Entropy

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22 pages, 560 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On Quantum Entropy

by Davi Geiger and Zvi M. Kedem

Entropy 2022, 24(10), 1341; https://doi.org/10.3390/e24101341 - 23 Sep 2022

Cited by 4 | Viewed by 5024

Abstract

Quantum physics, despite its intrinsically probabilistic nature, lacks a definition of entropy fully accounting for the randomness of a quantum state. For example, von Neumann entropy quantifies only the incomplete specification of a quantum state and does not quantify the probabilistic distribution of [...] Read more.

Quantum physics, despite its intrinsically probabilistic nature, lacks a definition of entropy fully accounting for the randomness of a quantum state. For example, von Neumann entropy quantifies only the incomplete specification of a quantum state and does not quantify the probabilistic distribution of its observables; it trivially vanishes for pure quantum states. We propose a quantum entropy that quantifies the randomness of a pure quantum state via a conjugate pair of observables/operators forming the quantum phase space. The entropy is dimensionless, it is a relativistic scalar, it is invariant under canonical transformations and under CPT transformations, and its minimum has been established by the entropic uncertainty principle. We expand the entropy to also include mixed states. We show that the entropy is monotonically increasing during a time evolution of coherent states under a Dirac Hamiltonian. However, in a mathematical scenario, when two fermions come closer to each other, each evolving as a coherent state, the total system’s entropy oscillates due to the increasing spatial entanglement. We hypothesize an entropy law governing physical systems whereby the entropy of a closed system never decreases, implying a time arrow for particle physics. We then explore the possibility that as the oscillations of the entropy must by the law be barred in quantum physics, potential entropy oscillations trigger annihilation and creation of particles. Full article

(This article belongs to the Special Issue Nature of Entropy and Its Direct Metrology)

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18 pages, 634 KiB

Open AccessArticle

Spin Entropy

by Davi Geiger and Zvi M. Kedem

Entropy 2022, 24(9), 1292; https://doi.org/10.3390/e24091292 - 14 Sep 2022

Cited by 4 | Viewed by 2864

Abstract

Two types of randomness are associated with a mixed quantum state: the uncertainty in the probability coefficients of the constituent pure states and the uncertainty in the value of each observable captured by the Born’s rule probabilities. Entropy is a quantification of randomness, [...] Read more.

Two types of randomness are associated with a mixed quantum state: the uncertainty in the probability coefficients of the constituent pure states and the uncertainty in the value of each observable captured by the Born’s rule probabilities. Entropy is a quantification of randomness, and we propose a spin-entropy for the observables of spin pure states based on the phase space of a spin as described by the geometric quantization method, and we also expand it to mixed quantum states. This proposed entropy overcomes the limitations of previously-proposed entropies such as von Neumann entropy which only quantifies the randomness of specifying the quantum state. As an example of a limitation, previously-proposed entropies are higher for Bell entangled spin states than for disentangled spin states, even though the spin observables are less constrained for a disentangled pair of spins than for an entangled pair. The proposed spin-entropy accurately quantifies the randomness of a quantum state, it never reaches zero value, and it is lower for entangled states than for disentangled states. Full article

(This article belongs to the Special Issue Nature of Entropy and Its Direct Metrology)

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44 pages, 2198 KiB

Open AccessEditor’s ChoiceArticle

Entropy and the Experience of Heat

by Hans U. Fuchs, Michele D’Anna and Federico Corni

Entropy 2022, 24(5), 646; https://doi.org/10.3390/e24050646 - 4 May 2022

Cited by 6 | Viewed by 2584

Abstract

We discuss how to construct a direct and experientially natural path to entropy as a extensive quantity of a macroscopic theory of thermal systems and processes. The scientific aspects of this approach are based upon continuum thermodynamics. We ask what the roots of [...] Read more.

We discuss how to construct a direct and experientially natural path to entropy as a extensive quantity of a macroscopic theory of thermal systems and processes. The scientific aspects of this approach are based upon continuum thermodynamics. We ask what the roots of an experientially natural approach might be—to this end we investigate and describe in some detail (a) how humans experience and conceptualize an extensive thermal quantity (i.e., an amount of heat), and (b) how this concept evolved during the early development of the science of thermal phenomena (beginning with the Experimenters of the Accademia del Cimento and ending with Sadi Carnot). We show that a direct approach to entropy, as the extensive quantity of models of thermal systems and processes, is possible and how it can be applied to the teaching of thermodynamics for various audiences. Full article

(This article belongs to the Special Issue Nature of Entropy and Its Direct Metrology)

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12 pages, 894 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

A Robust Protocol for Entropy Measurement in Mesoscopic Circuits

by Timothy Child, Owen Sheekey, Silvia Lüscher, Saeed Fallahi, Geoffrey C. Gardner, Michael Manfra and Joshua Folk

Entropy 2022, 24(3), 417; https://doi.org/10.3390/e24030417 - 17 Mar 2022

Cited by 13 | Viewed by 3238

Abstract

Previous measurements utilizing Maxwell relations to measure change in entropy, S, demonstrated remarkable accuracy in measuring the spin-1/2 entropy of electrons in a weakly coupled quantum dot. However, these previous measurements relied upon prior knowledge of the charge transition lineshape. This had [...] Read more.

