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789 KiB

Open AccessArticle

Some Trends in Quantum Thermodynamics

by Michael R. Von Spakovsky and Jochen Gemmer

Entropy 2014, 16(6), 3434-3470; https://doi.org/10.3390/e16063434 - 23 Jun 2014

Cited by 30 | Viewed by 7771

Abstract

Traditional answers to what the 2nd Law is are well known. Some are based on the microstate of a system wandering rapidly through all accessible phase space, while others are based on the idea of a system occupying an initial multitude of states [...] Read more.

Traditional answers to what the 2nd Law is are well known. Some are based on the microstate of a system wandering rapidly through all accessible phase space, while others are based on the idea of a system occupying an initial multitude of states due to the inevitable imperfections of measurements that then effectively, in a coarse grained manner, grow in time (mixing). What has emerged are two somewhat less traditional approaches from which it is said that the 2nd Law emerges, namely, that of the theory of quantum open systems and that of the theory of typicality. These are the two principal approaches, which form the basis of what today has come to be called quantum thermodynamics. However, their dynamics remains strictly linear and unitary, and, as a number of recent publications have emphasized, “testing the unitary propagation of pure states alone cannot rule out a nonlinear propagation of mixtures”. Thus, a non-traditional approach to capturing such a propagation would be one which complements the postulates of QM by the 2nd Law of thermodynamics, resulting in a possibly meaningful, nonlinear dynamics. An unorthodox approach, which does just that, is intrinsic quantum thermodynamics and its mathematical framework, steepest-entropy-ascent quantum thermodynamics. The latter has evolved into an effective tool for modeling the dynamics of reactive and non-reactive systems at atomistic scales. It is the usefulness of this framework in the context of quantum thermodynamics as well as the theory of typicality which are discussed here in some detail. A brief discussion of some other trends such as those related to work, work extraction, and fluctuation theorems is also presented. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

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238 KiB

Open AccessArticle

Gyarmati’s Variational Principle of Dissipative Processes

by József Verhás

Entropy 2014, 16(4), 2362-2383; https://doi.org/10.3390/e16042362 - 24 Apr 2014

Cited by 17 | Viewed by 6152

Abstract

Like in mechanics and electrodynamics, the fundamental laws of the thermodynamics of dissipative processes can be compressed into Gyarmati’s variational principle. This variational principle both in its differential (local) and in integral (global) forms was formulated by Gyarmati in 1965. The consistent application [...] Read more.

Like in mechanics and electrodynamics, the fundamental laws of the thermodynamics of dissipative processes can be compressed into Gyarmati’s variational principle. This variational principle both in its differential (local) and in integral (global) forms was formulated by Gyarmati in 1965. The consistent application of both the local and the global forms of Gyarmati’s principle provides all the advantages throughout explicating the theory of irreversible thermodynamics that are provided in the study of mechanics and electrodynamics by the corresponding classical variational principles, e.g., Gauss’ differential principle of least constraint or Hamilton’s integral principle. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

496 KiB

Open AccessArticle

Stochastic Dynamics of Proteins and the Action of Biological Molecular Machines

by Michal Kurzynski and Przemyslaw Chelminiak

Entropy 2014, 16(4), 1969-1982; https://doi.org/10.3390/e16041969 - 1 Apr 2014

Cited by 3 | Viewed by 6242

Abstract

It is now well established that most if not all enzymatic proteins display a slow stochastic dynamics of transitions between a variety of conformational substates composing their native state. A hypothesis is stated that the protein conformational transition networks, as just as higher-level [...] Read more.

