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2581 KiB

Open AccessArticle

A Novel Derivation of the Time Evolution of the Entropy for Macroscopic Systems in Thermal Non-Equilibrium

by Enrico Sciubba and Federico Zullo

Entropy 2017, 19(11), 594; https://doi.org/10.3390/e19110594 - 07 Nov 2017

Cited by 6 | Viewed by 3629

The paper discusses how the two thermodynamic properties, energy (U) and exergy (E), can be used to solve the problem of quantifying the entropy of non-equilibrium systems. Both energy and exergy are a priori concepts, and their formal dependence on thermodynamic state variables [...] Read more.

The paper discusses how the two thermodynamic properties, energy (U) and exergy (E), can be used to solve the problem of quantifying the entropy of non-equilibrium systems. Both energy and exergy are a priori concepts, and their formal dependence on thermodynamic state variables at equilibrium is known. Exploiting the results of a previous study, we first calculate the non-equilibrium exergy E_n-eq can be calculated for an arbitrary temperature distributions across a macroscopic body with an accuracy that depends only on the available information about the initial distribution: the analytical results confirm that E_n-eq exponentially relaxes to its equilibrium value. Using the Gyftopoulos-Beretta formalism, a non-equilibrium entropy S_n-eq(x,t) is then derived from E_n-eq(x,t) and U(x,t). It is finally shown that the non-equilibrium entropy generation between two states is always larger than its equilibrium (herein referred to as “classical”) counterpart. We conclude that every iso-energetic non-equilibrium state corresponds to an infinite set of non-equivalent states that can be ranked in terms of increasing entropy. Therefore, each point of the Gibbs plane corresponds therefore to a set of possible initial distributions: the non-equilibrium entropy is a multi-valued function that depends on the initial mass and energy distribution within the body. Though the concept cannot be directly extended to microscopic systems, it is argued that the present formulation is compatible with a possible reinterpretation of the existing non-equilibrium formulations, namely those of Tsallis and Grmela, and answers at least in part one of the objections set forth by Lieb and Yngvason. A systematic application of this paradigm is very convenient from a theoretical point of view and may be beneficial for meaningful future applications in the fields of nano-engineering and biological sciences. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

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1712 KiB

Open AccessArticle

Eigentimes and Very Slow Processes

by Bjarne Andresen and Christopher Essex

Entropy 2017, 19(9), 492; https://doi.org/10.3390/e19090492 - 14 Sep 2017

Cited by 4 | Viewed by 4658

Abstract

We investigate the importance of the time and length scales at play in our descriptions of Nature. What can we observe at the atomic scale, at the laboratory (human) scale, and at the galactic scale? Which variables make sense? For every scale we [...] Read more.

We investigate the importance of the time and length scales at play in our descriptions of Nature. What can we observe at the atomic scale, at the laboratory (human) scale, and at the galactic scale? Which variables make sense? For every scale we wish to understand we need a set of variables which are linked through closed equations, i.e., everything can meaningfully be described in terms of those variables without the need to investigate other scales. Examples from physics, chemistry, and evolution are presented. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

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2491 KiB

Open AccessArticle

Physical Universality, State-Dependent Dynamical Laws and Open-Ended Novelty

by Alyssa M. Adams, Angelica Berner, Paul C. W. Davies and Sara I. Walker

Entropy 2017, 19(9), 461; https://doi.org/10.3390/e19090461 - 01 Sep 2017

Cited by 8 | Viewed by 6426

Abstract

A major conceptual step forward in understanding the logical architecture of living systems was advanced by von Neumann with his universal constructor, a physical device capable of self-reproduction. A necessary condition for a universal constructor to exist is that the laws of physics [...] Read more.

A major conceptual step forward in understanding the logical architecture of living systems was advanced by von Neumann with his universal constructor, a physical device capable of self-reproduction. A necessary condition for a universal constructor to exist is that the laws of physics permit physical universality, such that any transformation (consistent with the laws of physics and availability of resources) can be caused to occur. While physical universality has been demonstrated in simple cellular automata models, so far these have not displayed a requisite feature of life—namely open-ended evolution—the explanation of which was also a prime motivator in von Neumann’s formulation of a universal constructor. Current examples of physical universality rely on reversible dynamical laws, whereas it is well-known that living processes are dissipative. Here we show that physical universality and open-ended dynamics should both be possible in irreversible dynamical systems if one entertains the possibility of state-dependent laws. We demonstrate with simple toy models how the accessibility of state space can yield open-ended trajectories, defined as trajectories that do not repeat within the expected Poincaré recurrence time and are not reproducible by an isolated system. We discuss implications for physical universality, or an approximation to it, as a foundational framework for developing a physics for life. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

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275 KiB

Open AccessArticle

The Thermodynamical Arrow and the Historical Arrow; Are They Equivalent?

by Martin Tamm

Entropy 2017, 19(9), 455; https://doi.org/10.3390/e19090455 - 30 Aug 2017

Cited by 1 | Viewed by 4670

Abstract

In this paper, the relationship between the thermodynamic and historical arrows of time is studied. In the context of a simple combinatorial model, their definitions are made more precise and in particular strong versions (which are not compatible with time symmetric microscopic laws) [...] Read more.

