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23 pages, 3373 KiB

Open AccessFeature PaperArticle

Elucidating the Role of the Mixing Entropy in Equilibrated Nanoconfined Reactions

by Leonid Rubinovich and Micha Polak

Entropy 2025, 27(6), 564; https://doi.org/10.3390/e27060564 - 27 May 2025

Viewed by 296

By introducing the concept of nanoreaction-based fluctuating mixing entropy, the challenge posed by the smallness of a closed molecular system is addressed through equilibrium statistical–mechanical averaging over fluctuating reaction extent. Based on the canonical partition function, the interplay between the mixing entropy and fluctuations in the reaction extent in nanoscale environments is unraveled while maintaining consistency with macroscopic behavior. The nanosystem size dependence of the mixing entropy, the reaction extent, and a concept termed the “reaction extent entropy” are modeled for the combination reactions

A + B \leftrightarrow 2 C

and the specific case of

H_{2} + I_{2} \leftrightarrow 2 H I

. A distinct inverse correlation is found between the first two properties, revealing consistency with the nanoconfinement entropic effect on chemical equilibrium (NCECE). To obtain the time dependence of the instantaneous mixing entropy following equilibration, the Stochastic Simulation (Gillespie) Algorithm is employed. In particular, the smallest nanosystems exhibit a step-like behavior that deviates significantly from the smooth mean values and is associated with the discrete probability distribution of the reaction extent. As illustrated further for molecular adsorption and spin polarization, the current approach can be extended beyond nanoreactions to other confined systems with a limited number of species. Full article

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21 pages, 614 KiB

Open AccessArticle

Generalizations of Talagrand Inequality for Sinkhorn Distance Using Entropy Power Inequality

by Shuchan Wang, Photios A. Stavrou and Mikael Skoglund

Entropy 2022, 24(2), 306; https://doi.org/10.3390/e24020306 - 21 Feb 2022

Cited by 4 | Viewed by 4165

Abstract

The distance that compares the difference between two probability distributions plays a fundamental role in statistics and machine learning. Optimal transport (OT) theory provides a theoretical framework to study such distances. Recent advances in OT theory include a generalization of classical OT with an extra entropic constraint or regularization, called entropic OT. Despite its convenience in computation, entropic OT still lacks sufficient theoretical support. In this paper, we show that the quadratic cost in entropic OT can be upper-bounded using entropy power inequality (EPI)-type bounds. First, we prove an HWI-type inequality by making use of the infinitesimal displacement convexity of the OT map. Second, we derive two Talagrand-type inequalities using the saturation of EPI that corresponds to a numerical term in our expressions. These two new inequalities are shown to generalize two previous results obtained by Bolley et al. and Bai et al. Using the new Talagrand-type inequalities, we also show that the geometry observed by Sinkhorn distance is smoothed in the sense of measure concentration. Finally, we corroborate our results with various simulation studies. Full article

(This article belongs to the Special Issue Distance in Information and Statistical Physics III)

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10 pages, 1580 KiB

Open AccessProceeding Paper

Quantum Trajectories in Entropic Dynamics

by Nicholas Carrara

Proceedings 2019, 33(1), 25; https://doi.org/10.3390/proceedings2019033025 - 13 Dec 2019

Viewed by 1217

Abstract

Entropic Dynamics is a framework for deriving the laws of physics from entropic inference. In an (ED) of particles, the central assumption is that particles have definite yet unknown positions. By appealing to certain symmetries, one can derive a quantum mechanics of scalar particles and particles with spin, in which the trajectories of the particles are given by a stochastic equation. This is much like Nelson’s stochastic mechanics which also assumes a fluctuating particle as the basis of the microstates. The uniqueness of ED as an entropic inference of particles allows one to continuously transition between fluctuating particles and the smooth trajectories assumed in Bohmian mechanics. In this work we explore the consequences of the ED framework by studying the trajectories of particles in the continuum between stochastic and Bohmian limits in the context of a few physical examples, which include the double slit and Stern-Gerlach experiments. Full article

(This article belongs to the Proceedings of The 39th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)

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37 pages, 397 KiB

Open AccessFeature PaperArticle

The Entropic Dynamics Approach to Quantum Mechanics

by Ariel Caticha

Entropy 2019, 21(10), 943; https://doi.org/10.3390/e21100943 - 26 Sep 2019

Cited by 21 | Viewed by 4045

Abstract

Entropic Dynamics (ED) is a framework in which Quantum Mechanics is derived as an application of entropic methods of inference. In ED the dynamics of the probability distribution is driven by entropy subject to constraints that are codified into a quantity later identified as the phase of the wave function. The central challenge is to specify how those constraints are themselves updated. In this paper we review and extend the ED framework in several directions. A new version of ED is introduced in which particles follow smooth differentiable Brownian trajectories (as opposed to non-differentiable Brownian paths). To construct ED we make use of the fact that the space of probabilities and phases has a natural symplectic structure (i.e., it is a phase space with Hamiltonian flows and Poisson brackets). Then, using an argument based on information geometry, a metric structure is introduced. It is shown that the ED that preserves the symplectic and metric structures—which is a Hamilton-Killing flow in phase space—is the linear Schrödinger equation. These developments allow us to discuss why wave functions are complex and the connections between the superposition principle, the single-valuedness of wave functions, and the quantization of electric charges. Finally, it is observed that Hilbert spaces are not necessary ingredients in this construction. They are a clever but merely optional trick that turns out to be convenient for practical calculations. Full article

(This article belongs to the Special Issue Entropy in Foundations of Quantum Physics)

15 pages, 420 KiB

Open AccessArticle

On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests

by Aaditya Ramdas, Nicolás García Trillos and Marco Cuturi

Entropy 2017, 19(2), 47; https://doi.org/10.3390/e19020047 - 26 Jan 2017

Cited by 317 | Viewed by 15713

Abstract

Nonparametric two-sample or homogeneity testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. The literature is old and rich, with a wide variety of statistics having being designed and analyzed, both for the unidimensional and the multivariate setting. In this short survey, we focus on test statistics that involve the Wasserstein distance. Using an entropic smoothing of the Wasserstein distance, we connect these to very different tests including multivariate methods involving energy statistics and kernel based maximum mean discrepancy and univariate methods like the Kolmogorov–Smirnov test, probability or quantile (PP/QQ) plots and receiver operating characteristic or ordinal dominance (ROC/ODC) curves. Some observations are implicit in the literature, while others seem to have not been noticed thus far. Given nonparametric two-sample testing’s classical and continued importance, we aim to provide useful connections for theorists and practitioners familiar with one subset of methods but not others. Full article

(This article belongs to the Special Issue Statistical Significance and the Logic of Hypothesis Testing)

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18 pages, 557 KiB

Open AccessArticle

Finite Key Size Analysis of Two-Way Quantum Cryptography

by Jesni Shamsul Shaari and Stefano Mancini

Entropy 2015, 17(5), 2723-2740; https://doi.org/10.3390/e17052723 - 30 Apr 2015

Cited by 7 | Viewed by 5138

Abstract

Quantum cryptographic protocols solve the longstanding problem of distributing a shared secret string to two distant users by typically making use of one-way quantum channel. However, alternative protocols exploiting two-way quantum channel have been proposed for the same goal and with potential advantages. Here, we overview a security proof for two-way quantum key distribution protocols, against the most general eavesdropping attack, that utilize an entropic uncertainty relation. Then, by resorting to the “smooth” version of involved entropies, we extend such a proof to the case of finite key size. The results will be compared to those available for one-way protocols showing some advantages. Full article

(This article belongs to the Special Issue Quantum Cryptography)

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