d
(l)
(u)
?
Ziegler orthog.
(j)
? (i) Conservation laws (mass, momentum, energy, charge)
(r) MinEP (ac) design (Bejan)
? MinEP steady state (e.g., Paulus/Gaggioli, Yoshida, Niven)
MaxEP steady state (Paltridge)
?
Finite-time (z) steady-state MinEP limit
Global
st
(q)
Upper bound theory (Malkus, etc.)
Time-variant flows
Local
Open (flow) systems (variable , )
MaxEnt on modes (e.g., Fourier, Galerkin) (k)
?
Open systems Suniv = eq
univ
?
Quantum thermodynamics (Beretta) (aa) Finite-time (g) thermodynamics
(e)
(m) MaxEnt on fluxes
(o) Steady-state analogues of four laws of thermodynamics
Prob. dynamics (h) (Liouville, Hamitonian, Fokker–Planck and Master eqs; ? Fisher information)
(t) Linear regime (Onsager)
?
?
?
(s) Onsager “min dissip”
(w)
Min/max power (networks) ?
Prigogine MinEP
(v)
Figure 32.1 Schematic diagram of relationships between extremum and other allied principles in equilibrium and nonequilibrium systems.
number (Ozawa et al., 2001, 2003), of a turbulent flow system based on Reynolds number (Niven, 2010b), or of a chemical degrading system based on chemical or biological processes (Meysman and Bruers, 2007). Furthermore, the type of extremum (MinEP or MaxEP) appears to depend on the choice of constraint from a given conjugate pair. For example: • For flow in series or parallel pipes, the choice of laminar or turbulent flow regime is consistent with MaxEP for a fixed flow rate, but with MinEP for a fixed head loss (Paulus and Gaggioli, 2004; Martyushev, 2007; Niven, 2010b). • For a zonal flow heat transfer system, the choice of convection regime is consistent with MaxEP for a flux-driven system, but with MinEP for a temperature-driven system (Kawazura and Yoshida, 2010, 2012). • For a heat convection system with parallel connections such as a Bénard cell, the choice of convection regime is consistent with MinEP for a flux-driven system, but with MaxEP for a temperature-driven system (Kawazura and Yoshida, 2012). The full spectrum of this MaxEP/MinEP inversion for different flow systems has not been properly delineated. Apart from the MaxEnt analysis based on fluxes given herein (see Sec. 32.3), few adherents of the Paltridge MaxEP principle have paid much attention to its observed conjugate MinEP form in the literature. 6. A so-called “minimum dissipation” principle was developed by Onsager and Machlup (1953), building on work by Helmholtz and Rayleigh, involving maximizing the entropy production less a dissipation function (s). This method applies only in the linear transport regime (t), i.e., in which the fluxes are linear functions of the forces jr = ∑ Lrk fk , where Lrk k
is the phenomenological coefficient between the rth and kth processes (Onsager, 1931). 7. A well-known theorem was given by Prigogine and coworkers (e.g., Prigogine, 1967), involving MinEP with respect to certain fluxes or forces, constrained by their conjugate parameters (v). The theorem can be derived in the linear transport regime (t), and selects the stationary state from the set of transient states of the system. However, since the
32_Singh_ch32_p32.1-32.8.indd 3
