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We introduce a generalized micro–macro Markov chain Monte Carlo (mM-MCMC) method with pseudo-marginal approximation to the free energy that is able to accelerate sampling of the microscopic Gibbs distributions when there is a time-scale separation between the macroscopic dynamics of a reaction coordinate and the remaining microscopic degrees of freedom. The mM-MCMC method attains this efficiency by iterating four steps: (i) propose a new value of the reaction coordinate, (ii) accept or reject the macroscopic sample, (iii) run a biased simulation that creates a microscopic molecular instance that lies close to the newly sampled macroscopic reaction coordinate value, and (iv) microscopic accept/reject step for the new microscopic sample. In the present paper, we eliminate the main computational bottleneck of earlier versions of this method: the necessity to have an accurate approximation of free energy. We show that the introduction of a pseudo-marginal approximation significantly reduces the computational cost of the microscopic accept/reject step while still providing unbiased samples. We illustrate the method's behavior on several molecular systems with low-dimensional reaction coordinates. [ABSTRACT FROM AUTHOR]

The article explores the metabolic rates of non-avian dinosaurs, crocodiles, and birds during embryonic development by calculating the activation energy of their metabolic reactions. The study uses conservation of energy principles and the Boltzmann factor to determine how metabolism changes with temperature. Results show that birds had the highest metabolism in the egg, with Troodon dinosaurs having the highest metabolism among the three dinosaur species studied. The findings suggest a link between metabolic rate and incubation temperature, with implications for sex determination in crocodilians. [Extracted from the article]

We introduce and analyze a natural class of nonlinear dynamics for spin systems such as the Ising model. This class of dynamics is based on the framework of mass action kinetics, which models the evolution of systems of entities under pairwise interactions, and captures a number of important nonlinear models from various fields, including chemical reaction networks, Boltzmann's model of an ideal gas, recombination in population genetics and genetic algorithms. In the context of spin systems, it is a natural generalization of linear dynamics based on Markov chains, such as Glauber dynamics and block dynamics, which are by now well understood. However, the inherent nonlinearity makes the dynamics much harder to analyze, and rigorous quantitative results so far are limited to processes which converge to essentially trivial stationary distributions that are product measures. In this paper we provide the first quantitative convergence analysis for natural nonlinear dynamics in a combinatorial setting where the stationary distribution contains non-trivial correlations, namely spin systems at high temperatures. We prove that nonlinear versions of both the Glauber dynamics and the block dynamics converge to the Gibbs distribution of the Ising model (with given external fields) in times O (n log n) and O (log n) respectively, where n is the size of the underlying graph (number of spins). Given the lack of general analytical methods for such nonlinear systems, our analysis is unconventional, and combines tools such as information percolation (due in the linear setting to Lubetzky and Sly), a novel coupling of the Ising model with Erdős-Rényi random graphs, and non-traditional branching processes augmented by a "fragmentation" process. Our results extend immediately to any spin system with a finite number of spins and bounded interactions. [ABSTRACT FROM AUTHOR]

In modern warfare, the use of UAVs for reconnaissance, search and rescue missions is very common, and it is essential to plan the flight path of UAVs. However, in the face of complex battlefield environment, the existing flight path planning algorithms have the problems of long time consumption and unstable path. Therefore, this paper studies the UAV flight path planning optimization in complex battlefield environment. First, we construct the battlefield environment model. Then, by analyzing the UAV flight constraints existing in battlefield environment, the objective function is obtained. And the problem of UAV flight path planning optimization is transformed into a nonlinear combinatorial optimization problem. On this basis, an Adaptive Adjustment Flight Path Planning algorithm (AA-FPP) is proposed. The AA-FPP algorithm adaptively adjusts the absorption coefficient of fireflies by using chaotic strategy. It adjusts the control position updating formula by using time-varying inertia weight to enhance its global searching ability. Then, random factors based on Boltzmann selection strategy are introduced to perturb the iterative solutions in AA-FPP. It expands the search space of the path and enhances the convergence efficiency. Finally, simulation results show that the AA-FPP algorithm can successfully plan a flight path that reduces static/dynamic threat intensity. And it has greater advantages in path stability and planning time consumption. [ABSTRACT FROM AUTHOR]

The first step in comprehending the properties of Au10 clusters is understanding the lowest energy structure at low and high temperatures. Functional materials operate at finite temperatures; however, energy computations employing density functional theory (DFT) methodology are typically carried out at zero temperature, leaving many properties unexplored. This study explored the potential and free energy surface of the neutral Au10 nanocluster at a finite temperature, employing a genetic algorithm coupled with DFT and nanothermodynamics. Furthermore, we computed the thermal population and infrared Boltzmann spectrum at a finite temperature and compared it with the validated experimental data. Moreover, we performed the chemical bonding analysis using the quantum theory of atoms in molecules (QTAIM) approach and the adaptive natural density partitioning method (AdNDP) to shed light on the bonding of Au atoms in the low-energy structures. In the calculations, we take into consideration the relativistic effects through the zero-order regular approximation (ZORA), the dispersion through Grimme's dispersion with Becke–Johnson damping (D3BJ), and we employed nanothermodynamics to consider temperature contributions. Small Au clusters prefer the planar shape, and the transition from 2D to 3D could take place at atomic clusters consisting of ten atoms, which could be affected by temperature, relativistic effects, and dispersion. We analyzed the energetic ordering of structures calculated using DFT with ZORA and single-point energy calculation employing the DLPNO-CCSD(T) methodology. Our findings indicate that the planar lowest energy structure computed with DFT is not the lowest energy structure computed at the DLPN0-CCSD(T) level of theory. The computed thermal population indicates that the 2D elongated hexagon configuration strongly dominates at a temperature range of 50–800 K. Based on the thermal population, at a temperature of 100 K, the computed IR Boltzmann spectrum agrees with the experimental IR spectrum. The chemical bonding analysis on the lowest energy structure indicates that the cluster bond is due only to the electrons of the 6 s orbital, and the Au d orbitals do not participate in the bonding of this system. [ABSTRACT FROM AUTHOR]

