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Entropy, Volume 22; doi:10.3390/e22111209

**Abstract:**

The subject of this paper deals with the mathematical formulation of the Heisenberg Indeterminacy Principle in the framework of Quantum Gravity. The starting point is the establishment of the so-called time-conjugate momentum inequalities holding for non-relativistic and relativistic Quantum Mechanics. The validity of analogous Heisenberg inequalities in quantum gravity, which must be based on strictly physically observable quantities (i.e., necessarily either 4-scalar or 4-vector in nature), is shown to require the adoption of a manifestly covariant and unitary quantum theory of the gravitational field. Based on the prescription of a suitable notion of Hilbert space scalar product, the relevant Heisenberg inequalities are established. Besides the coordinate-conjugate momentum inequalities, these include a novel proper-time-conjugate extended momentum inequality. Physical implications and the connection with the deterministic limit recovering General Relativity are investigated.

Entropy, Volume 22; doi:10.3390/e22111215

**Abstract:**

A triangular lattice model for pattern formation by core-shell particles at fluid interfaces is introduced and studied for the particle to core diameter ratio equal to 3. Repulsion for overlapping shells and attraction at larger distances due to capillary forces are assumed. Ground states and thermodynamic properties are determined analytically and by Monte Carlo simulations for soft outer- and stiffer inner shells, with different decay rates of the interparticle repulsion. We find that thermodynamic properties are qualitatively the same for slow and for fast decay of the repulsive potential, but the ordered phases are stable for temperature ranges, depending strongly on the shape of the repulsive potential. More importantly, there are two types of patterns formed for fixed chemical potential—one for a slow and another one for a fast decay of the repulsion at small distances. In the first case, two different patterns—for example clusters or stripes—occur with the same probability for some range of the chemical potential. For a fixed concentration, an interface is formed between two ordered phases with the closest concentration, and the surface tension takes the same value for all stable interfaces. In the case of degeneracy, a stable interface cannot be formed for one out of four combinations of the coexisting phases, because of a larger surface tension. Our results show that by tuning the architecture of a thick polymeric shell, many different patterns can be obtained for a sufficiently low temperature.

Entropy, Volume 22; doi:10.3390/e22111219

**Abstract:**

Weak invariants are time-dependent observables with conserved expectation values. Their fluctuations, however, do not remain constant in time. On the assumption that time evolution of the state of an open quantum system is given in terms of a completely positive map, the fluctuations monotonically grow even if the map is not unital, in contrast to the fact that monotonic increases of both the von Neumann entropy and Rényi entropy require the map to be unital. In this way, the weak invariants describe temporal asymmetry in a manner different from the entropies. A formula is presented for time evolution of the covariance matrix associated with the weak invariants in cases where the system density matrix obeys the Gorini–Kossakowski–Lindblad–Sudarshan equation.

Entropy, Volume 22; doi:10.3390/e22111212

**Abstract:**

The purpose of this study is to analyze the dynamic properties of gas hydrate development from a large hydrate simulator through numerical simulation. A mathematical model of heat transfer and entropy production of methane hydrate dissociation by depressurization has been established, and the change behaviors of various heat flows and entropy generations have been evaluated. Simulation results show that most of the heat supplied from outside is assimilated by methane hydrate. The energy loss caused by the fluid production is insignificant in comparison to the heat assimilation of the hydrate reservoir. The entropy generation of gas hydrate can be considered as the entropy flow from the ambient environment to the hydrate particles, and it is favorable from the perspective of efficient hydrate exploitation. On the contrary, the undesirable entropy generations of water, gas and quartz sand are induced by the irreversible heat conduction and thermal convection under notable temperature gradient in the deposit. Although lower production pressure will lead to larger entropy production of the whole system, the irreversible energy loss is always extremely limited when compared with the amount of thermal energy utilized by methane hydrate. The production pressure should be set as low as possible for the purpose of enhancing exploitation efficiency, as the entropy production rate is not sensitive to the energy recovery rate under depressurization.

