Research

19 pages, 448 KiB

Open AccessFeature PaperArticle

Geometrical Aspects in the Analysis of Microcanonical Phase-Transitions

by Ghofrane Bel-Hadj-Aissa, Matteo Gori, Vittorio Penna, Giulio Pettini and Roberto Franzosi

Entropy 2020, 22(4), 380; https://doi.org/10.3390/e22040380 - 26 Mar 2020

Cited by 7 | Viewed by 2522

In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian [...] Read more.

In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviours of thermodynamic observables at a phase transition point are rooted in more fundamental changes of the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of phase-transitions are shaped in the special case of

ϕ^{4}

models with either nearest-neighbours and mean-field interactions. Full article

(This article belongs to the Special Issue The Ubiquity of Entropy)

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15 pages, 820 KiB

Open AccessArticle

Möbius Transforms, Cycles and q-triplets in Statistical Mechanics

by Jean Pierre Gazeau and Constantino Tsallis

Entropy 2019, 21(12), 1155; https://doi.org/10.3390/e21121155 - 26 Nov 2019

Cited by 10 | Viewed by 2127

Abstract

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analysed sequences of q-triplets, or q-doublets if one of them was the unity, in terms of cycles of successive Möbius transforms of the line preserving [...] Read more.

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analysed sequences of q-triplets, or q-doublets if one of them was the unity, in terms of cycles of successive Möbius transforms of the line preserving unity (

q = 1

corresponds to the BG theory). Such transforms have the form

q \mapsto (a q + 1 - a) / [(1 + a) q - a]

, where a is a real number; the particular cases

a = - 1

and

a = 0

yield, respectively,

q \mapsto (2 - q)

and

q \mapsto 1 / q

, currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties. Full article

(This article belongs to the Special Issue The Ubiquity of Entropy)

11 pages, 840 KiB

Open AccessArticle

Reduced Data Sets and Entropy-Based Discretization

by Jerzy W. Grzymala-Busse, Zdzislaw S. Hippe and Teresa Mroczek

Entropy 2019, 21(11), 1051; https://doi.org/10.3390/e21111051 - 28 Oct 2019

Cited by 1 | Viewed by 2320

Abstract

Results of experiments on numerical data sets discretized using two methods—global versions of Equal Frequency per Interval and Equal Interval Width-are presented. Globalization of both methods is based on entropy. For discretized data sets left and right reducts were computed. For each discretized [...] Read more.

Results of experiments on numerical data sets discretized using two methods—global versions of Equal Frequency per Interval and Equal Interval Width-are presented. Globalization of both methods is based on entropy. For discretized data sets left and right reducts were computed. For each discretized data set and two data sets, based, respectively, on left and right reducts, we applied ten-fold cross validation using the C4.5 decision tree generation system. Our main objective was to compare the quality of all three types of data sets in terms of an error rate. Additionally, we compared complexity of generated decision trees. We show that reduction of data sets may only increase the error rate and that the decision trees generated from reduced decision sets are not simpler than the decision trees generated from non-reduced data sets. Full article

(This article belongs to the Special Issue The Ubiquity of Entropy)

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18 pages, 792 KiB

Open AccessArticle

Dynamical Transitions in a One-Dimensional Katz–Lebowitz–Spohn Model

by Alessandro Pelizzola, Marco Pretti and Francesco Puccioni

Entropy 2019, 21(11), 1028; https://doi.org/10.3390/e21111028 - 23 Oct 2019

Cited by 4 | Viewed by 2269

Abstract

Dynamical transitions, already found in the high- and low-density phases of the Totally Asymmetric Simple Exclusion Process and a couple of its generalizations, are singularities in the rate of relaxation towards the Non-Equilibrium Stationary State (NESS), which do not correspond to any transition [...] Read more.

Dynamical transitions, already found in the high- and low-density phases of the Totally Asymmetric Simple Exclusion Process and a couple of its generalizations, are singularities in the rate of relaxation towards the Non-Equilibrium Stationary State (NESS), which do not correspond to any transition in the NESS itself. We investigate dynamical transitions in the one-dimensional Katz–Lebowitz–Spohn model, a further generalization of the Totally Asymmetric Simple Exclusion Process where the hopping rate depends on the occupation state of the 2 nodes adjacent to the nodes affected by the hop. Following previous work, we choose Glauber rates and bulk-adapted boundary conditions. In particular, we consider a value of the repulsion which parameterizes the Glauber rates such that the fundamental diagram of the model exhibits 2 maxima and a minimum, and the NESS phase diagram is especially rich. We provide evidence, based on pair approximation, domain wall theory and exact finite size results, that dynamical transitions also occur in the one-dimensional Katz–Lebowitz–Spohn model, and discuss 2 new phenomena which are peculiar to this model. Full article