stationary state can be calculated by other methods, it is not very useful (Jaynes, 1980); indeed, there does not appear to be any application of Prigogine’s method in any branch of engineering. Unfortunately, there is widespread confusion between Prigogine’s and other MinEP principles in the literature, and also an incorrect view that Prigogine’s principle “contradicts” other principles, such as of Paltridge. 8. For decades, a MinEP or minimum power principle has been applied to the analysis of flow networks subject to fixed flow rates, primarily electrical circuits (e.g., Jeans, 1925) and more recently fluid flow networks (Paulus and Gaggioli 2004) (w). A conjugate MaxEP or maximum power principle also exists under conjugate constraints of fixed potential differences (e.g., Županović et al., 2004), often confused with the Paltridge principle (see Niven, 2010b). However, these two principles are of the same character, and can be proven to be valid only in networks with a linear or power-law relationship between flow rates and potential differences, in the power-low case with a common power exponent (Niven, 2010b). 9. In the field of “upper bound theory” within turbulent fluid mechanics, the steady state is identified by maximizing a functional of the total, mean, or turbulent dissipation under various constraints (e.g., Malkus, 1956, 2003; Busse, 1970; Howard, 1972) (y). While enjoying some success, such methods have been criticized for their seemingly ad hoc choice of extremum functional and lack of theoretical justification. 10. A minimum entropy or MinEP principle of rather different character has been derived in finite-time thermodynamics, as a fundamental bound to the “cost” of moving a thermodynamic system between two equilibrium states at a specified (finite) rate (e.g., Salamon and Berry, 1983; Nulton et al., 1985) (g). The same theoretical framework has also been applied to derive the MinEP bound for moving a flow system between two steady states at a finite rate (Niven and Andresen, 2009) (z). 11. A quantum formulation of thermodynamics is given by some authors (e.g., Beretta et al., 1984) (aa). Using this formulation, or more direct approaches, a “steepest entropy ascent” principle has been formulated to predict the transient state of an isolated thermodynamic system (Beretta, 2006) (ab).
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32-4 Entropy Production Extremum Principles
12. A quite different MinEP principle involves minimizing the entropy production of an engineered system to obtain the most efficient design (e.g., Bejan, 1996) (ac). This principle serves a different purpose to the predictive principles examined previously. 32.3 MAXIMUM ENTROPY ANALYSIS
We now embark on MaxEnt analyses of a nonequilibrium flow system, respectively for an infinitesimal (local) or macroscopic control volume. 32.3.1 Analysis of an Infinitesimal Fluid Element
Consider an open infinitesimal fluid element subject to instantaneous fluxes of heat, chemical species, and momentum, as well as instantaneous rates of various chemical reactions. The uncertainty in the system is expressed by the joint probability of these quantities. Departing from previous treatments (Niven 2009, 2010a), we here adopt the continuum assumption for each flux or rate, giving the joint probability density function (pdf) defined by
jQ ≤ ϒ Q ≤ jQ + djQ jc ≤ ϒ c ≤ jc + djc , ∀c p( X )dX = Prob. τ ≤ ϒ τ ≤ τ + dτ ˆ ˆ ˆ ξd ≤ ϒ d ≤ ξd + dξd , ∀d
ˆ where X = { jQ ,{ jc }, τ ,{ξd }}, dX = djQ ∏ djc dτ c
(32.8)
ˆ
∏ dξd , and ϒ x is the random d
variable of quantity x. The relative entropy, here termed the local flux entropy, then is p( X ) H = − ∫ d X p( X )ln ( X ) (32.9) q Ω X
where Ω X represents the domain of all variables. Adopting the global expectation notation 〈gr 〉 =
∫ d X p( X ) g r (X )
ΩX
(32.10)
for any function g r , we see that maximizing (32.9) subject to normalization (〈1〉 = 1) and specified values of each mean flux and reaction rate {〈 jQ 〉, 〈 jc 〉, 〈 τ 〉, ˆ 〈ξd 〉 } gives
p* ( X ) =
ˆ q exp −ζ Q ⋅ jQ − ∑ζ c ⋅ jc − ζ τ : τ − ∑ζ d ⋅ ξd (32.11) Z c d
where Z is the partition function and ζ x is the Lagrangian multiplier for quantity x. From (32.9) and (32.11) the maximum entropy is
