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2014 | OriginalPaper | Buchkapitel

1. Beyond the Second Law: An Overview

verfasst von : Roderick C. Dewar, Charles H. Lineweaver, Robert K. Niven, Klaus Regenauer-Lieb

Erschienen in: Beyond the Second Law

Verlag: Springer Berlin Heidelberg

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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.

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

Equilibrium is used here in the thermodynamic sense, and not in the dynamic sense of stationarity.

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

However, Paltridge’s energy balance model still contained a number of ad hoc assumptions and parameterizations (see Herbert and Paillard Chap. 9).

The ensemble average is over the probability distribution of microscopic trajectories in phase space (see Sect. 1.4.1).

Here L _ij is the inverse of the matrix R _ij in Table 1.1.

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

The regime of linear force-flux relations.

In Paltridge’s zonally-averaged climate model [4], thermal dissipation is given by σ = Σ_i J _i X _i where J _i = material heat transport between meridional zones i and i + 1, and X _i = 1/T _i+1−1/T _i is the corresponding inverse temperature difference.

The notation is simplified here for clarity. In terms of the notation of Chap. 2, for example, p _Γ = p(Γ(0), 0) is the probability of observing the system in an infinitesmal region around Γ(0) at time t = 0, where Γ(t) denotes the phase space vector at time t, and p _Γ* = p(Γ*(τ), 0) where Γ*(τ) is obtained from Γ(τ) by reversing all the particle velocities.

If δ(i,j) denotes the Kronecker delta function [0 if i ≠ j, 1 if i = j], Eq. (1.2) follows from p(Ω_Γ = A) = Σ_Γ p _Γδ(Ω_Γ, A) = [change of variable] Σ_Γ p _Γ*δ(Ω_Γ*, A) = (from Eq. 1.1) Σ_Γ p _Γexp(–Ω_Γ)δ(–Ω_Γ, A) = e ^AΣ_Γ p _Γδ(Ω_Γ, –A) = e ^A p(Ω_Γ = –A). Equation (1.3) follows from Gibbs’ inequality: –Σ_i p _ilnp _i ≤ –Σ_i p _ilnq _i for any probability distributions p _i and q _i, with equality if and only if p _i = q _i for all i.

In Chap. 3, MaxEnt is used to construct p(f), the probability distribution of macroscopic fluxes f, rather than p _Γ; the formalism can be re-expressed in terms of p _Γ as shown in [42].

Specifically (see Chap. 3), when the non-equilibrium driving force is such that \( \left\langle \Upomega \right\rangle^{\text{C}} > \left\langle \Upomega \right\rangle^{\text{C}}_{ \hbox{min} } , \) then MaxEnt implies \( \left\langle \Upomega \right\rangle = \left\langle \Upomega \right\rangle_{ \hbox{max} } \). Here C denotes a restricted set of stationarity constraints, the nature of which determines the physical nature of \( \left\langle \Upomega \right\rangle^{\text{C}} \) as an entropy production or dissipation functional; \( \left\langle \Upomega \right\rangle^{\text{C}}_{ \hbox{min} } \) and \( \left\langle \Upomega \right\rangle^{\text{C}}_{ \hbox{max} } \) are the lower and upper bounds on \( \left\langle \Upomega \right\rangle^{\text{C}} \).

In Chap. 8 and [28], only a spatially-averaged momentum balance constraint is imposed, rather than the full Navier–Stokes equation.

The restriction of the analysis in [43] to near-equilibrium systems was pointed out in [46, 47].

For an alternative perspective, see Chap. 5.

The minimum is along a path in the space of flux states.

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Titel: Beyond the Second Law: An Overview
verfasst von: Roderick C. Dewar
Charles H. Lineweaver
Robert K. Niven
Klaus Regenauer-Lieb
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_1