(This article belongs to the Special Issue The Ubiquity of Entropy)

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21 pages, 1016 KiB

Open AccessArticle

Entropy Distribution in a Quantum Informational Circuit of Tunable Szilard Engines

by Jose Diazdelacruz

Entropy 2019, 21(10), 980; https://doi.org/10.3390/e21100980 - 08 Oct 2019

Cited by 1 | Viewed by 2144

Abstract

This paper explores the possibility of extending the existing model of a single-particle Quantum Szilard Engine to take advantage of some features of quantum information for driving typical mechanical systems. It focuses on devices that output mechanical work, extracting energy from a single [...] Read more.

This paper explores the possibility of extending the existing model of a single-particle Quantum Szilard Engine to take advantage of some features of quantum information for driving typical mechanical systems. It focuses on devices that output mechanical work, extracting energy from a single thermal reservoir at the cost of increasing the entropy of a qubit; the reverse process is also considered. In this alternative, several engines may share the information carried by the same qubit, although its interception will prove completely worthless for any illegitimate user. To this end, multi-partite quantum entanglement is employed. Besides, some changes in the cycle of the standard single-particle Quantum Szilard Engine are described, which lend more flexibility to meeting additional requirements in typical mechanical systems. The modifications allow having qubit input and output states of adjustable entropy. This feature enables the possibility of chaining the qubit between engines so that its output state from one can be used as an input state for another. Finally, another tweak is presented that allows for tuning the average output force of the engine. Full article

(This article belongs to the Special Issue The Ubiquity of Entropy)

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Graphical abstract

9 pages, 804 KiB

Open AccessArticle

Frequency Dependence of the Entanglement Entropy Production in a System of Coupled Driven Nonlinear Oscillators

by Shi-Hui Zhang and Zhan-Yuan Yan

Entropy 2019, 21(9), 889; https://doi.org/10.3390/e21090889 - 13 Sep 2019

Cited by 2 | Viewed by 2261

Abstract

Driven nonlinear systems have attracted great interest owing to their applications in quantum technologies such as quantum information. In quantum information, entanglement is a vital resource and can be measured by entropy in bipartite systems. In this paper, we carry out an investigation [...] Read more.

Driven nonlinear systems have attracted great interest owing to their applications in quantum technologies such as quantum information. In quantum information, entanglement is a vital resource and can be measured by entropy in bipartite systems. In this paper, we carry out an investigation to study the impact of driving frequency on the entanglement with a bipartite system of two coupled driven nonlinear oscillators. It is numerically found that the time evolution of the entanglement entropy between the subsystems significantly depends on the driving frequency. The dependence curve of the entropy production on the driving frequency exhibits a pronounced peak. This means the entanglement between the subsystems can be greatly increased by tuning the driving frequency. Further analyses show that the enhancement of the entropy production by the driving frequency is closely related to the energy levels involved in the quantum evolution. This is confirmed by the results related to the quantum spectrum and the dispersion of the wave function in the phase space. Our work gives a convenient way to enhance the entanglement in driven nonlinear systems and throws light on the role of driven nonlinear systems in quantum information technologies. Full article

(This article belongs to the Special Issue The Ubiquity of Entropy)

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9 pages, 263 KiB

Open AccessArticle

The Subtle Unphysical Hypothesis of the Firewall Theorem

by Carlo Rovelli

Entropy 2019, 21(9), 839; https://doi.org/10.3390/e21090839 - 27 Aug 2019

Cited by 13 | Viewed by 2996

Abstract

The black-hole firewall theorem derives a suspicious consequence (large energy-momentum density at the horizon of a black hole) from a set of seemingly reasonable hypotheses. I point out the hypothesis that is likely to be unrealistic (a hypothesis not always sufficiently made explicit) [...] Read more.

The black-hole firewall theorem derives a suspicious consequence (large energy-momentum density at the horizon of a black hole) from a set of seemingly reasonable hypotheses. I point out the hypothesis that is likely to be unrealistic (a hypothesis not always sufficiently made explicit) and discuss the subtle confusion at its origin: mixing-up of two different notions of entropy and misusing the entropy bound. Full article

(This article belongs to the Special Issue The Ubiquity of Entropy)

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14 pages, 8753 KiB

Open AccessArticle

On Shock Propagation through Double-Bend Ducts by Entropy-Generation-Based Artificial Viscosity Method

by Arnab Chaudhuri

Entropy 2019, 21(9), 837; https://doi.org/10.3390/e21090837 - 26 Aug 2019

Cited by 1 | Viewed by 2771

Abstract

Shock-wave propagation through obstacles or internal ducts involves complex shock dynamics, shock-wave shear layer interactions and shock-wave boundary layer interactions arising from the associated diffraction phenomenon. This work addresses the applicability and effectiveness of the high-order numerical scheme for such complex viscous compressible [...] Read more.