It is now well established that most if not all enzymatic proteins display a slow stochastic dynamics of transitions between a variety of conformational substates composing their native state. A hypothesis is stated that the protein conformational transition networks, as just as higher-level biological networks, the protein interaction network, and the metabolic network, have evolved in the process of self-organized criticality. Here, the criticality means that all the three classes of networks are scale-free and, moreover, display a transition from the fractal organization on a small length-scale to the small-world organization on the large length-scale. Good mathematical models of such networks are stochastic critical branching trees extended by long-range shortcuts. Biological molecular machines are proteins that operate under isothermal conditions and hence are referred to as free energy transducers. They can be formally considered as enzymes that simultaneously catalyze two chemical reactions: the free energy-donating (input) reaction and the free energy-accepting (output) one. The far-from-equilibrium degree of coupling between the output and the input reaction fluxes have been studied both theoretically and by means of the Monte Carlo simulations on model networks. For single input and output gates the degree of coupling cannot exceed unity. Study simulations of random walks on model networks involving more extended gates indicate that the case of the degree of coupling value higher than one is realized on the mentioned above critical branching trees extended by long-range shortcuts. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

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297 KiB

Open AccessArticle

Information in Biological Systems and the Fluctuation Theorem

by Yaşar Demirel

Entropy 2014, 16(4), 1931-1948; https://doi.org/10.3390/e16041931 - 1 Apr 2014

Cited by 18 | Viewed by 8174

Abstract

Some critical trends in information theory, its role in living systems and utilization in fluctuation theory are discussed. The mutual information of thermodynamic coupling is incorporated into the generalized fluctuation theorem by using information theory and nonequilibrium thermodynamics. Thermodynamically coupled dissipative structures in [...] Read more.

Some critical trends in information theory, its role in living systems and utilization in fluctuation theory are discussed. The mutual information of thermodynamic coupling is incorporated into the generalized fluctuation theorem by using information theory and nonequilibrium thermodynamics. Thermodynamically coupled dissipative structures in living systems are capable of degrading more energy, and processing complex information through developmental and environmental constraints. The generalized fluctuation theorem can quantify the hysteresis observed in the amount of the irreversible work in nonequilibrium regimes in the presence of information and thermodynamic coupling. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

295 KiB

Open AccessArticle

Entropy Principle and Recent Results in Non-Equilibrium Theories

by Vito Antonio Cimmelli, David Jou, Tommaso Ruggeri and Péter Ván

Entropy 2014, 16(3), 1756-1807; https://doi.org/10.3390/e16031756 - 24 Mar 2014

Cited by 91 | Viewed by 10890

Abstract

We present the state of the art on the modern mathematical methods of exploiting the entropy principle in thermomechanics of continuous media. A survey of recent results and conceptual discussions of this topic in some well-known non-equilibrium theories (Classical irreversible thermodynamics CIT, Rational [...] Read more.

We present the state of the art on the modern mathematical methods of exploiting the entropy principle in thermomechanics of continuous media. A survey of recent results and conceptual discussions of this topic in some well-known non-equilibrium theories (Classical irreversible thermodynamics CIT, Rational thermodynamics RT, Thermodynamics of irreversible processes TIP, Extended irreversible thermodynamics EIT, Rational Extended thermodynamics RET) is also summarized. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

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355 KiB

Open AccessArticle

Contact Geometry of Mesoscopic Thermodynamics and Dynamics

by Miroslav Grmela

Entropy 2014, 16(3), 1652-1686; https://doi.org/10.3390/e16031652 - 21 Mar 2014

Cited by 88 | Viewed by 8428

Abstract

The time evolution during which macroscopic systems reach thermodynamic equilibrium states proceeds as a continuous sequence of contact structure preserving transformations maximizing the entropy. This viewpoint of mesoscopic thermodynamics and dynamics provides a unified setting for the classical equilibrium and nonequilibrium thermodynamics, kinetic [...] Read more.

The time evolution during which macroscopic systems reach thermodynamic equilibrium states proceeds as a continuous sequence of contact structure preserving transformations maximizing the entropy. This viewpoint of mesoscopic thermodynamics and dynamics provides a unified setting for the classical equilibrium and nonequilibrium thermodynamics, kinetic theory, and statistical mechanics. One of the illustrations presented in the paper is a new version of extended nonequilibrium thermodynamics with fluxes as extra state variables. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

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293 KiB

Open AccessArticle

Recent Progress in the Definition of Thermodynamic Entropy

by Enzo Zanchini and Gian Paolo Beretta

Entropy 2014, 16(3), 1547-1570; https://doi.org/10.3390/e16031547 - 19 Mar 2014

Cited by 27 | Viewed by 7611

Abstract

The principal methods for the definition of thermodynamic entropy are discussed with special reference to those developed by Carathéodory, the Keenan School, Lieb and Yngvason, and the present authors. An improvement of the latter method is then presented. Seven basic axioms are employed: [...] Read more.

