Contribution
A synthesis of the maximum-entropy-production (MEP) programme for Earth science. Its four parts:
- an introduction to nonequilibrium entropy budgets through box models;
- an order-of-magnitude entropy budget of the whole Earth;
- a survey of MEP applications to turbulent, hydrological and biotic processes;
- an argument that MEP gives the Gaia hypothesis a thermodynamic basis, with negative feedbacks at an MEP state and life driving the planet toward higher entropy production.
Key insight
Wherever a flux depletes the gradient that drives it, entropy production (flux × force) is zero at both extremes: no flux, or a flux so strong it leaves no gradient. In between there is a maximum. If a system’s internal configuration is free to vary, MEP says it sits at that maximum. The freedom has to come from somewhere, for example turbulent eddies, vegetation or soil heterogeneity. The thermodynamic picture is then about which boundary conditions are “flexible”.
Assumptions
- MEP’s own condition. “If there are sufficient degrees of freedom within the system, it will adopt a steady state at which σ is maximized” (p. 655). “Sufficient” is not defined.
- MEP is taken as given for the implications section. That section is explicitly “under the presumption that MEP is justified sufficiently well” (p. 670).
- Isothermal or near-uniform temperature where dissipation is identified with entropy production times T, since 1/T “usually does not vary by much” (p. 655).
- The box models use linear conductive fluxes, F_ab = k(T_a − T_b), with k reinterpreted as an eddy diffusivity when MEP is demonstrated (case 3, p. 658).
- Radiative entropy is computed without the 4/3 factor, judged “of no relevance to the energy exchanges on Earth” (p. 661).
Key results
- MEP vs Prigogine (pp. 655–656, 673). “Prigogine’s principle states that the steady state is one of minimum entropy production compared to transient states”. Once a steady state is assumed it “makes no further predictions”, in particular among several possible steady states. MEP “acts as a selection principle of the most probable out of many possible steady states”. The text illustrates the two with Fig. 1 (relaxation) and Fig. 2 (selection among steady states). Applying Prigogine’s principle to steady-state configurations, as in geomorphology, is called “possibly for the wrong reasons”.
- Box models (Figs. 1–2).
- An isolated two-box system relaxes to equilibrium with σ → 0 and S → S_max.
- A forced system reaches a steady state with S < S_max. It moves away from equilibrium when the forcing time scale is shorter than the mixing time scale.
- With alternative heat-loss pathways, σ_ab peaks at an intermediate conductivity. In that model the MEP state is also the state of minimum entropy S, furthest from equilibrium. The author warns this is “clearly no definite proof that this is necessarily always so” (p. 659).
- Earth’s entropy budget (Fig. 3, §“Entropy production by Earth system
processes”), in mW m⁻² K⁻¹:
- absorption of sunlight: 555 at the surface and 258 in the atmosphere;
- scattering: about 26;
- terrestrial radiative exchange: about 26;
- the hydrologic cycle: 23;
- the carbon cycle: about 6, 70% of it at photosynthesis;
- ground heat diffusion: about 1.
- Turbulent processes are small in the budget. Their point is that they enhance radiative entropy export.
- Surveyed MEP applications.
- Atmosphere: Paltridge’s climate model and its successors; Lorenz et al. on Mars and Titan; GCM sensitivity studies in which entropy production saturates at a maximum.
- Ocean: Shimokawa and Ozawa, where the MEP state is the most stable of several steady states.
- Mantle and earthquakes.
- Land evapotranspiration (Wang et al.).
- Sediment decomposition (Meysman and Bruers).
- Each is reported, not reproduced.
- Implications, given MEP.
- An MEP state answers perturbations with negative feedback. A change in boundary conditions first gives positive feedback until the new optimum is reached (Fig. 7).
- Planetary MEP mediated by life can give Gaia-like homeostasis (Kleidon 2004a).
- The Earth may have evolved stepwise toward higher entropy production.
- MEP “provides the basis to unite” Odum’s maximum power principle, Bejan’s constructal law and self-organized criticality.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | MEP and Prigogine’s minimum principle answer different questions: approach to a steady state vs selection among steady states | moderate | argued with the box models (Figs. 1–2); no formal statement of either principle’s conditions |
| C2 | Flux–force trade-offs give MEP states in heat transport, convection and hydrology | strong for the existence of a maximum in the box models; the claim that real systems sit at it is the empirical MEP literature, reported | Figs. 2, 5; surveyed studies |
| C3 | The Earth’s entropy budget is dominated by absorption of solar radiation | strong (order of magnitude) | Fig. 3 and text; compiled from earlier budgets; three figures carry unit slips (Corrections) |
| C4 | MEP has a theoretical foundation in Dewar’s information-theoretic work | weak, and the author says so | ”not yet been firmly established”, “work in progress” (pp. 654–655) |
| C5 | Biotic activity raises planetary entropy production by enhancing absorption of sunlight | moderate in the models cited; untested as a general claim | Kleidon 2004a, 2006 and Kleidon and Fraedrich 2005 model studies; “would still need to be further explored” (p. 670) |
| C6 | MEP implies negative feedbacks at steady state, so a planet at MEP behaves as Gaia postulates | weak | conditional on MEP; Fig. 7 is schematic |
| C7 | MEP unites maximum power, the constructal law and SOC | assertion | p. 673; the SOC link rests on Dewar 2003, which Grinstein and Linsker refuted |
Concepts
- Net entropy exchange (NEE): the divergence of entropy fluxes across the boundary, with export positive. In steady state σ = NEE (p. 655).