Based on the theory of the thermodynamic equilibrium in a system of quantum vortices in superfluids in the presence of a counterflow, the influence of a vortex tangle on various thermodynamic phenomena in quantum liquids is studied. Using the early calculated partition function, we study some of the properties of He II related to counterflow, such as the distribution of vortex loops in their length, the suppression of the superfluid density ρ s , and the shift T λ . The physics behind this issue is related to the fact that the partition function describing the ensemble of chaotic vortex filaments depends on the relative velocity v n s . The partition function, in turn, depends on relative velocity due to the Gibbs distribution with the specific velocity-dependent Hamiltonian. Good agreement with the earlier obtained results is a fairly strong argument in favor of the point of view that a collection of chaotic quantum vortices can, indeed, be considered as a kind of gas of quasiparticles at high temperatures, especially near a phase transition. The work is closely related to nonlinear physics, which studies chaotic processes, and is currently in the stage of active development, resulting in many meaningful and expressive results. The application of the developed formalism to the theory of quantum turbulence is briefly discussed. [ABSTRACT FROM AUTHOR]

Understanding natural protein evolution and designing novel proteins are motivating interest in development of high-throughput methods to explore large sequence spaces. In this work, we demonstrate the application of multisite λ dynamics (MSλD), a rigorous free energy simulation method, and chemical denaturation experiments to quantify evolutionary selection pressure from sequence-stability relationships and to address questions of design. This study examines a mesophilic phylogenetic clade of ribonuclease H (RNase H), furthering its extensive characterization in earlier studies, focusing on E. coli RNase H (ecRNH) and a more stable consensus sequence (AncCcons) differing at 15 positions. The stabilities of 32,768 chimeras between these two sequences were computed using the MSλD framework. The most stable and least stable chimeras were predicted and tested along with several other sequences, revealing a designed chimera with approximately the same stability increase as AncCcons, but requiring only half the mutations. Comparing the computed stabilities with experiment for 12 sequences reveals a Pearson correlation of 0.86 and root mean squared error of 1.18 kcal/mol, an unprecedented level of accuracy well beyond less rigorous computational design methods. We then quantified selection pressure using a simple evolutionary model in which sequences are selected according to the Boltzmann factor of their stability. Selection temperatures from 110 to 168 K are estimated in three ways by comparing experimental and computational results to evolutionary models. These estimates indicate selection pressure is high, which has implications for evolutionary dynamics and for the accuracy required for design, and suggests accurate high-throughput computational methods likeMSλDmay enable more effective protein design. [ABSTRACT FROM AUTHOR]

Matter grows and self-assembles to produce complex structures such as virus capsids, carbon fullerenes, proteins, glasses, etc. Due to its complexity, performing pen-and-paper calculations to explain and describe such assemblies is cumbersome. Many years ago, Richard Kerner presented a pen-and-paper path integral approach to understanding self-organized matter. Although this approach successfully addressed many important problems, including the yield of fullerene formation, the glass transition temperature of doped chalcogenide glasses, the fraction of boroxol rings in B 2 O 3 glasses, the first theoretical explanation for the empirical recipe of window and Pyrex glass and the understanding of virus capsid self-assembly, it still is not the primary choice when tackling similar problems. The reason lies in the fact that it diverges from mainstream approaches based on the energy landscape paradigm and non-equilibrium thermodynamics. In this context, a critical review is presented, demonstrating that the Richard Kerner method is, in fact, a clever way to identify relevant configurations. Its equations are simplified common physical sense versions of those found in the energy landscape kinetic equations. Subsequently, the utilization of equilibrium Boltzmann factors in the transition Markov chain probabilities is analyzed within the context of local two-level energy landscape models kinetics. This analysis demonstrates that their use remains valid when the local energy barrier between reaction coordinate states is small compared to the thermal energy. This finding places the Richard Kerner model on par with other more sophisticated methods and, hopefully, will promote its adoption as an initial and useful choice for describing the self-agglomeration of matter. [ABSTRACT FROM AUTHOR]

The author considers a practical algorithm for estimating an unknown parameter of the mathematical model of a Markov process with local interaction based on the Gibbs distribution. It is proposed to apply the stochastic quasi-gradient method to the maximum likelihood function, which is constructed from the observations of the realizations of the Gibbs field. The obtained results have a wide application scope in the modeling of stochastic processes. [ABSTRACT FROM AUTHOR]

The author considers a practical algorithm for estimating an unknown parameter of the mathematical model of a Markov process with local interaction based on the Gibbs distribution. It is proposed to apply the stochastic quasi-gradient method to the maximum likelihood function, which is constructed from the observations of the realizations of the Gibbs field. The obtained results have a wide application scope in the modeling of stochastic processes. [ABSTRACT FROM AUTHOR]

