Special Issue "Entropy, Time and Evolution"

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Time".

Deadline for manuscript submissions: closed (30 June 2017).

Special Issue Editor

Prof. Dr. Leonid M. Martyushev
E-Mail Website
Guest Editor
1. Institute of Industrial Ecology, S. Kovalevskaya str. 20, 620219 Ekaterinburg, Russia;
2. Ural Federal University, Mira str.19, 620002 Ekaterinburg, Russia
Interests: fundamental problems of nature (irreversibility, asymmetry and scale of time, evolution, etc.); non-equilibrium thermodynamics; the second law of thermodynamics and entropy; maximum entropy production in physics, chemistry, and biology; growth processes in nature (experiment, theory, and simulation); morphological stability (crystal growth and fluid flow); pattern formation (dendrites, viscous fingers, etc.)
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Special Issue Information

Dear Colleagues,

We, and the world around us, constantly develop and evolve. The Universe and stars as well as societies and living beings pass through sequential stages from birth to maturity and death. This is a continuous process: one is replaced with the other. The one that has appeared repeats the old in some way, while being new in the other. We, and the world around us, develop directionally and irreversibly. From long ago, humanity has used time to describe this movement and, specifically, its directionality and duration. The greatest minds of the past were interested in this concept and studied it: St. Augustine, Newton, Kant, Bergson, Einstein, et al. However, time—one of the most complex and controversial concepts used by people—is still not fully defined and understood.

Entropy, another concept, appeared more than 150 years ago in thermodynamics and then penetrated and developed in other branches of science. This quantity is used to study the evolution of various objects by representatives of numerous sciences: physics, chemistry, biology, linguistics and economics, among others. There are a number of important statements formulated for this quantity in science. Most notably, these are the second law of thermodynamics and the principles of minimum and maximum entropy production. Entropy is considered to be a measure of irreversibility, directionality of a process, and it is similar to time in this respect. However, despite being difficult to introduce and measure for some systems, entropy is simpler than the concept of time.

The following questions arise in this regard: (1) Can time be understood and defined through entropy (or maybe vice versa)? (2) To what extent are these concepts related? (3) Can such a relation be used to understand the existing mysteries and regularities of the evolution of the surrounding world, and us therein? Authors of articles for this Special Issue are invited to answer these and related questions.

Prof. Dr. Leonid M. Martyushev
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Time in natural sciences
  • Temporal asymmetries
  • Entropy, entropy production
  • Second law of thermodynamics
  • Maximum and minimum entropy production
  • Evolution of the universe, star, planet system, climate, etc.
  • Evolution of ecological systems, biological objects, etc.

Published Papers (11 papers)

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Research

Article
A Novel Derivation of the Time Evolution of the Entropy for Macroscopic Systems in Thermal Non-Equilibrium
Entropy 2017, 19(11), 594; https://doi.org/10.3390/e19110594 - 07 Nov 2017
Cited by 5 | Viewed by 1830
Abstract
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 En-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 En-eq exponentially relaxes to its equilibrium value. Using the Gyftopoulos-Beretta formalism, a non-equilibrium entropy Sn-eq(x,t) is then derived from En-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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Article
Eigentimes and Very Slow Processes
Entropy 2017, 19(9), 492; https://doi.org/10.3390/e19090492 - 14 Sep 2017
Cited by 1 | Viewed by 2572
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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Article
Physical Universality, State-Dependent Dynamical Laws and Open-Ended Novelty
Entropy 2017, 19(9), 461; https://doi.org/10.3390/e19090461 - 01 Sep 2017
Cited by 5 | Viewed by 3460
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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Article
The Thermodynamical Arrow and the Historical Arrow; Are They Equivalent?
Entropy 2017, 19(9), 455; https://doi.org/10.3390/e19090455 - 30 Aug 2017
Cited by 1 | Viewed by 2075
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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Article
Stochastic Thermodynamics of Brownian Motion
Entropy 2017, 19(9), 434; https://doi.org/10.3390/e19090434 - 23 Aug 2017
Cited by 9 | Viewed by 3420
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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Article
Solutions to the Cosmic Initial Entropy Problem without Equilibrium Initial Conditions
Entropy 2017, 19(8), 411; https://doi.org/10.3390/e19080411 - 10 Aug 2017
Cited by 4 | Viewed by 2730
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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Article
Kinetic Theory beyond the Stosszahlansatz
Entropy 2017, 19(8), 381; https://doi.org/10.3390/e19080381 - 25 Jul 2017
Cited by 4 | Viewed by 3559
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)
Article
Nonequilibrium Entropy in a Shock
Entropy 2017, 19(7), 368; https://doi.org/10.3390/e19070368 - 19 Jul 2017
Cited by 12 | Viewed by 2070
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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Article
Cosmological Time, Entropy and Infinity
Entropy 2017, 19(7), 357; https://doi.org/10.3390/e19070357 - 14 Jul 2017
Viewed by 3022
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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Article
On Interrelation of Time and Entropy
Entropy 2017, 19(7), 345; https://doi.org/10.3390/e19070345 - 10 Jul 2017
Cited by 4 | Viewed by 2020
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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Article
On Quantum Collapse as a Basis for the Second Law of Thermodynamics
Entropy 2017, 19(3), 106; https://doi.org/10.3390/e19030106 - 09 Mar 2017
Cited by 9 | Viewed by 3493
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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