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# Special Issue "Residual Entropy and Nonequilibrium States"

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (31 December 2017).

## Special Issue Editor

Prof. Dr. Purushottam D. Gujrati
E-Mail Website
Guest Editor
Department of Physics, The University of Akron, Akron, OH 44325, USA
Interests: theoretical physics; classical nonequilibrium thermodynamics and statistical mechanics; stochastic physics; polymer physics
Special Issues and Collections in MDPI journals

## Special Issue Information

Dear Colleagues,

One of the basic axioms of classical thermodynamics is the Third Law. Even though absolute zero T=0 is unreachable, the law has played an important role in equilibrium thermodynamics. Away from equilibrium, it is no longer applicable and the limiting entropy at absolute zero, known as the residual entropy SR(T=0), is a measure of how far a system is out of equilibrium at T=0. Assuming equilibrium entropy Seq(T=0)=0, SR(T=0) is simply the nonequilibrium entropy at T=0. As long as entropy can be defined for any state, SR(T=0) or SR(T) at T>0 can also be identified. The systems we have in mind range from conventional thermodynamic systems usually studied in physics, chemistry, engineering, biology, etc., to black holes and systems characterized by quantum field theories (low-dimensional many-body systems), where entropy is usually called entanglement entropy. Several questions can be asked including the following. How does SR(T) depend on the process of preparing a state? How could it be measured or calculated? What can be said about SR and Seq at the same energy E? How does SR(T) relate to Seq(T)? What can be said about the energy E at T=0 for the equilibrium and nonequilibrium states? A question that has not been answered is: Can it be used as a measure of the “amount” of entanglement with the surroundings?

Aim: The hope of the Special Issue is to bring together contributors from a wide variety of fields to unravel the mystery of SR(T) to better prepare us to deal with nonequilibrium states in general.

Prof. Dr. Purushottam D. Gujrati
Guest Editor

Manuscript Submission Information

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## Keywords

• Entropy
• Entanglement Entropy
• Residual Entropy
• Metastability and Nonequilibrium States
• Classical and Quantum Systems, Entanglement