Logit dynamics is a form of randomized game dynamics in which players have a bias toward strategic deviations that give a higher improvement in cost. It is used extensively in practice. In congestion (or potential) games, the dynamics converge to the so-called Gibbs distribution over the set of all strategy profiles when interpreted as a Markov chain. In general, logit dynamics can converge slowly to the Gibbs distribution, but beyond that, not much is known about its algorithmic aspects, nor that of the Gibbs distribution. In this work, we are interested in the following two questions for congestion games: (i) Is there an efficient algorithm for sampling from the Gibbs distribution? (ii) If yes, does there also exist natural randomized dynamics that converge quickly to the Gibbs distribution? We first study these questions in extension parallel congestion games, a well-studied special case of symmetric network congestion games. As our main result, we show that there is a simple variation on the logit dynamics that converges quickly to the Gibbs distribution in such games. We also address the first question for the class of so-called capacitated uniform congestion games and the second question for max cut games played on a complete graph. To prove our results, we rely on the recent breakthrough work of Anari et al. (2019) regarding the approximate sampling of a base of a matroid according to a strongly log-concave probability distribution. [ABSTRACT FROM AUTHOR]

In this paper, we obtain weak convergence results for a family of Gibbs measures depending on the parameter θ > 0 in the following form d P θ (x) = Z θ exp − H θ (x) / θ d Q (x) , where we show that the limit distribution is concentrated in the set of the global minima of the limit Gibbs potential. We also give an explicit calculus for the limit distribution. Here, we use the above as an alternative to Lyapunov's function or to direct methods for stationary probability convergence and apply it to the repairman problem. Finally, we illustrate this method with a numerical example. [ABSTRACT FROM AUTHOR]

The American Community Survey (ACS) is an ongoing program conducted by the US Census Bureau that publishes estimates of important demographic statistics over pre-specified administrative areas. ACS provides spatially referenced count-valued outcomes that are paired with finite populations. For example, the number of people below the poverty line and the total population for each county are estimated by ACS. One common assumption is that the spatially referenced count-valued outcome given the finite population is binomial distributed. This conditionally specified (CS) model does not define the joint relationship between the count-valued outcome and the finite population. Thus, we consider a joint model for the count-valued outcome and the finite population. When cross-dependence in our joint model can be leveraged to 'improve spatial prediction' we say that the finite population is 'informative.' We model the count given the finite population as binomial and the finite population as negative binomial and use multivariate logit-beta prior distributions. This leads to closed-form expressions of the full-conditional distributions for an efficient Gibbs sampler. We illustrate our model through simulations and our motivating application of ACS poverty estimates. These empirical analyses show the benefits of using our proposed model over the more traditional CS binomial model. [ABSTRACT FROM AUTHOR]

Determination of the absolute configuration of chiral molecules is a prerequisite for obtaining a fundamental understanding in any chirality‐related field. The interaction with polarised light has proven to be a powerful means to determine this absolute configuration, but its application rests on the comparison between experimental and computed spectra for which the inherent uncertainty in conformational Boltzmann factors has proven to be extremely hard to tackle. Here we present a novel approach that overcomes this issue by combining a genetic algorithm that identifies the relevant conformers by accounting for the uncertainties in DFT relative energies, and a hierarchical clustering algorithm that analyses the trends in the spectra of the considered conformers and identifies on‐the‐fly when a given chiroptical technique is not able to make reliable predictions. The effectiveness of this approach is demonstrated by considering the challenging cases of papuamine and haliclonadiamine, two bis‐indane natural products with eight chiral centres and considerable conformational heterogeneity that could not be assigned unambiguously with current approaches. [ABSTRACT FROM AUTHOR]

The quantum statistical properties of the proton subsystem in hydrogen-bonded crystals (HBC) are investigated. Based on the non-stationary Liouville operator equation (taking into account a number of assumptions established in the experiment), a quantum kinetic equation is constructed for the ensemble of non-interacting protons (an ideal proton gas) moving in the crystal potential image perturbed by the external electric field. The balanced density matrix for the unperturbed proton subsystem is constructed using the quantum canonical Gibbs distribution, and the non-balanced density matrix is calculated from the solutions of the nonlinear quantum kinetic equation by methods in linear approximation of perturbation theory for the blocking electrode model. Full quantum mechanical averaging of the polarization operator makes it possible to study the theoretical frequency-temperature spectra of the complex dielectric permittivity (CDP) calculated using quantum relaxation parameters that differ significantly from their semiclassical counterparts. A scheme is presented for an analytical study of the dielectric loss tangent in the region of quantum nonlinear relaxation in HBC. The results obtained in the given paper are of scientific interest in developing the theoretical foundations of proton conduction processes in energy-independent memory elements (with anomalously high residual polarization) based on thin films of ferroelectric materials in the ultralow temperature range (1–10 K). The theoretical results obtained have a direct application to the study of the tunneling mechanisms of spontaneous polarization in ferroelectric HBC with a rectangular hysteresis loop, in particular in crystals of potassium dideutrophosphate (KDP), widely used in nonlinear optics and laser technology. The quantum properties of proton relaxation in HBC can be applied in the future to the study of solid-state electrolytes with high proton conductivity for hydrogen energy, capacitor technology (superionics, varicodes), and elements of MIS and MSM structures in the development of resonant tunnel diodes for microelectronics and computer technology. [ABSTRACT FROM AUTHOR]