Entropy, Volume 22; doi:10.3390/e22111218

**Abstract:**

This paper presents an extensive and practical study of the estimation of stable channel bank shape and dimensions using the maximum entropy principle. The transverse slope (St) distribution of threshold channel bank cross-sections satisfies the properties of the probability space. The entropy of St is subject to two constraint conditions, and the principle of maximum entropy must be applied to find the least biased probability distribution. Accordingly, the Lagrange multiplier (λ) as a critical parameter in the entropy equation is calculated numerically based on the maximum entropy principle. The main goal of the present paper is the investigation of the hydraulic parameters influence governing the mean transverse slope

Entropy, Volume 22; doi:10.3390/e22111217

**Abstract:**

How much different genders contribute to citations and whether we see different gender patterns between STEM and non-STEM researchers are questions that have long been studied in academia. Here we analyze the research output in terms of citations collected from the Web of Science of males and females from the largest Croatian university, University of Zagreb. Applying the Mann–Whitney statistical test, for most faculties, we demonstrate no gender difference in research output except for seven faculties, where males are significantly better than females on six faculties. We find that female STEM full professors are significantly more cited than male colleagues, while male non-STEM assistant professors are significantly more cited than their female colleagues. There are ten faculties where females have the larger average citations than their male colleagues and eleven faculties where the most cited researcher is woman. For the most cited researchers, our Zipf plot analyses demonstrate that both genders follow power laws, where the exponent calculated for male researchers is moderately larger than the exponent for females. The exponent for STEM citations is slightly larger than the exponent obtained for non-STEM citations, implying that compared to non-STEM, STEM research output leads to fatter tails and so larger citations inequality than non-STEM.

Entropy, Volume 22; doi:10.3390/e22111213

**Abstract:**

A novel approach to solve optimal control problems dealing simultaneously with fractional differential equations and time delay is proposed in this work. More precisely, a set of global radial basis functions are firstly used to approximate the states and control variables in the problem. Then, a collocation method is applied to convert the time-delay fractional optimal control problem to a nonlinear programming one. By solving the resulting challenge, the unknown coefficients of the original one will be finally obtained. In this way, the proposed strategy introduces a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The algorithm’s performance has been analyzed for several non-trivial examples, and the obtained results have shown that this scheme is more accurate, robust, and efficient than most previous methods.

Entropy, Volume 22; doi:10.3390/e22111208

**Abstract:**

The thermodynamic Ricci curvature scalar R has been applied in a number of contexts, mostly for systems characterized by 2D thermodynamic geometries. Calculations of R in thermodynamic geometries of dimension three or greater have been very few, especially in the fluid regime. In this paper, we calculate R for two examples involving binary fluid mixtures: a binary mixture of a van der Waals (vdW) fluid with only repulsive interactions, and a binary vdW mixture with attractive interactions added. In both of these examples, we evaluate R for full 3D thermodynamic geometries. Our finding is that basic physical patterns found for R in the pure fluid are reproduced to a large extent for the binary fluid.

Entropy, Volume 22; doi:10.3390/e22111210

**Abstract:**

We discuss a possibility that the entire universe on its most fundamental level is a neural network. We identify two different types of dynamical degrees of freedom: “trainable” variables (e.g., bias vector or weight matrix) and “hidden” variables (e.g., state vector of neurons). We first consider stochastic evolution of the trainable variables to argue that near equilibrium their dynamics is well approximated by Madelung equations (with free energy representing the phase) and further away from the equilibrium by Hamilton–Jacobi equations (with free energy representing the Hamilton’s principal function). This shows that the trainable variables can indeed exhibit classical and quantum behaviors with the state vector of neurons representing the hidden variables. We then study stochastic evolution of the hidden variables by considering D non-interacting subsystems with average state vectors, x¯1, …, x¯D and an overall average state vector x¯0. In the limit when the weight matrix is a permutation matrix, the dynamics of x¯μ can be described in terms of relativistic strings in an emergent D+1 dimensional Minkowski space-time. If the subsystems are minimally interacting, with interactions that are described by a metric tensor, and then the emergent space-time becomes curved. We argue that the entropy production in such a system is a local function of the metric tensor which should be determined by the symmetries of the Onsager tensor. It turns out that a very simple and highly symmetric Onsager tensor leads to the entropy production described by the Einstein–Hilbert term. This shows that the learning dynamics of a neural network can indeed exhibit approximate behaviors that were described by both quantum mechanics and general relativity. We also discuss a possibility that the two descriptions are holographic duals of each other.

Entropy, Volume 22; doi:10.3390/e22111211

**Abstract:**

Selective assembly is the method of obtaining high precision assemblies from relatively low precision components. For precision instruments, the geometric error on mating surface is an important factor affecting assembly accuracy. Different from the traditional selective assembly method, this paper proposes an optimization method of selective assembly for shafts and holes based on relative entropy and dynamic programming. In this method, relative entropy is applied to evaluate the clearance uniformity between shafts and holes, and dynamic programming is used to optimize selective assembly of batches of shafts and holes. In this paper, the case studied has 8 shafts and 20 holes, which need to be assembled into 8 products. The results show that optimal combinations are selected, which provide new insights into selective assembly optimization and lay the foundation for selective assembly of multi-batch precision parts.