## Published Papers (10 papers)

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# Research

Open AccessArticle
Partition Function and Configurational Entropy in Non-Equilibrium States: A New Theoretical Model
Entropy 2018, 20(4), 218; https://doi.org/10.3390/e20040218 - 23 Mar 2018
Cited by 2 | Viewed by 2973
Abstract
A new model of non-equilibrium thermodynamic states has been investigated on the basis of the fact that all thermodynamic variables can be derived from partition functions. We have thus attempted to define partition functions for non-equilibrium conditions by introducing the concept of pseudo-temperature [...] Read more.
A new model of non-equilibrium thermodynamic states has been investigated on the basis of the fact that all thermodynamic variables can be derived from partition functions. We have thus attempted to define partition functions for non-equilibrium conditions by introducing the concept of pseudo-temperature distributions. These pseudo-temperatures are configurational in origin and distinct from kinetic (phonon) temperatures because they refer to the particular fragments of the system with specific energies. This definition allows thermodynamic states to be described either for equilibrium or non-equilibrium conditions. In addition; a new formulation of an extended canonical partition function; internal energy and entropy are derived from this new temperature definition. With this new model; computational experiments are performed on simple non-interacting systems to investigate cooling and two distinct relaxational effects in terms of the time profiles of the partition function; internal energy and configurational entropy. Full article
(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)
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Open AccessFeature PaperArticle
Hierarchy of Relaxation Times and Residual Entropy: A Nonequilibrium Approach
Entropy 2018, 20(3), 149; https://doi.org/10.3390/e20030149 - 26 Feb 2018
Cited by 10 | Viewed by 1640
Abstract
We consider nonequilibrium (NEQ) states such as supercooled liquids and glasses that are described with the use of internal variables. We classify the latter by the state-dependent hierarchy of relaxation times to assess their relevance for irreversible contributions. Given an observation time $τ$ [...] Read more.
We consider nonequilibrium (NEQ) states such as supercooled liquids and glasses that are described with the use of internal variables. We classify the latter by the state-dependent hierarchy of relaxation times to assess their relevance for irreversible contributions. Given an observation time $τ obs$ , we determine the window of relaxation times that divide the internal variables into active and inactive groups, the former playing a central role in the NEQ thermodynamics. Using this thermodynamics, we determine (i) a bound on the NEQ entropy and on the residual entropy and (ii) the nature of the isothermal relaxation of the entropy and the enthalpy in accordance with the second law. A theory that violates the second law such as the entropy loss view is shown to be internally inconsistent if we require it to be consistent with experiments. The inactive internal variables still play an indirect role in determining the temperature $T ( t )$ and the pressure $P ( t )$ of the system, which deviate from their external values. Full article
(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)
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Open AccessFeature PaperArticle
Attraction Controls the Entropy of Fluctuations in Isosceles Triangular Networks
Entropy 2018, 20(2), 122; https://doi.org/10.3390/e20020122 - 12 Feb 2018
Cited by 3 | Viewed by 1729
Abstract
We study two-dimensional triangular-network models, which have degenerate ground states composed of straight or randomly-zigzagging stripes and thus sub-extensive residual entropy. We show that attraction is responsible for the inversion of the stable phase by changing the entropy of fluctuations around the ground-state [...] Read more.
We study two-dimensional triangular-network models, which have degenerate ground states composed of straight or randomly-zigzagging stripes and thus sub-extensive residual entropy. We show that attraction is responsible for the inversion of the stable phase by changing the entropy of fluctuations around the ground-state configurations. By using a real-space shell-expansion method, we compute the exact expression of the entropy for harmonic interactions, while for repulsive harmonic interactions we obtain the entropy arising from a limited subset of the system by numerical integration. We compare these results with a three-dimensional triangular-network model, which shows the same attraction-mediated selection mechanism of the stable phase, and conclude that this effect is general with respect to the dimensionality of the system. Full article
(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)
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Open AccessFeature PaperArticle
Glass Transition, Crystallization of Glass-Forming Melts, and Entropy
Entropy 2018, 20(2), 103; https://doi.org/10.3390/e20020103 - 01 Feb 2018
Cited by 27 | Viewed by 2742
Abstract
A critical analysis of possible (including some newly proposed) definitions of the vitreous state and the glass transition is performed and an overview of kinetic criteria of vitrification is presented. On the basis of these results, recent controversial discussions on the possible values [...] Read more.
A critical analysis of possible (including some newly proposed) definitions of the vitreous state and the glass transition is performed and an overview of kinetic criteria of vitrification is presented. On the basis of these results, recent controversial discussions on the possible values of the residual entropy of glasses are reviewed. Our conclusion is that the treatment of vitrification as a process of continuously breaking ergodicity with entropy loss and a residual entropy tending to zero in the limit of zero absolute temperature is in disagreement with the absolute majority of experimental and theoretical investigations of this process and the nature of the vitreous state. This conclusion is illustrated by model computations. In addition to the main conclusion derived from these computations, they are employed as a test for several suggestions concerning the behavior of thermodynamic coefficients in the glass transition range. Further, a brief review is given on possible ways of resolving the Kauzmann paradox and its implications with respect to the validity of the third law of thermodynamics. It is shown that neither in its primary formulations nor in its consequences does the Kauzmann paradox result in contradictions with any basic laws of nature. Such contradictions are excluded by either crystallization (not associated with a pseudospinodal as suggested by Kauzmann) or a conventional (and not an ideal) glass transition. Some further so far widely unexplored directions of research on the interplay between crystallization and glass transition are anticipated, in which entropy may play—beyond the topics widely discussed and reviewed here—a major role. Full article
(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)
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Open AccessArticle
Residual Entropy and Critical Behavior of Two Interacting Boson Species in a Double Well
Entropy 2018, 20(2), 84; https://doi.org/10.3390/e20020084 - 25 Jan 2018
Cited by 10 | Viewed by 1546
Abstract
Motivated by the importance of entanglement and correlation indicators in the analysis of quantum systems, we study the equilibrium and the bipartite residual entropy in a two-species Bose–Hubbard dimer when the spatial phase separation of the two species takes place. We consider both [...] Read more.
Motivated by the importance of entanglement and correlation indicators in the analysis of quantum systems, we study the equilibrium and the bipartite residual entropy in a two-species Bose–Hubbard dimer when the spatial phase separation of the two species takes place. We consider both the zero and non-zero-temperature regime. We present different kinds of residual entropies (each one associated with a different way of partitioning the system), and we show that they strictly depend on the specific quantum phase characterizing the two species (supermixed, mixed or demixed) even at finite temperature. To provide a deeper physical insight into the zero-temperature scenario, we apply the fully-analytical variational approach based on su(2) coherent states and provide a considerably good approximation of the entanglement entropy. Finally, we show that the effectiveness of bipartite residual entropy as a critical indicator at non-zero temperature is unchanged when considering a restricted combination of energy eigenstates. Full article
(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)
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Open AccessArticle
Entropy of Entanglement between Quantum Phases of a Three-Level Matter-Radiation Interaction Model
Entropy 2018, 20(2), 72; https://doi.org/10.3390/e20020072 - 24 Jan 2018
Cited by 1 | Viewed by 1788
Abstract
We show that the entropy of entanglement is sensitive to the coherent quantum phase transition between normal and super-radiant regions of a system of a finite number of three-level atoms interacting in a dipolar approximation with a one-mode electromagnetic field. The atoms are [...] Read more.
We show that the entropy of entanglement is sensitive to the coherent quantum phase transition between normal and super-radiant regions of a system of a finite number of three-level atoms interacting in a dipolar approximation with a one-mode electromagnetic field. The atoms are treated as semi-distinguishable using different cooperation numbers and representations of SU(3), variables which are relevant to the sensitivity of the entropy with the transition. The results are computed for all three possible configurations ( $Ξ$ , $Λ$ and V) of the three-level atoms. Full article
(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)
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Open AccessArticle
Entanglement Entropy of the Spin-1 Condensates at Zero Temperature
Entropy 2018, 20(1), 80; https://doi.org/10.3390/e20010080 - 22 Jan 2018
Cited by 1 | Viewed by 1531
Abstract
For spin-1 condensates, the spatial degrees of freedom can be considered as being frozen at temperature zero, while the spin-degrees of freedom remain free. Under this condition, the entanglement entropy has been derived exactly with an analytical form. The entanglement entropy is found [...] Read more.
For spin-1 condensates, the spatial degrees of freedom can be considered as being frozen at temperature zero, while the spin-degrees of freedom remain free. Under this condition, the entanglement entropy has been derived exactly with an analytical form. The entanglement entropy is found to decrease monotonically with the increase of the magnetic polarization as expected. However, for the ground state in polar phase, an extremely steep fall of the entropy is found when the polarization emerges from zero. Then the fall becomes a gentle descent after the polarization exceeds a turning point. Full article
(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)
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# Other