ˆ H* = ln Z + ζ Q ⋅ 〈 jQ 〉 + ∑ζ c ⋅ 〈 jc 〉 + ζ τ : 〈 τ 〉 + ∑ζ d ⋅ 〈ξd 〉 (32.12) d
c
Comparing the latter to the local entropy production (32.5), we see that each Lagrangian multiplier must be proportional to the mean gradient or potential difference conjugate to its flux or rate (Niven, 2009): 1 µ ∇v G ˆ H* = ln Z − κ −1 〈∇ 〉⋅ 〈 jQ 〉 − ∑ 〈∇ c 〉⋅ 〈 jc 〉 − 〈 〉 : 〈 τ 〉 − ∑ 〈∆ d 〉⋅ 〈ξ d 〉 T T T T c d −1 ˆ = ln Z − κ σ (32.13) where κ is a physical constant (of units J K−1 s−1 m−3), and σˆ = ∑ 〈 jr 〉⋅〈fr 〉 is r
the local entropy production in the mean. Strictly, the latter is not the same as the total entropy production, since by the Reynolds decomposition
〈σˆ 〉 = ∑ 〈 jr ⋅ fr 〉 = ∑ 〈 jr 〉⋅〈fr 〉 + ∑ 〈 jr ' ⋅ fr '〉 = σˆ + ∑ 〈 jr ' ⋅ fr '〉 (32.14) r
r
r
32_Singh_ch32_p32.1-32.8.indd 4
Φst = − ln Z = − H* − κ −1 σˆ
32.3.2 Analysis of a (Macroscopic) Control Volume
We now examine an open macroscopic control volume, in which instantaneous fluxes of fluid, heat, and chemical species cross its control surface. From (32.2) at the steady state, it is not necessary to account separately for dissipation or internal chemical reactions, since their entropy production must be manifested as heat or material fluxes through the boundary (Jaynes, 1980). It is, however, necessary to account for the entropy carried by the fluid velocity v through the boundary, since this is transported into or out ˆ = { j ,{ j }, v }, the total uncertainty is now expressed of the system. Writing X Q c by the joint probability of the fluxes, each as a function of position x on the boundary: jQ ( x ) ≤ ϒ Q ( x ) ≤ jQ ( x ) + djQ ( x ) ˆ ( x ), x ) dX ˆ dx = Prob. jc ( x ) ≤ ϒ c ( x ) ≤ jc ( x ) + djc ( x ), ∀c p( X v ( x ) ≤ ϒ ( x ) ≤ v ( x ) + dv ( x ) (32.16) v x ≤ ϒ x ≤ x + dx The control surface flux entropy can then be defined by integration around the boundary: HCV = − ∫ d x CS
∫
Ω Xˆ ( x )
ˆ ˆ ˆ p( X ˆ , x )ln p( X , x ) ˆ p( X ˆ , x )ln p( X , x ) = − d A dX dX ∫∫ ∫ ˆ ˆ q( X , x ) q( X , x ) Ωˆ CS X(x )
(32.17) where Ω Xˆ ( x ) is the domain of all fluxes at each position on the control surface. By an analogous argument to the previous section, maximizing (32.17) subject to normalization and constraints on the mean value of each flux around the boundary {〈 jQ ( x )〉,〈 jc ( x )〉,〈 v ( x )〉}, the inferred pdf and maximum entropy are ˆ , x ) = q exp −η ( x ) ⋅ j ( x ) − ∑η ( x ) ⋅ j ( x ) − η ( x ) ⋅ v ( x ) (32.18) p* ( X Q c c v Q ZCS c
(
* HCS = ln ZCS + ∫∫ dA ηQ (x ) ⋅〈 jQ (x )〉 + ∑ηc (x ) ⋅〈 jc (x )〉 + ηv (x ) ⋅〈 v (x )〉 CS
c
)
(32.19)
r
where the last term is the mean-fluctuating component (Niven and Noack, 2014). Equation (32.13) can be further rearranged in terms of a local flux potential:
We see that for an open system, the differential dΦst accounts (in a negative sense) for the interplay between the change of (flux) entropy within the system, dH* , and the transfer of (thermodynamic) entropy (in the mean) from the system to the environment, d σˆ . From this we conclude that (1) the flux and thermodynamic entropy concepts, while distinct, are explicitly connected through (32.15); and (2) we can interpret Φst = − ln Z as a thermodynamiclike potential which should be minimized at the stationary state of the system. The flux potential Φst therefore governs the state of the system. Furthermore, depending on the states accessible to the system, the interplay between changes in H* and σˆ to minimize Φst can be manifested in three outcomes: 1. The system could increase both H*and σˆ , to give ∆H > 0 and ∆ σˆ / κ > 0 2. The system could increase H* and decreaseσˆ , provided ∆H ≥| ∆ σˆ / κ |≥ 0 3. The system could decrease H* and increase σˆ , provided ∆ σˆ / κ ≥| ∆H |≥ 0 These outcomes are analogous to what is observed in chemical thermodynamics, in the competition between (so-called) “entropic” and “enthalpic” processes (Atkins, 1982). Finally, we see that if one is unaware of the flux entropy H* concept, the first and third outcomes above might be identified heuristically as “MaxEP processes,” while the middle outcome would be identified as a “MinEP process.” We also note these processes are formulated in the same manner as the Paltridge MaxEP or conjugate MinEP form, based on products of mean fluxes and mean thermodynamic forces, rather than the total entropy production (Niven and Noack, 2014). We therefore establish a connection between the MaxEnt analysis of an infinitesimal fluid element and the two Paltridge extremum principles, united by a thermodynamic-like potential which is minimized at the steady-state flow.
(32.15)
where ηr are the Lagrangian multipliers and ZCS is the global partition function. Comparing (32.19) to (32.2) and (32.4), we recognize each multiplier as the mean conjugate intensive variable or entropy density in the direction of the unit normal, i.e., ηr ( x ) = n 〈λr ( x )〉 for jr ∈{ jQ , jc } and η v ( x ) = n 〈 ρ( x )s( x )〉. From (32.19) and the steady-state form of (32.2)
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REFERENCES 32-5
1 µ (x ) * = ln ZCS − κ −1 HCS ∫∫ dA n T (x ) ⋅ 〈 jQ (x )〉 − ∑ n Tc(x ) ⋅ CS c
(32.20)
〈 jc ( x )〉+ n 〈 ρ ( x )s( x )〉⋅ 〈 v ( x )〉 −1 = ln ZCS − κ ∫∫ dAω ( x )
CS
where ω ( x ) is the mean half-boundary entropy production per unit area at position x, due to the absolute fluxes out of the enclosed volume through the boundary [for more details see Niven and Noack (2014)]. Rearrangement gives the control surface flux potential:
* ΦCS = − ln ZCS = − HCS − κ −1 ∫∫ dA ω (x ) CS
(32.21)
As with (32.15), this again expresses the (negative) interplay between changes of (control surface flux) entropy within the system and the transfer of (thermodynamic) entropy from the system to the environment. We again recover three possibilities, of which two involve apparent maximization of an entropy production term, while one involves its apparent minimization. Once again, we obtain a connection between the MaxEnt analysis of a (macroscopic) control volume and the two conjugate Paltridge extremum principles, united by minimization of a thermodynamic-like potential at the steady-state flow. 32.4 REVIEW OF APPLICATIONS IN HYDROLOGY AND HYDRAULICS
Applications of the Paltridge MaxEP or MinEP principles to hydrology and hydraulics only commenced recently. Kleidon and Schymanski (2008), Kleidon et al., (2009), and Westhoff and Zehe (2013) developed detailed MaxEP models for hydrological water balance, while Porada et al., (2011) and Zehe et al., (2013) have applied MaxEP to hillslope drainage. Zehe et al., (2010) proposed a maximum dissipation model for soil infiltration. Westoff et al., (2014) and Wang et al., (2015) found the empirical relations for the partitioning of precipitation between runoff and evaporation—and subsequent runoff and soil wetting—to be consistent with MaxEP. Paik and Kumar (2010) and Beven (2015) examined the possibility of an optimality principle for landscape evolution, while Hergarten et al., (2014) proposed a minimum dissipation principle for surface drainage networks. Quijano and Lin (2014) review a variety of optimality principles for the critical zone (rock-soil-waterair-organism interface). Other analyses based directly on MaxEnt are examined in Chap. 31. As mentioned, MaxEP