Beyond the Second Law

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.

This chapter is about the role of the dissipation rate function—and other functions derived from it—in determining the constitutive behaviour of dissipative materials. It consists of a discussion of some general theory, followed by examples. We address the class of materials for which knowledge of the functional form of the dissipation rate function supplies the complete constitutive response, without recourse to further assumptions. A careful distinction is drawn between functions that are true potentials and those that are pseudopotentials (defined in the chapter), in order to clarify some aspects of terminology that the Author has elsewhere found confusing. In plasticity theory the intimate relationship between the dissipation rate function and the yield surface is explored. The chapter is illustrated by examples of simple one- and two-dimensional conceptual models, as well as full continuum models. Both rate independent (plastic) and rate-dependent (viscous, or viscoplastic) models are addressed.

This chapter discusses two current interpretations of the maximum entropy production principle—as a physical principle and as an inference procedure. A simple model of relaxation of an isolated system towards equilibrium is considered for this purpose.

General characteristics of entropy production in a fluid system are investigated from a thermodynamic viewpoint. A basic expression for entropy production due to irreversible transport of heat or momentum is formulated together with balance equations of energy and momentum in a fluid system. It is shown that entropy production always decreases with time when the system is of a pure diffusion type without advection of heat or momentum. The minimum entropy production (MinEP) property is thus intrinsic to a pure diffusion-type system. However, this MinEP property disappears when the system is subject to advection of heat or momentum due to dynamic motion. When the rate of advection exceeds the rate of diffusion of heat or momentum, entropy production tends to increase over time. The maximum entropy production (MaxEP), suggested as a selection principle for steady states of nonlinear non-equilibrium systems, can therefore be understood as a characteristic feature of systems with dynamic instability. The observed mean state of vertical convection of the atmosphere is consistent with the condition for MaxEP presented in this study.

This chapter concerns “control volume analysis”, the standard engineering tool for the analysis of flow systems, and its application to entropy balance calculations. Firstly, the principles of control volume analysis are enunciated and applied to flows of conserved quantities (e.g. mass, momentum, energy) through a control volume, giving integral (Reynolds transport theorem) and differential forms of the conservation equations. Several definitions of steady state are discussed. The concept of “entropy” is then established using Jaynes’ maximum entropy method, both in general and in equilibrium thermodynamics. The thermodynamic entropy then gives the “entropy production” concept. Equations for the entropy production are then derived for simple, integral and infinitesimal flow systems. Some technical aspects are examined, including discrete and continuum representations of volume elements, the effect of radiation, and the analysis of systems subdivided into compartments. A Reynolds decomposition of the entropy production equation then reveals an “entropy production closure problem” in fluctuating dissipative systems: even at steady state, the entropy production based on mean flow rates and gradients is not necessarily in balance with the outward entropy fluxes based on mean quantities. Finally, a direct analysis of an infinitesimal element by Jaynes’ maximum entropy method yields a theoretical framework with which to predict the steady state of a flow system. This is cast in terms of a “minimum flux potential” principle, which reduces, in different circumstances, to maximum or minimum entropy production (MaxEP or MinEP) principles. It is hoped that this chapter inspires others to attain a deeper understanding and higher technical rigour in the calculation and extremisation of the entropy production in flow systems of all types.

Planet Earth is a thermodynamic system far from equilibrium and its functioning—obviously—obeys the second law of thermodynamics, at the detailed level of processes, but also at the planetary scale of the whole system. Here, we describe the dynamics of the Earth system as the consequence of sequences of energy conversions that are constrained by thermodynamics. We first describe the well-established Carnot limit and show how it results in a maximum power limit when interactions with the boundary conditions are being allowed for. To understand how the dynamics within a system can achieve this limit, we then explore with a simple model how different configurations of flow structures are associated with different intensities of dissipation. When the generation of power and these different configuration of flow structures are combined, one can associate the dynamics towards the maximum power limit with a fast, positive and a slow, negative feedback that compensate each other at the maximum power state. We close with a discussion of the importance of a planetary, thermodynamic view of the whole Earth system, in which thermodynamics limits the intensity of the dynamics, interactions strongly shape these limits, and the spatial organization of flow represents the means to reach these limits.

In this chapter, we show how the MaxEP hypothesis may be used to build simple climate models without representing explicitly the energy transport by the atmosphere. The purpose is twofold. First, we assess the performance of the MaxEP hypothesis by comparing a simple model with minimal input data to a complex, state-of-the-art General Circulation Model. Next, we show how to improve the realism of MaxEP climate models by including climate feedbacks, focusing on the case of the water-vapour feedback. We also discuss the dependence of the entropy production rate and predicted surface temperature on the resolution of the model.

A detailed thermodynamic, sensitivity analysis of the steady state climate system is performed with respect to the solar constant S ^* and the carbon dioxide concentration of the atmosphere, [CO₂]. Using PlaSim, an Earth-like general circulation model of intermediate complexity, S ^* is modulated between 1,160 and 1,510 Wm⁻² for values of [CO₂] ranging from 90 to 2,880 ppm. It is observed that in a wide parameter range, which includes the present climate conditions, the climate is multistable, i.e. there are two coexisting attractors, one characterised by warm, moist climates (W) and the other by a completely frozen sea surface (Snowball Earth, SB). For both sets of states, empirical relationships for surface temperature, material entropy production, meridional energy transport, Carnot efficiency and dissipation of kinetic energy are constructed in the parametric plane ([CO₂], S ^* ). Linear relationships are found for the two transition lines (W → SB and SB → W) in ([CO₂], S ^* ) between S ^* and the logarithm of [CO₂]. The dynamical and thermodynamical properties of W and SB are completely different. W states are dominated by the hydrological cycle and latent heat is prominent in the material entropy production. The SB states are mainly dry climates where heat transport is realized through sensible heat fluxes and entropy mostly generated by dissipation of kinetic energy. It is also shown that the Carnot-like efficiency regularly increases towards each transition between W and SB and that each transition is associated with a large decrease of the Carnot efficiency indicating a restabilisation of the system. Furthermore, it has been found that in SB states, changes in the vertical temperature structure are responsible for the observed changes in the meridional transport.

Distributions of temperature and longwave radiation are predicted from a state of maximum entropy production (MaxEP) due to meridional heat flux in the atmospheres of the Earth, Mars, Titan and Venus, and the predicted distributions are compared with observational results. In the predictions, we use a multi-box energy balance model that takes into account the effects of obliquity and latitudinal variation of albedo on shortwave absorption. It is found that the predicted distributions are generally in agreement with observations of the Earth, Titan and Venus, suggesting the validity of the MaxEP state for these planets. In the case of Mars, the predicted distributions do not agree well with the observations when compared with those predicted from a state of no meridional heat flux. A simple analysis on advective heat flux using a two-box model shows that the Martian atmosphere is so scant that it cannot carry the heat energy that is necessary for the MaxEP state by advection. These results suggest that the validity of the MaxEP state for a planetary atmosphere is limited when the total amount of atmosphere is not enough to sustain the advective heat flux that is necessary for the MaxEP state.

Heat radiation in gases or plasmas is usually out of local thermodynamic equilibrium (LTE) even if the underlying matter is in LTE. Radiative transfer can then be described with the radiative transfer equation (RTE) for the radiation intensity. A common approach to solve the RTE consists in a moment expansion of the radiation intensity, which leads to an infinite set of coupled hyperbolic partial differential equations for the moments. A truncation of the moment equations requires the definition of a closure. We recommend to use a closure based on a constrained minimum entropy production rate principle. It yields transport coefficients (e.g., effective mean absorption coefficients and Eddington factor) in accordance with the analytically known limit cases. In particular, it corrects errors and drawbacks from other closures often used, like the maximum entropy principle (e.g., the M1 approximation) and the isotropic diffusive P1 approximation. This chapter provides a theoretical overview on the entropy production closure, with results for an illustrative artificial example and for a realistic air plasma.