Reply to “Comment on ‘Glass Transition, Crystallization of Glass-Forming Melts, and Entropy”’ by Zanotto and Mauro
Entropy 2018, 20(9), 704; https://doi.org/10.3390/e20090704 - 13 Sep 2018
Cited by 3 | Viewed by 1263
Abstract
A response is given to a comment of Zanotto and Mauro on our paper published in Entropy 20, 103 (2018). Our arguments presented in this paper are widely ignored by them, and no new considerations are outlined in the comment, which would [...] Read more.
A response is given to a comment of Zanotto and Mauro on our paper published in Entropy 20, 103 (2018). Our arguments presented in this paper are widely ignored by them, and no new considerations are outlined in the comment, which would require a revision of our conclusions. For this reason, we restrict ourselves here to a brief response, supplementing it by some additional arguments in favor of our point of view not included in our above-cited paper. Full article
(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)
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Open AccessComment
Comment on “Glass Transition, Crystallization of Glass-Forming Melts, and Entropy” Entropy 2018, 20, 103.
Entropy 2018, 20(9), 703; https://doi.org/10.3390/e20090703 - 13 Sep 2018
Cited by 3 | Viewed by 1264
Abstract
In a recent article, Schmelzer and Tropin [Entropy 2018, 20, 103] presented a critique of several aspects of modern glass science, including various features of glass transition and relaxation, crystallization, and the definition of glass itself. We argue that these [...] Read more.
In a recent article, Schmelzer and Tropin [Entropy 2018, 20, 103] presented a critique of several aspects of modern glass science, including various features of glass transition and relaxation, crystallization, and the definition of glass itself. We argue that these criticisms are at odds with well-accepted knowledge in the field from both theory and experiments. The objective of this short comment is to clarify several of these issues. Full article
(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)
Open AccessDiscussion
On the Possibility of Calculating Entropy, Free Energy, and Enthalpy of Vitreous Substances
Entropy 2018, 20(3), 187; https://doi.org/10.3390/e20030187 - 11 Mar 2018
Cited by 6 | Viewed by 2236
Abstract
A critical analysis for the arguments in support of, and against, the traditional approach to thermodynamics of vitreous state is provided. In this approach one presumes that there is a continuous variation of the entropy in the glass-liquid transition temperature range, or a [...] Read more.
A critical analysis for the arguments in support of, and against, the traditional approach to thermodynamics of vitreous state is provided. In this approach one presumes that there is a continuous variation of the entropy in the glass-liquid transition temperature range, or a “continuous entropy approach” towards 0 K which produces a positive value of the entropy at T → 0 K. I find that arguments given against this traditional approach use a different understanding of the thermodynamics of glass transition on cooling a liquid, because it suggests a discontinuity or “entropy loss approach” in the variation of entropy in the glass-liquid transition range. That is based on: (1) an unjustifiable use of the classical Boltzmann statistics for interpreting the value of entropy at absolute zero; (2) the rejection of thermodynamic analysis of systems with broken ergodicity, even though the possibility of analogous analysis was proposed already by Gibbs; (3) the possibility of a finite change in entropy of a system without absorption or release of heat; and, (4) describing the thermodynamic properties of glasses in the framework of functions, instead of functionals. The last one is necessary because for glasses the entropy and enthalpy are not functions of the state, but functionals, as defined by Gibbs’ in his classification. Full article
(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)
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