Previous measurements utilizing Maxwell relations to measure change in entropy, S, demonstrated remarkable accuracy in measuring the spin-1/2 entropy of electrons in a weakly coupled quantum dot. However, these previous measurements relied upon prior knowledge of the charge transition lineshape. This had the benefit of making the quantitative determination of entropy independent of scale factors in the measurement itself but at the cost of limiting the applicability of the approach to simple systems. To measure the entropy of more exotic mesoscopic systems, a more flexible analysis technique may be employed; however, doing so requires a precise calibration of the measurement. Here, we give details on the necessary improvements made to the original experimental approach and highlight some of the common challenges (along with strategies to overcome them) that other groups may face when attempting this type of measurement. Full article

(This article belongs to the Special Issue Nature of Entropy and Its Direct Metrology)

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27 pages, 883 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Coupled Transport Effects in Solid Oxide Fuel Cell Modeling

by Aydan Gedik, Nico Lubos and Stephan Kabelac

Entropy 2022, 24(2), 224; https://doi.org/10.3390/e24020224 - 31 Jan 2022

Cited by 6 | Viewed by 3180

Abstract

With its outstanding performance characteristics, the SOFC represents a promising technology for integration into the current energy supply system. For cell development and optimization, a reliable quantitative description of the transport mechanisms and the resulting losses are relevant. The local transport processes are [...] Read more.

With its outstanding performance characteristics, the SOFC represents a promising technology for integration into the current energy supply system. For cell development and optimization, a reliable quantitative description of the transport mechanisms and the resulting losses are relevant. The local transport processes are calculated by a 1D model based on the non-equilibrium thermodynamics (NET). The focus of this study is the mass transport in the gas diffusion layers (GDL), which was described as simplified by Fick’s law in a previously developed model. This is first replaced by the Dusty-Gas model (DGM) and then by the thermal diffusion (Soret effect) approach. The validation of the model was performed by measuring

U, j

-characteristics resulting in a maximum deviation of experimental to simulated cell voltage to up to 0.93%. It is shown that, under the prevailing temperature, gradients the Soret effect can be neglected, but the extension to the DGM has to be considered. The temperature and heat flow curves illustrate the relevance of the Peltier effects. At

T = 1123.15

K and

j = 8000

A/m², 64.44% of the total losses occur in the electrolyte. The exergetic efficiency for this operating point is 0.42. Since lower entropy production rates can be assumed in the GDL, the primary need is to investigate alternative electrolyte materials. Full article

(This article belongs to the Special Issue Nature of Entropy and Its Direct Metrology)

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17 pages, 763 KiB

Open AccessArticle

Which Physical Quantity Deserves the Name “Quantity of Heat”?

by Friedrich Herrmann and Michael Pohlig

Entropy 2021, 23(8), 1078; https://doi.org/10.3390/e23081078 - 19 Aug 2021

Cited by 8 | Viewed by 3213

Abstract

“What is heat?” was the title of a 1954 article by Freeman J. Dyson, published in Scientific American. Apparently, it was appropriate to ask this question at that time. The answer is given in the very first sentence of the article: heat is [...] Read more.

“What is heat?” was the title of a 1954 article by Freeman J. Dyson, published in Scientific American. Apparently, it was appropriate to ask this question at that time. The answer is given in the very first sentence of the article: heat is disordered energy. We will ask the same question again, but with a different expectation for its answer. Let us imagine that all the thermodynamic knowledge is already available: both the theory of phenomenological thermodynamics and that of statistical thermodynamics, including quantum statistics, but that the term “heat” has not yet been attributed to any of the variables of the theory. With the question “What is heat?” we now mean: which of the physical quantities deserves this name? There are several candidates: the quantities

Q

,

H

,

E_{therm}

and

S

. We can then formulate a desideratum, or a profile: What properties should such a measure of the quantity or amount of heat ideally have? Then, we evaluate all the candidates for their suitability. It turns out that the winner is the quantity

S

, which we know by the name of entropy. In the second part of the paper, we examine why entropy has not succeeded in establishing itself as a measure for the amount of heat, and we show that there is a real chance today to make up for what was missed. Full article

(This article belongs to the Special Issue Nature of Entropy and Its Direct Metrology)

14 pages, 2907 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

A Thermodynamic Approach to Measuring Entropy in a Few-Electron Nanodevice

by Eugenia Pyurbeeva and Jan A. Mol

Entropy 2021, 23(6), 640; https://doi.org/10.3390/e23060640 - 21 May 2021

Cited by 12 | Viewed by 3548

Abstract

The entropy of a system gives a powerful insight into its microscopic degrees of freedom; however, standard experimental ways of measuring entropy through heat capacity are hard to apply to nanoscale systems, as they require the measurement of increasingly small amounts of heat. [...] Read more.