We review the experimental and theoretical literature on the steady terminal orientation of a body as it settles in a viscous fluid. The terminal orientation of a rigid body is a classic example of a system out of equilibrium. While the dynamical equations are effective in deriving the equilibrium states, they are far too complex and intractable as of yet to resolve questions about the nature of stability of the solutions. The maximum entropy production principle is therefore invoked, as a selection principle, to understand the stable, steady state patterns. Some on-going work and inherent complexities of fluid solid systems are also discussed.

The principle of Maximum Entropy Production (MaxEP) has been successfully used to reproduce the steady states of a range of non-equilibrium systems. Here we investigate MaxEP and maximum heat flux extremum principles directly via the simulation of a Rayleigh-Bèrnard convective system implemented as a lattice gas model. Heat flux and entropy production emerges in this system via the resolution of particle interactions. In the spirit of other related works, we use a reductionist approach, creating a lattice-Boltzmann model to produce steady-convective states between reservoirs of different temperatures. Convection cells emerge that show meta-stability where a given lattice size is able to support a range of convective states. Slow expansion and contraction of the model lattice, implemented by addition and subtraction of vertices, shows hysteresis loops where stable convection cells are expanded to regions wherein they become meta-stable, and eventually transition into more stable configurations. The maximally stable state is found to be that which maximises the rate of heat transfer, which is only equivalent to maximum internal entropy production in a strong forcing regime, while it is consistent with minimising entropy production in a weak forcing case. These results demonstrate the utility of lattice-Boltzmann models for future studies of non-equilibrium systems, and highlight the importance of dissipation and forcing rates in disambiguating proposed extremum principles.

The self-organization of macroscopic structure apparently contradicts the second law of thermodynamics. However, disorder can still develop on microscopic scales. In nonlinear systems, order and disorder may thus coexist on different scales. Here we study the self-organization of macroscopic structures in driven, nonlinear systems. Using a simple phenomenological transport model (of current in an electric circuit, or heat transport in a turbulent fluid/plasma) with linear and nonlinear impedances, we analyze the behavior of the rate of entropy production (σ) as a macroscopic system undergoes a bifurcation between linear and nonlinear operating points. Here σ acts as a generating function for a Legendre transformation between flux-driven and force-driven systems, the thermodynamic potential for which is a generalization of Onsager’s dissipation function. We derive a duality relation that implies min/max-σ behavior depending on the connectivity of the impedances (series or parallel) and the type of driving.

Recent investigations on scale-invariant processes such as topography and modeling evapotranspiration demonstrate the usefulness and potential of the principles of maximum entropy (MaxEnt) and maximum entropy production (MaxEP) in the study of Earth systems. MaxEnt allows theoretical predictions of probability distributions of geophysical multifractal processes based on a small number of geometric parameters. MaxEP leads to the prediction of evapotranspiration and heat fluxes using fewer input variables than existing process based models. Encouraging progress in the application of MaxEnt and MaxEP, viewed as unified principles of inference, paves the way for more approaches of understanding, characterizing and predicting the behavior of the complex Earth systems.

The ability to understand and predict how thermal, hydrological, mechanical and chemical (THMC) processes interact is fundamental to many research initiatives and industrial applications. We present (1) a new Thermal–Hydrological–Mechanical–Chemical (THMC) coupling formulation, based on non-equilibrium thermodynamics; (2) show how THMC feedback is incorporated in the thermodynamic approach; (3) suggest a unifying thermodynamic framework for multi-scaling; and (4) formulate a new rationale for assessing upper and lower bounds of dissipation for THMC processes. The technique is based on deducing time and length scales suitable for separating processes using a macroscopic finite time thermodynamic approach. We show that if the time and length scales are suitably chosen, the calculation of entropic bounds can be used to describe three different types of material and process uncertainties: geometric uncertainties, stemming from the microstructure; process uncertainty, stemming from the correct derivation of the constitutive behavior; and uncertainties in time evolution, stemming from the path dependence of the time integration of the irreversible entropy production. Although the approach is specifically formulated here for THMC coupling we suggest that it has a much broader applicability. In a general sense it consists of finding the entropic bounds of the dissipation defined by the product of thermodynamic force times thermodynamic flux which in material sciences corresponds to generalized stress and generalized strain rates, respectively.