We consider the problem of high-dimensional Bayesian nonparametric variable selection using an aggregation of so-called "weak learners." The most popular variant of this is the Bayesian additive regression trees (BART) model, which is the natural Bayesian analog to boosting decision trees. In this article, we use Gibbs distributions on random partitions to induce sparsity in ensembles of weak learners. Looking at BART as a special case, we show that the class of Gibbs priors includes two recently proposed models—the Dirichlet additive regression trees (DART) model and the spike-and-forest model—as extremal cases, and we show that certain Gibbs priors are capable of achieving the benefits of both the DART and spike-and-forest models while avoiding some of their key drawbacks. We then show the promising performance of Gibbs priors for other classes of weak learners, such as tensor products of spline basis functions. A Pólya Urn scheme is developed for efficient computations. for this article are available online. [ABSTRACT FROM AUTHOR]

The Helmholtz free energy, A, and the Gibbs free energy, G, of polyethylene (PE) were determined based on its empirical pressure–volume–temperature–entropy equation of state, P-V-T-S Eos, and the empirical partition function, which we reported previously. The PA-VA-TA-SA Eos for the iso-Helmholtz free energy, iso-A, was determined based on the P-V-T-S Eos where the pressure (PA), volume (VA), temperature (TA), and entropy (SA) satisfied the condition that A is constant. The PG-VG-TG-SG Eos for the G being constant was also determined based on the P-V-T-S Eos. The three-dimensional (3D) plots, 3D plot, with the three axes, x = TA, y = PA, z = VA expressed by the 3D (TA, PA, VA) and the 3D (TA, VA, SA) at constant A in PE were determined, where the iso-A processes were characterized assuming that the volume and the entropy in the iso-A process decreased with increasing pressure and temperature, while in the 3D (TG, PG, VG) and 3D (TG, VG, SG) plots at constant G in PE, the volume and the entropy in the iso-G process were increasing with increasing pressure and temperature. The empirical partition function for the Helmholtz free energy was expressed by the summation of the terms of T, T2, and T4, the Boltzmann factors e − i θ 1 T with i = 1, 2, 3, and 6 and θ 1 = 71.5 and the U(V,T = 0 K) term for the zero Kelvin internal energy where the T4 term and the Boltzmann factors e − i θ 1 T with i = 6 and 3 were dominant negative factors in the A-T relation at the constant V = 1.04 cm3/g. In case of the Gibbs free energy where the partition function was expressed by the summation of the terms of T, T2, and T4, the Boltzmann factors e − i θ 1 T with i = 1, 2, 3, and 6, the U(V,T = 0 K) term, PV and PV(T = 0) terms, the T2 term and the Boltzmann factors e − i θ 1 T with i = 6 and 3 were dominant negative factors in the G-T relation at 1 atm (1.01 × 10−4 GPa), while at high pressure, 1.0 GPa, the PV and PV(T = 0) terms were dominant positive factors. [ABSTRACT FROM AUTHOR]

In this article, we investigate the long-time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential. We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitably weighted sequence space. This allows us to prove the existence of global solutions which all stabilize toward the grand canonical equilibrium probability distribution as the time variable becomes large, some of them doing so exponentially rapidly but not all. When we set the chemical potential equal to zero, the stability statements continue to hold in the sense that all solutions converge toward the Gibbs probability distribution of the canonical ensemble which characterizes the equilibrium of the given system with a heat bath at fixed temperature. [ABSTRACT FROM AUTHOR]

Youden's index cutoff is a classifier mapping a patient's diagnostic test outcome and available covariate information to a diagnostic category. Typically the cutoff is estimated indirectly by first modeling the conditional distributions of test outcomes given diagnosis and then choosing the optimal cutoff for the estimated distributions. Here we present a Gibbs posterior distribution for direct inference on the cutoff. Our approach makes incorporating prior information about the cutoff much easier compared to existing methods, and does so without specifying probability models for the data, which may be misspecified. The proposed Gibbs posterior distribution is robust with respect to data distributions, is supported by large-sample theory, and performs well in simulations compared to alternative Bayesian and bootstrap-based methods. In addition, two real data sets are examined which illustrate the flexibility of the Gibbs posterior approach and its ability to utilize direct prior information about the cutoff. [ABSTRACT FROM AUTHOR]

A new trend in deep learning, represented by Mutual Information Neural Estimation (MINE) and Information Noise Contrast Estimation (InfoNCE), is emerging. In this trend, similarity functions and Estimated Mutual Information (EMI) are used as learning and objective functions. Coincidentally, EMI is essentially the same as Semantic Mutual Information (SeMI) proposed by the author 30 years ago. This paper first reviews the evolutionary histories of semantic information measures and learning functions. Then, it briefly introduces the author's semantic information G theory with the rate-fidelity function R(G) (G denotes SeMI, and R(G) extends R(D)) and its applications to multi-label learning, the maximum Mutual Information (MI) classification, and mixture models. Then it discusses how we should understand the relationship between SeMI and Shannon's MI, two generalized entropies (fuzzy entropy and coverage entropy), Autoencoders, Gibbs distributions, and partition functions from the perspective of the R(G) function or the G theory. An important conclusion is that mixture models and Restricted Boltzmann Machines converge because SeMI is maximized, and Shannon's MI is minimized, making information efficiency G/R close to 1. A potential opportunity is to simplify deep learning by using Gaussian channel mixture models for pre-training deep neural networks' latent layers without considering gradients. It also discusses how the SeMI measure is used as the reward function (reflecting purposiveness) for reinforcement learning. The G theory helps interpret deep learning but is far from enough. Combining semantic information theory and deep learning will accelerate their development. [ABSTRACT FROM AUTHOR]