The principal methods for the definition of thermodynamic entropy are discussed with special reference to those developed by Carathéodory, the Keenan School, Lieb and Yngvason, and the present authors. An improvement of the latter method is then presented. Seven basic axioms are employed: three Postulates, which are considered as having a quite general validity, and four Assumptions, which identify the domains of validity of the definitions of energy (Assumption 1) and entropy (Assumptions 2, 3, 4). The domain of validity of the present definition of entropy is not restricted to stable equilibrium states. For collections of simple systems, it coincides with that of the proof of existence and uniqueness of an entropy function which characterizes the relation of adiabatic accessibility proposed by Lieb and Yngvason. However, our treatment does not require the formation of scaled copies so that it applies not only to collections of simple systems, but also to systems contained in electric or magnetic fields and to small and few-particle systems. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

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496 KiB

Open AccessArticle

Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations

by Ramon F. Álvarez-Estrada

Entropy 2014, 16(3), 1426-1461; https://doi.org/10.3390/e16031426 - 11 Mar 2014

Cited by 3 | Viewed by 5103

Abstract

We treat the non-equilibrium evolution of an open one-particle statistical system, subject to a potential and to an external “heat bath” (hb) with negligible dissipation. For the classical equilibrium Boltzmann distribution, W_c,eq, a non-equilibrium three-term hierarchy for moments fulfills [...] Read more.

We treat the non-equilibrium evolution of an open one-particle statistical system, subject to a potential and to an external “heat bath” (hb) with negligible dissipation. For the classical equilibrium Boltzmann distribution, W_c,eq, a non-equilibrium three-term hierarchy for moments fulfills Hermiticity, which allows one to justify an approximate long-time thermalization. That gives partial dynamical support to Boltzmann’s Wc_,eq, out of the set of classical stationary distributions, Wc;st, also investigated here, for which neither Hermiticity nor that thermalization hold, in general. For closed classical many-particle systems without hb (by using W_c,eq), the long-time approximate thermalization for three-term hierarchies is justified and yields an approximate Lyapunov function and an arrow of time. The largest part of the work treats an open quantum one-particle system through the non-equilibrium Wigner function, W. Weq for a repulsive finite square well is reported. W’s (< 0 in various cases) are assumed to be quasi-definite functionals regarding their dependences on momentum (q). That yields orthogonal polynomials, H_Q,n(q), for Weq (and for stationary W_st), non-equilibrium moments, W_n, of W and hierarchies. For the first excited state of the harmonic oscillator, its stationary W_st is a quasi-definite functional, and the orthogonal polynomials and three-term hierarchy are studied. In general, the non-equilibrium quantum hierarchies (associated with Weq) for the W_n’s are not three-term ones. As an illustration, we outline a non-equilibrium four-term hierarchy and its solution in terms of generalized operator continued fractions. Such structures also allow one to formulate long-time approximations, but make it more difficult to justify thermalization. For large thermal and de Broglie wavelengths, the dominant W_eq and a non-equilibrium equation for W are reported: the non-equilibrium hierarchy could plausibly be a three-term one and possibly not far from Gaussian, and thermalization could possibly be justified. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

256 KiB

Open AccessArticle

A Characterization of Conserved Quantities in Non-Equilibrium Thermodynamics

by Ignacio Romero

Entropy 2013, 15(12), 5580-5596; https://doi.org/10.3390/e15125580 - 17 Dec 2013

Cited by 8 | Viewed by 5654

Abstract

The well-known Noether theorem in Lagrangian and Hamiltonian mechanics associates symmetries in the evolution equations of a mechanical system with conserved quantities. In this work, we extend this classical idea to problems of non-equilibrium thermodynamics formulated within the GENERIC (General Equations for Non-Equilibrium [...] Read more.