The entropy of a system gives a powerful insight into its microscopic degrees of freedom; however, standard experimental ways of measuring entropy through heat capacity are hard to apply to nanoscale systems, as they require the measurement of increasingly small amounts of heat. Two alternative entropy measurement methods have been recently proposed for nanodevices: through charge balance measurements and transport properties. We describe a self-consistent thermodynamic framework for applying thermodynamic relations to few-electron nanodevices—small systems, where fluctuations in particle number are significant, whilst highlighting several ongoing misconceptions. We derive a relation (a consequence of a Maxwell relation for small systems), which describes both existing entropy measurement methods as special cases, while also allowing the experimentalist to probe the intermediate regime between them. Finally, we independently prove the applicability of our framework in systems with complex microscopic dynamics—those with many excited states of various degeneracies—from microscopic considerations. Full article

(This article belongs to the Special Issue Nature of Entropy and Its Direct Metrology)

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Review

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21 pages, 2646 KiB

Open AccessReview

On the Thermal Capacity of Solids

by Armin Feldhoff

Entropy 2022, 24(4), 479; https://doi.org/10.3390/e24040479 - 29 Mar 2022

Cited by 4 | Viewed by 3206

Abstract

The term thermal capacity appears to suggest a storable thermal quantity. However, this claim is not redeemed when thermal capacity is projected onto “heat”, which, like all energy forms, exits only in transit and is not a part of internal energy. The storable [...] Read more.

The term thermal capacity appears to suggest a storable thermal quantity. However, this claim is not redeemed when thermal capacity is projected onto “heat”, which, like all energy forms, exits only in transit and is not a part of internal energy. The storable thermal quantity is entropy, and entropy capacity is a well-defined physical coefficient which has the advantage of being a susceptibility. The inverse of the entropy capacity relates the response of the system (change of temperature) to a stimulus (change of entropy) such as the fluid level responses to a change in amount of fluid contained in a vessel. Frequently, entropy capacity has been used implicitly, which is clarified in examples of the low-temperature analysis of phononic and electronic contributions to the thermal capacity of solids. Generally, entropy capacity is used in the estimation of the entropy of a solid. Implicitly, the thermoelectric figure of merit refers to entropy capacity. The advantage of the explicit use of entropy capacity comes with a descriptive fundamental understanding of the thermal behaviour of solids, which is made clear by the examples of the Debye model of phonons in solids, the latest thermochemical modelling of carbon allotropes (diamond and graphite) and not least caloric materials. An electrocaloric cycle of barium titanate close to its paraelectric–ferroelectric phase transition is analysed by means of entropy capacity. Entropy capacity is a key to intuitively understanding thermal processes. Full article

(This article belongs to the Special Issue Nature of Entropy and Its Direct Metrology)

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32 pages, 1035 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Quantum Transport of Particles and Entropy

by Christoph Strunk

Entropy 2021, 23(12), 1573; https://doi.org/10.3390/e23121573 - 25 Nov 2021

Cited by 5 | Viewed by 2714

Abstract

A unified view on macroscopic thermodynamics and quantum transport is presented. Thermodynamic processes with an exchange of energy between two systems necessarily involve the flow of other balancable quantities. These flows are first analyzed using a simple drift-diffusion model, which includes the thermoelectric [...] Read more.

A unified view on macroscopic thermodynamics and quantum transport is presented. Thermodynamic processes with an exchange of energy between two systems necessarily involve the flow of other balancable quantities. These flows are first analyzed using a simple drift-diffusion model, which includes the thermoelectric effects, and connects the various transport coefficients to certain thermodynamic susceptibilities and a diffusion coefficient. In the second part of the paper, the connection between macroscopic thermodynamics and quantum statistics is discussed. It is proposed to employ not particles, but elementary Fermi- or Bose-systems as the elementary building blocks of ideal quantum gases. In this way, the transport not only of particles but also of entropy can be derived in a concise way, and is illustrated both for ballistic quantum wires, and for diffusive conductors. In particular, the quantum interference of entropy flow is in close correspondence to that of electric current. Full article

(This article belongs to the Special Issue Nature of Entropy and Its Direct Metrology)

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