We consider the problem of testing for two Gibbs probabilities $ \mu_0 $ and $ \mu_1 $ defined for a dynamical system $ (\Omega, T) $. Due to the fact that in general full orbits are not observable or computable, one needs to restrict to subclasses of tests defined by a finite time series $ h(x_0), h(x_1) = h(T(x_0)),..., h(x_n) = h(T^n(x_0)) $, $ x_0\in \Omega $, $ n\ge 0 $, where $ h:\Omega\to\mathbb R $ denotes a suitable measurable function. We determine in each class the Neyman-Pearson tests, the minimax tests, and the Bayes solutions and show the asymptotic decay of their risk functions as $ n\to\infty $. In the case of $ \Omega $ being a symbolic space, for each $ n\in \mathbb{N} $, these optimal tests rely on the information of the measures for cylinder sets of size $ n $. [ABSTRACT FROM AUTHOR]

The Ising model with a transverse field and an antiferromagnetic transverse interaction is represented as a matrix in the computational basis with non-zero off-diagonal elements with both positive and negative signs and thus may be regarded to be non-stoquastic. We show that the local Boltzmann factors of such a system under an appropriate Suzuki–Trotter representation can be chosen non-negative and thus may potentially be simulated classically without a sign problem if the parameter values are limited to a subspace of the whole parameter space. We then derive conditions for parameters to satisfy asymptotically in order that simulated quantum annealing of this system converges to thermal equilibrium in the long-time limit. [ABSTRACT FROM AUTHOR]

Within the framework of the Gibbs statistical theory, the issue of the size distribution of particles forming a statistical system and the moments of this distribution are considered. The size distribution of particles and the moments of this distribution are determined from probabilistic considerations. The particle size depends on the interactions in the system, the compressibility factor, the number of interacting particles, and the volume of the system. The relations for the average particle size are substituted into expressions for the intrinsic volume of particles in the equations of state written using excluded volume theory for various expressions for the exclusion factor. The equations of state obtained in this way can be considered as a refinement of the equation of state for dense systems, that is, as a transition to a higher level of description. [ABSTRACT FROM AUTHOR]

This article is a preface and editorial note for an issue of the journal Kinetics & Catalysis. The issue is dedicated to the translation of a monograph by V.V. Azatyan, originally published in Russian in 2020. The monograph explores the chain reactions of combustion, explosion, and detonation of gases, focusing on the chemical methods of control. The revised and extended version of the book includes new chapters on the theory of ignition induction periods and the role of heterogeneous chain termination in flame propagation. The book addresses the fundamental problems and patterns of these combustion processes and their importance in various fields. It emphasizes the role of free atoms and radicals in gas-phase processes and the necessity of chain reactions for combustion at different pressures. The book also discusses the kinetic mechanism of high reaction rates, the phenomenon of the induction period, and the role of impurities and inhibitors in controlling combustion and explosion. It highlights the importance of understanding the chain nature of reactions for studying and controlling these processes. The book is intended for researchers, engineers, university teachers, and graduate students in the field of chemical kinetics and combustion. [Extracted from the article]

Abstract polymer models are systems of weighted objects, called polymers, equipped with an incompatibility relation. An important quantity associated with such models is the partition function, which is the weighted sum over all sets of compatible polymers. Various approximation problems reduce to approximating the partition function of a polymer model. Central to the existence of such approximation algorithms are weight conditions of the respective polymer model. Such conditions are derived either via complex analysis or via probabilistic arguments. We follow the latter path and establish a new condition—the clique dynamics condition—, which is less restrictive than the ones in the literature. We introduce a new Markov chain where the clique dynamics condition implies rapid mixing by utilizing cliques of incompatible polymers that naturally arise from the translation of algorithmic problems into polymer models. This leads to improved parameter ranges for several approximation algorithms, such as a factor of at least 2 1 / α for the hard-core model on bipartite α -expanders. [ABSTRACT FROM AUTHOR]

In this study, oxidative degradation of Orange G (OG) and Eosin Y (EY) dyes with TiO2 as the catalyst was explored in air-equilibrated aqueous dispersions under UV light irradiation. To determine the optimal operating conditions for degradation, various effects were investigated, such as the pH of the dye dispersion, the addition of cationic surfactants and the specific additive. The photodegradation efficiency of both dyes was significantly enhanced at alkaline pH, particularly in the presence of tetraalkylammonium bromide surfactants, as they promote a mutual interaction between the dye and TiO2 surface, otherwise prevented due to their negative charge under basic conditions. The surfactant concentration also played an important role, because it influences the state of surfactant aggregation on the semiconductor surface. The kinetic trend of the fragmentation process was strongly affected by different hydrophobic and electrostatic interactions that OG and EY and their derivative intermediates are able to establish with surfactant/TiO2 in various forms of aggregation, as well as by their redox properties. Density functional theory (DFT) calculations of charge distribution and Gibbs free energy changes of solvation, performed on optimized molecular geometry, were useful to support and rationalize the surfactant involvement in the degradation process. [ABSTRACT FROM AUTHOR]

