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2014 | OriginalPaper | Buchkapitel

3. A Theoretical Basis for Maximum Entropy Production

verfasst von : Roderick C. Dewar, Amos Maritan

Erschienen in: Beyond the Second Law

Verlag: Springer Berlin Heidelberg

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Abstract

Maximum entropy production (MaxEP) is a conjectured selection criterion for the stationary states of non-equilibrium systems. In the absence of a firm theoretical basis, MaxEP has largely been applied in an ad hoc manner. Consequently its apparent successes remain something of a curiosity while the interpretation of its apparent failures is fraught with ambiguity. Here we show how Jaynes’ maximum entropy (MaxEnt) formulation of statistical mechanics provides a theoretical basis for MaxEP which answers two outstanding questions that have so far hampered its wider application: What do the apparent successes and failures of MaxEP actually mean physically? And what is the appropriate entropy production that is maximized in any given problem? As illustrative examples, we show how MaxEnt underpins previous applications of MaxEP to planetary climates and fluid turbulence. We also discuss the relationship of MaxEP to the fluctuation theorem and Ziegler’s maximum dissipation principle.

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Vorheriges Kapitel The Dissipation Function: Its Relationship to Entropy Production, Theorems for Nonequilibrium Systems and Observations on Its Extrema

Nächstes Kapitel Dissipation Rate Functions, Pseudopotentials, Potentials and Yield Surfaces

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The constraints themselves may be perfectly accurate (e.g. global energy balance). In that case it is the completeness of the chosen set that is being tested.

If instead we had I ₀ > I _max(C) then, since I ≤ I _max(C) by definition, we would have I < I ₀ and so μ = 0.

With μ = 0⁺ (Eq. 3.11), \( S = S(\boldsymbol{\uplambda },\boldsymbol{\upalpha }) \) is a functional of \( {\boldsymbol{\uplambda }} \,{\text{and}}\,{\boldsymbol{\upalpha }} \) alone. We have \( S({\boldsymbol{\uplambda }},{\boldsymbol{\upalpha}}) \le S({\boldsymbol{\uplambda}} = {\mathbf{0}},{\boldsymbol{\upalpha}}) \) because removing a constraint—in this case, the auxiliary constraint (3.7)—cannot lead to a decrease in S. Therefore the MaxEnt solution for p(f) in the case \( {\boldsymbol{\Upphi}^{\text{A}}} \ne {\mathbf{0}} \) has \( {\boldsymbol{\uplambda}} = {\mathbf{0}}. \) Note that we cannot set \( {\boldsymbol{\uplambda}} = {\mathbf{0}} \) in the case \( {\boldsymbol{\Upphi}^{\text{A}}} = {\mathbf{0}} \) (i.e. the case S/AS) because (3.13) would then give I(F) = 0 which contravenes I(F) > I _min(C); in this case I(F) is given by (3.15).

Paltridge [1] also maximized the vertical flux of latent and sensible heat from ground to atmosphere in each zone. However, this additional constraint does not alter the MaxEnt derivation of (3.19) as the appropriate EP function for predicting the meridional flux.

Note that under \( \varvec{u} \to -\varvec{u},\varvec{v} \to - 2U\left( z \right)\hat{\varvec{x}} - \varvec{v} \) so that v _x v _z → v _x v _z + 2U(z)v _z, and since ∥v _z∥ = 0 the terms involving v _x v _z do not contribute to Φ ₁ ^A (z) or Φ ₂ ^A .

In fact with d so defined, the fluctuation theorem is a purely mathematical result (see Chap. 1, Footnote 12). As we have seen, the physical interpretation of \( \langle d\rangle \) as entropy production emerges in MaxEnt through the physical constraints that determine p(f).

For an alternative perspective, see Chap. 5. See also Chap. 1, Sect. 1.4.5.

Here we assume units such that Boltzmann’s constant equals 1.

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Titel: A Theoretical Basis for Maximum Entropy Production
verfasst von: Roderick C. Dewar
Amos Maritan
Verlag: Springer Berlin Heidelberg
Buch: Beyond the Second Law
Print ISBN: 978-3-642-40153-4

Electronic ISBN: 978-3-642-40154-1

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-642-40154-1_3