In this paper, the relationship between the thermodynamic and historical arrows of time is studied. In the context of a simple combinatorial model, their definitions are made more precise and in particular strong versions (which are not compatible with time symmetric microscopic laws) and weak versions (which can be compatible with time symmetric microscopic laws) are given. This is part of a larger project that aims to explain the arrows as consequences of a common time symmetric principle in the set of all possible universes. However, even if we accept that both arrows may have the same origin, this does not imply that they are equivalent, and it is argued that there can be situations where one arrow may be well-defined but the other is not. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

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387 KiB

Open AccessArticle

Stochastic Thermodynamics of Brownian Motion

by Grégoire Nicolis and Yannick De Decker

Entropy 2017, 19(9), 434; https://doi.org/10.3390/e19090434 - 23 Aug 2017

Cited by 13 | Viewed by 8241

Abstract

A stochastic thermodynamics of Brownian motion is set up in which state functions are expressed in terms of state variables through the same relations as in classical irreversible thermodynamics, with the difference that the state variables are now random fields accounting for the [...] Read more.

A stochastic thermodynamics of Brownian motion is set up in which state functions are expressed in terms of state variables through the same relations as in classical irreversible thermodynamics, with the difference that the state variables are now random fields accounting for the effect of fluctuations. Explicit expressions for the stochastic analog of entropy production and related quantities are derived for a dilute solution of Brownian particles in a fluid of light particles. Their statistical properties are analyzed and, in the light of the insights afforded, the thermodynamics of a single Brownian particle is revisited and the status of the second law of thermodynamics is discussed. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

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1511 KiB

Open AccessArticle

Solutions to the Cosmic Initial Entropy Problem without Equilibrium Initial Conditions

by Vihan M. Patel and Charles H. Lineweaver

Entropy 2017, 19(8), 411; https://doi.org/10.3390/e19080411 - 10 Aug 2017

Cited by 7 | Viewed by 5698

Abstract

The entropy of the observable universe is increasing. Thus, at earlier times the entropy was lower. However, the cosmic microwave background radiation reveals an apparently high entropy universe close to thermal and chemical equilibrium. A two-part solution to this cosmic initial entropy problem [...] Read more.

The entropy of the observable universe is increasing. Thus, at earlier times the entropy was lower. However, the cosmic microwave background radiation reveals an apparently high entropy universe close to thermal and chemical equilibrium. A two-part solution to this cosmic initial entropy problem is proposed. Following Penrose, we argue that the evenly distributed matter of the early universe is equivalent to low gravitational entropy. There are two competing explanations for how this initial low gravitational entropy comes about. (1) Inflation and baryogenesis produce a virtually homogeneous distribution of matter with a low gravitational entropy. (2) Dissatisfied with explaining a low gravitational entropy as the product of a ‘special’ scalar field, some theorists argue (following Boltzmann) for a “more natural” initial condition in which the entire universe is in an initial equilibrium state of maximum entropy. In this equilibrium model, our observable universe is an unusual low entropy fluctuation embedded in a high entropy universe. The anthropic principle and the fluctuation theorem suggest that this low entropy region should be as small as possible and have as large an entropy as possible, consistent with our existence. However, our low entropy universe is much larger than needed to produce observers, and we see no evidence for an embedding in a higher entropy background. The initial conditions of inflationary models are as natural as the equilibrium background favored by many theorists. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

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237 KiB

Open AccessArticle

Kinetic Theory beyond the Stosszahlansatz

by Gregor Chliamovitch, Orestis Malaspinas and Bastien Chopard

Entropy 2017, 19(8), 381; https://doi.org/10.3390/e19080381 - 25 Jul 2017

Cited by 6 | Viewed by 6595

Abstract

In a recent paper (Chliamovitch, et al., 2015), we suggested using the principle of maximum entropy to generalize Boltzmann’s Stosszahlansatz to higher-order distribution functions. This conceptual shift of focus allowed us to derive an analog of the Boltzmann equation for the two-particle distribution [...] Read more.

In a recent paper (Chliamovitch, et al., 2015), we suggested using the principle of maximum entropy to generalize Boltzmann’s Stosszahlansatz to higher-order distribution functions. This conceptual shift of focus allowed us to derive an analog of the Boltzmann equation for the two-particle distribution function. While we only briefly mentioned there the possibility of a hydrodynamical treatment, we complete here a crucial step towards this program. We discuss bilocal collisional invariants, from which we deduce the two-particle stationary distribution. This allows for the existence of equilibrium states in which the momenta of particles are correlated, as well as for the existence of a fourth conserved quantity besides mass, momentum and kinetic energy. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

660 KiB

Open AccessArticle

Nonequilibrium Entropy in a Shock

by L.G. Margolin

Entropy 2017, 19(7), 368; https://doi.org/10.3390/e19070368 - 19 Jul 2017

Cited by 16 | Viewed by 4553

Abstract

In a classic paper, Morduchow and Libby use an analytic solution for the profile of a Navier–Stokes shock to show that the equilibrium thermodynamic entropy has a maximum inside the shock. There is no general nonequilibrium thermodynamic formulation of entropy; the extension of [...] Read more.