The maximum entropy production (MaxEP) principle has been applied to steady state systems, but biogeochemical problems of interest are typically transient in nature. To apply MaxEP to biogeochemical reaction networks, we propose that living systems maximum entropy production over appropriate time horizons based on strategic information stored in their genomes, which differentiates them from inanimate chemistry, such as fire, that maximizes entropy production instantaneously. We develop a receding horizon optimal control procedure that maximizes internal entropy production over different intervals of time. This procedure involves optimizing the stoichiometry of a reaction network to determine how biological structure is partitioned to reactions over an interval of time. The modeling work is compared to a methanotrophic microcosm experiment that is being conducted to examine how microbial systems integrate entropy production over time when subject to time varying energy input attained by periodically cycling feed-gas composition. The MaxEP-based model agrees well with experimental results, and model analysis shows that increasing the optimization time horizon increases internal entropy production.

Accepted (July 2012) in: Beyond the Second Law: Entropy Production and Non-Equilibrium Systems. R. C. Dewar, C. H. Lineweaver, R. K. Niven and K. Regenauer-Lieb, Springer.

There have been many attempts to use optimization approaches to study the biological evolution of enzyme kinetics. Our basic assumption here is that the biological evolution of catalytic cycle fluxes between enzyme internal functional states is accompanied by increased entropy production of the fluxes and increased Shannon information entropy of the states. We use simplified models of enzyme catalytic cycles and bioenergetically important free-energy transduction cycles to examine the extent to which this assumption agrees with experimental data. We also discuss the relevance of Prigogine’s minimal entropy production theorem to biological evolution.

This chapter discusses morphological transitions during non-equilibrium crystallization and coexistence of crystals of different shapes from the viewpoint of the maximum entropy production principle (MEPP).

The dominant mode of entropy production enabled by the large-scale technological systems that power the world economy is the degradation of chemical energy in fossil fuels. One key parameter determining the rates of fossil fuel consumption and entropy production is the price of energy. The Rayleigh-Benard cell provides a laboratory analog in which, for a given driving force, the rate of entropy production is determined by the value of the thermal boundary layer thickness, whose inverse plays a role similar to that of price in large fossil fuel systems. In steady, serial systems like the diffusion-advection-diffusion Rayleigh-Benard cell or the oilfield-pipeline-city “technology cell”, an auto-control parameter like price or boundary layer thickness is required to coordinate spatially separated energy source and sink dynamics. For complex fossil fuel technologies the implicit and often unknown dependence of such control parameters on intrinsic system variables can hide internal constraints. If applied in the absence of knowledge of such constraints, the principle of Maximum Entropy Production (MaxEP) would yield, for sufficiently complex systems, an upper limit to rather than the actual value of the entropy production rate. Internal constraints on technology-enabled energy consumption, however, may represent only temporary hangups on the road toward a larger entropy production rate.

If the universe had been born in a high entropy, equilibrium state, there would be no stars, no planets and no life. Thus, the initial low entropy of the universe is the fundamental reason why we are here. However, we have a poor understanding of why the initial entropy was low and of the relationship between gravity and entropy. We are also struggling with how to meaningfully define the maximum entropy of the universe. This is important because the entropy gap between the maximum entropy of the universe and the actual entropy of the universe is a measure of the free energy left in the universe to drive all processes. I review these entropic issues and the entropy budget of the universe. I argue that the low initial entropy of the universe could be the result of the inflationary origin of matter from unclumpable false vacuum energy. The entropy of massive black holes dominates the entropy budget of the universe. The entropy of a black hole is proportional to the square of its mass. Therefore, determining whether the Maximum Entropy Production Principle (MaxEP) applies to the entropy of the universe is equivalent to determining whether the accretion disks around black holes are maximally efficient at dumping mass onto the central black hole. In an attempt to make this question more precise, I review the magnetic angular momentum transport mechanisms of accretion disks that are responsible for increasing the masses of black holes