We derive the PDE governing the hydrodynamic limit of a Metropolis rate crystal surface height process in the ``rough scaling"" regime introduced by Marzuola and Weare. The PDE takes the form of a continuity equation, and the expression for the current involves a numerically computed multiplicative correction term similar to a mobility. The correction accounts for the fact that, unusually, the local equilibrium distribution of the process is not a local Gibbs measure even though the global equilibrium distribution is Gibbs. We give definitive numerical evidence of this fact, originally suggested by Y. Gao, A. E. Katsevich, J.-G. Liu, J. Lu, and J. L. Marzuola [Pure Appl. Anal., 3 (2021), pp. 595-612]. In that paper, an approximate PDE-our PDE, but without the correction term-was derived for the limit of the Metropolis rate process under the assumption of a local Gibbs distribution. Our main contribution is to present a numerical method to compute the corrected macroscopic current, which is given by a function of the third spatial derivative of the height profile. Our method exploits properties of the local equilibrium (LE) state of the third order finite difference process. We find that the LE state of this process is not only useful for deriving the PDE, it also enjoys nonstandard properties which are interesting in their own right. Namely, we demonstrate that the LE state is a ``rough LE,"" a novel kind of LE state discovered in our recent work on an Arrhenius rate crystal surface process. [ABSTRACT FROM AUTHOR]

We prove interpolating estimates providing a bound for the oscillation of a function in terms of two Lp norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems. [ABSTRACT FROM AUTHOR]

We analyze the large deviations for a discrete energy Kac-like walk. In particular, we exhibit a path, with probability exponentially small in the number of particles, that looses energy. [ABSTRACT FROM AUTHOR]

We discuss dynamics of periodically driven open quantum systems. The time evolution of the quantum state is described by the quantum master equation, and the form of the dissipator is chosen so that the instantaneous stationary state is given by the Gibbs distribution. We find that the correlation between the population part and the coherence part of the density operator is induced by an adiabatic gauge potential. Although the introduction of the counterdiabatic term eliminates the correlation, additional correlations prevent convergence to the Gibbs distribution. We study the performance of the control by the counterdiabatic term. The system has three different scales and the performance strongly depends on the relations between their magnitudes. This article is part of the theme issue 'Shortcuts to adiabaticity: theoretical, experimental and interdisciplinary perspectives'. [ABSTRACT FROM AUTHOR]

We define degree of mutual dependence of three events in a probability space by using Boltzmann-Shannon entropy function of an appropriate variable distribution produced by these events and depending on four parameters varying, in general, within of a polytope. It turns out that the entropy function attains its absolute maximum exactly when the three events are mutually independent and its absolute minimum at some vertices of the polytope where the events are "maximally" dependent. By composing the entropy function with an appropriate linear function we obtain a continuous "degree of mutual dependence" function with the same domain and the interval [0, 1] as a target. It attains value 0 when the events are mutually independent (the entropy is maximal) and value 1 when they are "maximally" dependent (the entropy is minimal). A link is available for downloading a Java code which evaluates the degree of mutual dependence of three events in the classical case of a sample space with equally likely outcomes. [ABSTRACT FROM AUTHOR]

We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth. In particular, in graphs with subexponential neighborhood growth like ℤd, our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing. As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer‐dimer models in such graphs. [ABSTRACT FROM AUTHOR]

This article describes a new quantitative evaluation method for the degree of dissociation of a compound depending on the curvature of the liquidus line curve. The main point of this method was to determine the specific expressions for dissociation parameters according to the mathematical model of the crystallization line of the compound on the phase diagram. This was based on generally accepted thermodynamic relations. Empirically, it has now been established that for many types of phase equilibria, a common feature is the presence of a correlation dependence between the ratios of the real Gibbs energy of the distribution of components to its ideal component. The osmotic coefficient of Bjerrum–Guggenheim was used as a measure of the deviation of the system from ideality. This coefficient can be in an analytical form depending on the temperature and composition of the phases. The obtained correlation dependence of the osmotic coefficient of Bjerrum–Guggenheim on the ratio of the activity of the liquid and solid phase has been used to develop a mathematical apparatus to analytically describe the lines and surfaces of the crystallization phases. Accordingly, a single analytical basis, as a universal dependence of the modified Schröder–Le Chatelier equation, has been applied. Based on this equation, the conditions have been defined to develop an equilibrium method with which to calculate the thermal dissociation of chemical compounds. The principle of this method has been examined based on the two-component iron-titanium system. The numerical results of the degree of dissociation of the congruent Fe2Ti compound have been obtained using the Gibbs energy of the dissociation reaction and the reaction constant. However, it has been found that the degree of dissociation in the compound Fe2Ti was 0.04%. Demonstrative material for the behavior of the osmotic coefficient of Bjerrum–Guggenheim under boundary conditions has been presented as an assessment criterion of melt structures. The diagrams of the Φi function of the crystallizing phases near the melting temperature of congruently melting compounds (Tm.) have been mathematically studied. Thus, this study demonstrates that Φ A m B n ′ diagrams tended to zero and approached the melting temperature of the compound T m , A m B n , i.e., above the melting temperature of this compound Φ A m B n ′ →∞; then, ln x A m B n →0, and, as a result, x A m B n →1. [ABSTRACT FROM AUTHOR]