Shock-wave propagation through obstacles or internal ducts involves complex shock dynamics, shock-wave shear layer interactions and shock-wave boundary layer interactions arising from the associated diffraction phenomenon. This work addresses the applicability and effectiveness of the high-order numerical scheme for such complex viscous compressible flows. An explicit Discontinuous Spectral Element Method (DSEM) equipped with entropy-generation-based artificial viscosity method was used to solve compressible Navier–Stokes system of equations for this purpose. The shock-dynamics and viscous interactions associated with a planar moving shock-wave through a double-bend duct were resolved by two-dimensional numerical simulations. The shock-wave diffraction patterns, the large-scale structures of the shock-wave-turbulence interactions, agree very well with previous experimental findings. For shock-wave Mach number

M_{s} = 1.3466

and reference Reynolds number

{Re}_{f} = 10^{6}

, the predicted pressure signal at the exit section of the duct is in accordance with the literature. The attenuation in terms of overpressure for

M_{s} = 1.53

is found to be ≈0.51. Furthermore, the effect of reference Reynolds number is studied to address the importance of viscous interactions. The shock-shear layer and shock-boundary layer dynamics strongly depend on the

{Re}_{f}

while the principal shock-wave patterns are generally independent of

{Re}_{f}

. Full article

(This article belongs to the Special Issue The Ubiquity of Entropy)

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14 pages, 330 KiB

Open AccessFeature PaperArticle

Canonical Divergence for Flat α-Connections: Classical and Quantum

by Domenico Felice and Nihat Ay

Entropy 2019, 21(9), 831; https://doi.org/10.3390/e21090831 - 25 Aug 2019

Cited by 1 | Viewed by 1957

Abstract

A recent canonical divergence, which is introduced on a smooth manifold

M

endowed with a general dualistic structure

(g, \nabla, \nabla^{*})

, is considered for flat

α

-connections. In the classical setting, we compute such a canonical divergence [...] Read more.

A recent canonical divergence, which is introduced on a smooth manifold

M

endowed with a general dualistic structure

(g, \nabla, \nabla^{*})

, is considered for flat

α

-connections. In the classical setting, we compute such a canonical divergence on the manifold of positive measures and prove that it coincides with the classical

α

-divergence. In the quantum framework, the recent canonical divergence is evaluated for the quantum

α

-connections on the manifold of all positive definite Hermitian operators. In this case as well, we obtain that the recent canonical divergence is the quantum

α

-divergence. Full article

(This article belongs to the Special Issue The Ubiquity of Entropy)

20 pages, 2219 KiB

Open AccessArticle

Maximum Entropy Methods for Loss Data Analysis: Aggregation and Disaggregation Problems

by Erika Gomes-Gonçalves, Henryk Gzyl and Silvia Mayoral

Entropy 2019, 21(8), 762; https://doi.org/10.3390/e21080762 - 06 Aug 2019

Viewed by 2486

Abstract

The analysis of loss data is of utmost interest in many branches of the financial and insurance industries, in structural engineering and in operation research, among others. In the financial industry, the determination of the distribution of losses is the first step to [...] Read more.

The analysis of loss data is of utmost interest in many branches of the financial and insurance industries, in structural engineering and in operation research, among others. In the financial industry, the determination of the distribution of losses is the first step to take to compute regulatory risk capitals; in insurance we need the distribution of losses to determine the risk premia. In reliability analysis one needs to determine the distribution of accumulated damage or the first time of occurrence of a composite event, and so on. Not only that, but in some cases we have data on the aggregate risk, but we happen to be interested in determining the statistical nature of the different types of events that contribute to the aggregate loss. Even though in many of these branches of activity one may have good theoretical descriptions of the underlying processes, the nature of the problems is such that we must resort to numerical methods to actually compute the loss distributions. Besides being able to determine numerically the distribution of losses, we also need to assess the dependence of the distribution of losses and that of the quantities computed with it, on the empirical data. It is the purpose of this note to illustrate the how the maximum entropy method and its extensions can be used to deal with the various issues that come up in the computation of the distribution of losses. These methods prove to be robust and allow for extensions to the case when the data has measurement errors and/or is given up to an interval. Full article

(This article belongs to the Special Issue The Ubiquity of Entropy)

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