may be important in controlling spatial vegetation patterns (Kleidon et al., 2007; del Jesus et al., 2012), while the optimization of the photosynthesis cycle and various whole-plant processes is consistent with MaxEP (e.g., Dewar et al., 2006; Dewar, 2010). Further research is needed on the connection between these phenomena and the MaxEnt analyses of Sec. 32.3. As discussed, analyses of simple series or parallel pipe networks show that the choice of laminar or turbulent flow is consistent with MaxEP if flowconstrained or MinEP if head-constrained (Niven, 2010b). Chung and Vaidya (2011) and Vaidya (2013) found the settling of various particles at low Reynolds number, including elongated particles, dual spheres or a sphere near a wall, to be consistent with MaxEP. Shimokawa and Ozawa (2010) and Ozawa and Shimokawa (2015) found that tropical cyclones evolve to a highEP steady state, and may tend to move along trajectories with higher rates of entropy production. Finally, as discussed, there is now a sizeable literature on MaxEP as a driving force for the Earth’s climate system (e.g., Paltridge, 1975, 1978; Ozawa and Ohmura 1997; Lorenz et al., 2001; Shimokawa and Ozawa, 2001, 2002; Ozawa et al., 2003; Kleidon and Lorenz, 2005), which could have many implications for global atmospheric and oceanic circulation patterns, water and energy cycles, and species distributions under climate change (Kleidon, 2004, 2009a-b, 2010a-b; Kleidon et al., 2007). 32.5 CLOSING REMARKS
This chapter presents a review of extremum principles for nonequilibrium systems, with emphasis on the MaxEP principle of Paltridge (1975, 1978) and its conjugate MinEP form. A crude “treasure map” of such principles, containing some hints on the known and unknown borders and connections between different formulations, is provided in Fig. 32.1. The MaxEnt method of Jaynes (1957) is then applied to local and global nonequilibrium systems, in each case giving a flux potential which is minimized to determine the steady state.
32_Singh_ch32_p32.1-32.8.indd 5
This furnishes subsidiary (pseudo-) MinEP and MaxEP principles, consistent with the two conjugate Paltridge principles. The relatively sparse literature on applications in hydrology and hydraulics is then reviewed. This chapter indicates the need for further substantial, rigorous research on the theoretical development of nonequilibrium extremum principles, and their application throughout hydrology and hydraulics. REFERENCES
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?
(p)
Turbulence cascade (Kolmogorov)
(k)
Fluctuation Theorem (steady state)
Steady-state analogues of four laws of thermodynamics
(o)
MaxEnt (Boltzmann, Jaynes)
(a)
(b)
(x)
(y)
(l)
Time-variant flows
(j)
MaxEnt on modes (eg Fourier, Galerkin)
Upper bound theory (Malkus etc)
?
MaxEnt on fluxes
(i)
?
?
=
(n)
Onsager “min dissip”
?
(s)
?
(r)
(t)
?
(w)
(v)
?
MinEP design (Bejan)
(z)
(ac )
Finite-time steady-state MinEP limit
(aa)
Quantum thermodynamics (Beretta)
Finite-time (g) thermodynamics
?
Min/max power (networks)
Prigogine MinEP
?
? MinEP steady state (eg Paulus/Gaggioli, Yoshida, Niven)
Global
Local
Open systems
(e)
Linear regime (Onsager)
MaxEP steady state (Paltridge)
(q)
(f) Empirical thermodynamics
Open (flow) systems (variable , )
Isolated systems max Ssys
(d)
?
Ziegler orthog.
(u)
Zeroth, first, second & third laws of thermodynamics
?
(ab) Steepest entropy ascent (Beretta)
(c)
Conservation laws (mass, momentum, energy, charge)
(m)
MaxEnt on contents
Extended irreversible thermodynamics
Prob. dynamics (h) (Liouville, Hamitonian, Fokker-Planck and Master eqs; ? Fisher information)