We introduce a novel approach for decomposing and learning every scale of a given multiscale objective function in R d , where d ⩾ 1. This approach leverages a recently demonstrated implicit bias of the optimization method of gradient descent [44] , which enables the automatic generation of data that nearly follow Gibbs distribution with an effective potential at any desired scale. One application of this automated effective potential modeling is to construct reduced-order models. For instance, a deterministic surrogate Hamiltonian model can be developed to substantially soften the stiffness that bottlenecks the simulation, while maintaining the accuracy of phase portraits at the scale of interest. Similarly, a stochastic surrogate model can be constructed at a desired scale, such that both its equilibrium and out-of-equilibrium behaviors (characterized by auto-correlation function and mean path) align with those of a damped mechanical system with the original multiscale function being its potential. The robustness and efficiency of our proposed approach in multi-dimensional scenarios have been demonstrated through a series of numerical experiments. A by-product of our development is a method for anisotropic noise estimation and calibration. More precisely, Langevin model of stochastic mechanical systems may not have isotropic noise in practice, and we provide a systematic algorithm to quantify its covariance matrix without directly measuring the noise. In this case, the system may not admit closed form expression of its invariant distribution either, but with this tool, we can design friction matrix appropriately to calibrate the system so that its invariant distribution has a closed form expression of Gibbs. • We are able to learn each component of V = V 0 + ∑ j = 1 K V j , ε j at different scales without explicit access to each scale or knowing ε j values a priori. • Once an effective potential U k (q) approximating V up to scale ε k is learnt, we construct surrogate models for both deterministic and stochastic dynamical systems which preserve the original properties. • We have developed a method to estimate and calibrate an isotropic stochastic forcings for kinetic Langevin systems by using equilibrium properties and adjusting the mass matrix. [ABSTRACT FROM AUTHOR]

Cesium-137 is a major byproduct of nuclear energy generation and is environmentally threatening due to its long half-life and affinity for naturally occurring micaceous clays. Recent experimental observations of illite and phlogopite mica indicate that Cs+ is capable of exchanging with K+ bound in the anhydrous interlayers of layered silicates, forming sharp exchange fronts, leading to interstratification of Cs- and K-illite. We present here a coarse-grained (CG) model of the anhydrous illite interlayer developed using iterative Boltzmann inversion that qualitatively and quantitatively reproduces features of a previously proposed feedback mechanism of ion exchange. The CG model represents a 70-fold speedup over all-atom models of clay systems and predicts interlayer expansion for K-illite near ion exchange fronts. Contrary to the longstanding theory that ion exchange in a neighboring layer increases the binding of K in lattice counterion sites leading to interstratification, we find that the presence of neighboring exchanged layers leads to short-range structural relaxations that increase basal spacing and decrease cohesion of the neighboring K-illite layers. We also provide evidence that the formation of alternating Cs- and K-illite interlayers (i.e., ordered interstratification) is both thermodynamically and mechanically favorable compared to exchange in adjacent interlayers. [ABSTRACT FROM AUTHOR]

In the paper, we consider the Cauchy problem for a Hamiltonian system consisting of a Klein–Gordon field and an infinite harmonic crystal. We assume that the initial data of the problem are a random function and study the convergence of the distributions of the solutions to a limiting measure for large times. Under the condition that the initial random function in the "left" and "right" parts of the space has the Gibbs distribution with different temperatures, we find the stationary states of the system in which the limiting energy current density does not vanish. Thus, for this system, a class of stationary non-equilibrium states is constructed. [ABSTRACT FROM AUTHOR]

In our previous reports, two types of the internal energy for polyethylene (PE) were derived, which were based on the thermodynamic relation that d U T H = C V d T + ( γ V T − P) d V and based on the partition function, that U P F = R T 2 ∂ ln f ∂ V where f = ∑ e − θ i T , CV is the isochoric heat capacity, γ V is the thermal pressure coefficient and e − θ i T is the Boltzmann factor. In this work, a revised internal energy based on the UTH, Urev.TH, was derived by introducing a zero Kelvin internal energy. A revised internal energy based on the UPF, Urev.PF, was also derived by introducing the zero Kelvin internal energy. The zero Kelvin internal energy, U(V, T = 0 K), was determined based on a zero Kelvin isotherm, which was determined from the P-V-T equation of state derived in our previous work. In this report, the three-dimensional (3D) plot, 3D plot, with the three axes, x = TU, y = PU, z = VU expressed by the 3D (TU, PU, VU) and the 3D (TU, VU. SU) at constant U in PE were determined by the Urev.TH where the processes were characterized by decreasing of the volume and the entropy with increasing of the pressure, while the temperature in the iso-internal process had a maximum with increasing the pressure, which were different from those determined by the UTH in our previous work. The revised internal energy Urev.PF was expressed by the summation of the terms of T2 and T4, the Boltzmann factors e − i θ 1 T and the U(V,T = 0 K) where the temperature dependence of the T4 term and the Boltzmann factors e − i θ 1 T with i = 6 and 3 were dominant in the Urev.PF at V = 1.04 cm3/g, while at V = 0.85 cm3/g in the range of high pressure, the U(V,T = 0 K) became a controlling factor. In the case of the Urev,PF at constant temperature T = 320 K, the volume dependence of the T2 term and the U(V,T = 0 K) were dominant factors. [ABSTRACT FROM AUTHOR]