In a classic paper, Morduchow and Libby use an analytic solution for the profile of a Navier–Stokes shock to show that the equilibrium thermodynamic entropy has a maximum inside the shock. There is no general nonequilibrium thermodynamic formulation of entropy; the extension of equilibrium theory to nonequililbrium processes is usually made through the assumption of local thermodynamic equilibrium (LTE). However, gas kinetic theory provides a perfectly general formulation of a nonequilibrium entropy in terms of the probability distribution function (PDF) solutions of the Boltzmann equation. In this paper I will evaluate the Boltzmann entropy for the PDF that underlies the Navier–Stokes equations and also for the PDF of the Mott–Smith shock solution. I will show that both monotonically increase in the shock. I will propose a new nonequilibrium thermodynamic entropy and show that it is also monotone and closely approximates the Boltzmann entropy. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

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365 KiB

Open AccessFeature PaperArticle

Cosmological Time, Entropy and Infinity

by Clémentine Hauret, Pierre Magain and Judith Biernaux

Entropy 2017, 19(7), 357; https://doi.org/10.3390/e19070357 - 14 Jul 2017

Cited by 2 | Viewed by 6066

Abstract

Time is a parameter playing a central role in our most fundamental modelling of natural laws. Relativity theory shows that the comparison of times measured by different clocks depends on their relative motion and on the strength of the gravitational field in which [...] Read more.

Time is a parameter playing a central role in our most fundamental modelling of natural laws. Relativity theory shows that the comparison of times measured by different clocks depends on their relative motion and on the strength of the gravitational field in which they are embedded. In standard cosmology, the time parameter is the one measured by fundamental clocks (i.e., clocks at rest with respect to the expanding space). This proper time is assumed to flow at a constant rate throughout the whole history of the universe. We make the alternative hypothesis that the rate at which the cosmological time flows depends on the dynamical state of the universe. In thermodynamics, the arrow of time is strongly related to the second law, which states that the entropy of an isolated system will always increase with time or, at best, stay constant. Hence, we assume that the time measured by fundamental clocks is proportional to the entropy of the region of the universe that is causally connected to them. Under that simple assumption, we find it possible to build toy cosmological models that present an acceleration of their expansion without any need for dark energy while being spatially closed and finite, avoiding the need to deal with infinite values. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

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918 KiB

Open AccessArticle

On Interrelation of Time and Entropy

by Leonid M. Martyushev

Entropy 2017, 19(7), 345; https://doi.org/10.3390/e19070345 - 10 Jul 2017

Cited by 5 | Viewed by 3819

Abstract

A measure of time is related to the number of ways by which the human correlates the past and the future for some process. On this basis, a connection between time and entropy (information, Boltzmann–Gibbs, and thermodynamic one) is established. This measure gives [...] Read more.

A measure of time is related to the number of ways by which the human correlates the past and the future for some process. On this basis, a connection between time and entropy (information, Boltzmann–Gibbs, and thermodynamic one) is established. This measure gives time such properties as universality, relativity, directionality, and non-uniformity. A number of issues of the modern science related to the finding of laws describing changes in nature are discussed. A special emphasis is made on the role of evolutionary adaptation of an observer to the surrounding world. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

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233 KiB

Open AccessArticle

On Quantum Collapse as a Basis for the Second Law of Thermodynamics

by Ruth E. Kastner

Entropy 2017, 19(3), 106; https://doi.org/10.3390/e19030106 - 09 Mar 2017

Cited by 13 | Viewed by 6528

Abstract

It was first suggested by David Z. Albert that the existence of a real, physical non-unitary process (i.e., “collapse”) at the quantum level would yield a complete explanation for the Second Law of Thermodynamics (i.e., the increase in entropy over time). The contribution [...] Read more.

It was first suggested by David Z. Albert that the existence of a real, physical non-unitary process (i.e., “collapse”) at the quantum level would yield a complete explanation for the Second Law of Thermodynamics (i.e., the increase in entropy over time). The contribution of such a process would be to provide a physical basis for the ontological indeterminacy needed to derive the irreversible Second Law against a backdrop of otherwise reversible, deterministic physical laws. An alternative understanding of the source of this possible quantum “collapse” or non-unitarity is presented herein, in terms of the Transactional Interpretation (TI). The present model provides a specific physical justification for Boltzmann’s often-criticized assumption of molecular randomness (Stosszahlansatz), thereby changing its status from an ad hoc postulate to a theoretically grounded result, without requiring any change to the basic quantum theory. In addition, it is argued that TI provides an elegant way of reconciling, via indeterministic collapse, the time-reversible Liouville evolution with the time-irreversible evolution inherent in so-called “master equations” that specify the changes in occupation of the various possible states in terms of the transition rates between them. The present model is contrasted with the Ghirardi–Rimini–Weber (GRW) “spontaneous collapse” theory previously suggested for this purpose by Albert. Full article

(This article belongs to the Special Issue Entropy, Time and Evolution)

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