- Thermodynamic equilibrium (TE) as reference state for each cycle:
- for the hydrologic cycle, a saturated atmosphere;
- for the carbon cycle, CO₂ in equilibrium with the hot interior;
- for motion, rest (Table 1).
- “Batteries”: processes that generate the free energy driving a cycle away from TE, such as atmospheric circulation for water, and photosynthesis and the geologic interior for carbon (Fig. 4).
Connections
- Foundation. MEP’s foundation in Dewar (LIT-521) is acknowledged as incomplete. The criticisms are named: Grinstein and Linsker (LIT-525), Bruers 2007, Rodgers 1976 and Goody 2007 on rotation, Volk 2007, and Caldeira 2007 (“may be true, but trivially so”).
- Physical foundation and the Prigogine question. Deferred to Martyushev and Seleznev 2006 (LIT-520) and Martyushev, Nazarova and Seleznev 2007, “On the problem of the minimum entropy production in the nonequilibrium stationary state”. Neither is read in the record.
- Biological lineage. Lotka, Odum, Ulanowicz and Hannon, Schneider and Kay, Schneider and Sagan. Schrödinger 1944 for the cell exporting entropy. Lovelock for disequilibrium as a sign of life and for Gaia.
Bearing on the record
- Is MEP established? No, on this text’s own account. “At this point, no one would argue that MEP is well established and that applications are without their limitations” (p. 673). The record should treat MEP as a proposed principle with a body of successful model fits and a contested derivation. Dewar 2003 and Grinstein and Linsker together are the derivation and the dispute. This review is the applications.
- The objection to Prigogine, as this literature states it. The minimum principle is not wrong but answers another question: it says nothing about which steady state is selected. So it cannot do the work Prigogine’s programme and its geomorphological users asked of it. The review does not state the conditions under which Prigogine’s theorem holds. Whether that objection agrees with Landauer’s 1975 objection (LIT-532), that no entropy-based quantity can characterise the steady state at all, cannot be checked: Landauer 1975 could not be read.
- Misreporting of the critics. The review says Grinstein and Linsker “point out some technical flaws in the derivation of the linkage of MEP and SOC”. Their Comment (NOTE-422) makes that point second. Its first and main point is that Dewar’s 2005 derivation of MaxEP fails for nonlinear constitutive relations, i.e. far from equilibrium, where the review applies MEP. The review understates the critique.
- Life as dissipation. The second half is where the record’s dissipative-structure and life literatures meet. See the LIT’s Standing for LIT-192 and LIT-211. The review offers no criterion separating a living dissipator from an abiotic one, and its biotic MEP claims are model sensitivity studies.
- No ML instruction. The anthology does not hold it.
Limitations
- Advocacy. The author is the principal proponent, and much of the surveyed evidence is his own model work.
- MEP is presumed in the implications section, by the author’s statement.
- “Sufficient degrees of freedom” is the condition MEP needs. It is never made operational, so a failure of MEP can always be read as too few degrees of freedom.
- The Prigogine comparison is by example. The review never states the conditions under which the minimum principle is a theorem.
- Box-model results such as MEP coinciding with minimum S are flagged by the author as not general.
Open questions
- A derivation of MEP that holds far from equilibrium. The author calls for “a more solid foundation” (p. 673).
- What makes a system’s boundary conditions flexible enough for MEP? The author speculates that this sets limits on emergent complexity (p. 659).
- Can planetary MEP be tested against Earth history, for example stepwise increases in entropy production at the oxidation events?
Corrections
- none to a seeded skim (there was no seed)
- Units in the entropy budget text (p. 662). Three estimates are given
in W m⁻² K⁻¹ that the stated fluxes and temperatures give in mW m⁻² K⁻¹:
- dry convection: 20 W m⁻² × (1/280 − 1/288) K⁻¹ ≈ 2.0 × 10⁻³ W m⁻² K⁻¹, printed “2 W m⁻² K⁻¹”;
- poleward heat transport: about 10 W m⁻² across about 45 K, printed “about 6 W m⁻² K⁻¹” for what comes to about 6 mW m⁻² K⁻¹;
- mantle convection: about 0.1 W m⁻² across about 5,000 K, printed “about 0.3 W m⁻² K⁻¹” for what comes to about 0.3 mW m⁻² K⁻¹. The surrounding budget is in mW m⁻² K⁻¹. The hydrologic (23) and carbon (6) figures recompute correctly from the text’s inputs.
- Grinstein and Linsker’s critique is reported as concerning only the MEP–SOC link. See Bearing.