The well-known Noether theorem in Lagrangian and Hamiltonian mechanics associates symmetries in the evolution equations of a mechanical system with conserved quantities. In this work, we extend this classical idea to problems of non-equilibrium thermodynamics formulated within the GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling) framework. The geometric meaning of symmetry is reviewed in this formal setting and then utilized to identify possible conserved quantities and the conditions that guarantee their strict conservation. Examples are provided that demonstrate the validity of the proposed definition in the context of finite and infinite dimensional thermoelastic problems. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

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6991 KiB

Open AccessArticle

Symmetry-Based Balance Equation for Local Entropy Density in a Dissipative Multibaker Chain System

by Hideshi Ishida

Entropy 2013, 15(10), 4345-4375; https://doi.org/10.3390/e15104345 - 16 Oct 2013

Cited by 1 | Viewed by 4661

Abstract

In this study, the balance equation for local entropy density defined on each partition is obtained by the decomposition of the time-evolution operator for local entropy density, on the level of the master equation, by using symmetric and antisymmetric properties for the inversion [...] Read more.

In this study, the balance equation for local entropy density defined on each partition is obtained by the decomposition of the time-evolution operator for local entropy density, on the level of the master equation, by using symmetric and antisymmetric properties for the inversion of partition, density pairs and a given drift velocity. The resultant equation includes the following terms: convection, diffusion, entropy flow due to a thermostat and entropy production. The averaging of the four terms recover the corresponding terms in a balance equation for the macroscopic entropy density of irreversible thermodynamics for a thermostated system. Moreover, an empirical law of order estimation is introduced to explain the limiting behavior of the averaged quantities in the macroscopic limit for the bulk system. The law makes it possible to separate some minor contributions from the major four terms and, for example, to explain the positive entropy production rate in a nonequilibrium state for volume-preserving systems, even if the state is far from steady state. They are numerically confirmed on an invertible, dissipative multibaker chain system, named a circuit model. These properties are independent of partitioning. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

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Review

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580 KiB

Open AccessReview

On Equivalence of Nonequilibrium Thermodynamic and Statistical Entropies

by Purushottam D. Gujrati

Entropy 2015, 17(2), 710-754; https://doi.org/10.3390/e17020710 - 5 Feb 2015

Cited by 21 | Viewed by 6935

Abstract

We review the concept of nonequilibrium thermodynamic entropy and observables and internal variables as state variables, introduced recently by us, and provide a simple first principle derivation of additive statistical entropy, applicable to all nonequilibrium states by treating thermodynamics as an experimental science. [...] Read more.

We review the concept of nonequilibrium thermodynamic entropy and observables and internal variables as state variables, introduced recently by us, and provide a simple first principle derivation of additive statistical entropy, applicable to all nonequilibrium states by treating thermodynamics as an experimental science. We establish their numerical equivalence in several cases, which includes the most important case when the thermodynamic entropy is a state function. We discuss various interesting aspects of the two entropies and show that the number of microstates in the Boltzmann entropy includes all possible microstates of non-zero probabilities even if the system is trapped in a disjoint component of the microstate space. We show that negative thermodynamic entropy can appear from nonnegative statistical entropy. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

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1057 KiB

Open AccessReview

Panel I: Connecting 2nd Law Analysis with Economics, Ecology and Energy Policy

by Richard Gaggioli and Mauro Reini

Entropy 2014, 16(7), 3903-3938; https://doi.org/10.3390/e16073903 - 16 Jul 2014

Cited by 16 | Viewed by 6122

Abstract

The present paper is a review of several papers from the Proceedings of the Joint European Thermodynamics Conference, held in Brescia, Italy, 1–5 July 2013, namely papers introduced by their authors at Panel I of the conference. Panel I was devoted to applications [...] Read more.

The present paper is a review of several papers from the Proceedings of the Joint European Thermodynamics Conference, held in Brescia, Italy, 1–5 July 2013, namely papers introduced by their authors at Panel I of the conference. Panel I was devoted to applications of the Second Law of Thermodynamics to social issues—economics, ecology, sustainability, and energy policy. The concept called Available Energy which goes back to mid-nineteenth century work of Kelvin, Rankine, Maxwell and Gibbs, is relevant to all of the papers. Various names have been applied to the concept when interactions between the system of interest and an environment are involved. Today, the name exergy is generally accepted. The scope of the papers being reviewed is wide and they complement one another well. Full article

(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)

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