In our previous report, an empirical partition function, f , for polyethylene (PE) was determined based on a pressure (PV) – temperature (TV) – volume (V) – entropy (SV) equation of state where the variables, PV, TV and SV, were isochoric variables with V = const. evaluated by using the experimental P-V-T data and the isobaric heat capacity, CP, at 1 atm. In this work we describe the proposal of a modified, reformed partition function of the Debye model, l n f m − R D , where the l n f m − R D is expressed by the summation of the terms of T, T−1 and T3 and the Boltzmann factors of e − i θ 1 T for i = 1, 2, 3 and 6. The modified reformed pressure, P m − R D , at constant volume calculated based on the l n f m − R D showed that the Boltzmann factors, e − i θ 1 T , for i = 3 and 6 were dominant and increased with increasing temperature and the T 2 and T 4 terms were negative and decreased with increasing temperature, while the modified reformed entropy, S m − R D , showed that the Boltzmann factors of e − i θ 1 T for i = 3 and 6 were also dominant and increased with increasing temperature and the T 3 term increased rapidly over a higher temperature range. All Boltzmann factors of e − i θ 1 T for i = 1, 2, 3 and 6 in the modified reformed isochoric heat capacity, CV,m-RD, calculated based on the l n f m − R D , were positive, which was different from that in the previous work where one negative Boltzmann factor was observed. [ABSTRACT FROM AUTHOR]

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The electron impact ionization is a crucial atomic process in the collisional radiative model and the study of ionization balance. The superstatistics theory, which was originally proposed for the study of non-equilibrium complex systems, has recently been extended to studies of small systems interacting with a finite environment due to their interesting statistical behavior. This paper introduces the superstatistics formalism in the case of ionization rates with different values of the dynamical parameter q and shows how it affects the calculation of the ionization rates for Li+. Moreover, the distribution function for the effective Boltzmann factor of superstatistics was swapped. [ABSTRACT FROM AUTHOR]

Supradegeneracy--degeneracy G(E) increasing with increasing energy E faster than the Boltzmann factor e-E/kT decreases with increasing E--has been investigated with respect to its possibly engendering challenges to the Second Law of Thermodynamics. Supradegeneracy alone does not challenge the Second Law: Systems manifesting supradegeneracy yet compliant with the Second Law are ubiquitous. If there is to be even the possibility that a system manifesting supradegeneracy can challenge the Second Law, additional requirements over and above supradegeneracy per se must also be fulfilled. We hypothesize what prima facie seem to be the two most obvious of these additional requirements. We then consider a simple system manifesting supradegeneracy and also fulfilling these two requirements. At least for the system that we consider, the answer seems to be negative: The Second Law seems not challenged. But understanding why the answer is at least apparently negative for the supradegenerate system that we consider may help in understanding of what at least prima facie seem to be positive results via analyses, including computer simulations but to the best knowledge of the author at the time of this writing not yet experimental tests, of other supradegenerate systems: of what is the minimal complete set of additional requirements--over and above supradegeneracy per se--that must be fulfilled by a supradegenerate system if it is to challenge the Second Law. Moreover, even if it turns out that all supradegenerate systems do not challenge the Second Law, they could still be useful even within its strictures. The same principles apply with respect to both supradegeneracy and anti-supradegeneracy [degeneracy G (E) decreasing with increasing energy E], so a brief discussion of anti-supradegeneracy suffices. It is followed by proposal of simple experimental tests of our system: I hope, albeit probably in vain, to be proven wrong: Only experiments--the final arbiter--can decide the issue for sure! Concluding remarks are provided describing implications if the Second Law could be violated by any means whatsoever (supradegeneracy, anti-supradegeneracy, and/or otherwise). [ABSTRACT FROM AUTHOR]

The Gibbs distribution universally characterizes states of thermal equilibrium. In order to extend the Gibbs distribution to non-equilibrium steady states, one must relate the self-information I (x) = − log ( P ss (x)) of microstate x to measurable physical quantities. This is a central problem in non-equilibrium statistical physics. By considering open systems described by stochastic dynamics which become deterministic in the macroscopic limit, we show that changes Δ I = I ( x t ) − I ( x 0 ) in steady state self-information along deterministic trajectories can be bounded by the macroscopic entropy production Σ. This bound takes the form of an emergent second law Σ + k b Δ I ≥ 0 , which contains the usual second law Σ ≥ 0 as a corollary, and is saturated in the linear regime close to equilibrium. We thus obtain a tighter version of the second law of thermodynamics that provides a link between the deterministic relaxation of a system and the non-equilibrium fluctuations at steady state. In addition to its fundamental value, our result leads to novel methods for computing non-equilibrium distributions, providing a deterministic alternative to Gillespie simulations or spectral methods. Contrary to states of thermal equilibrium, there is no universal characterization of non-equilibrium steady states displaying constant flows of energy and/or matter. Here, the authors make progress in this direction by deriving an emergent and stricter version of the second law of thermodynamics. [ABSTRACT FROM AUTHOR]

Rencently Chow, Huang, Li and Zhou proposed discrete forms of the Fokker-Planck equations on a finite graph. As a primary step, they constructed Riemann metrics on the graph by endowing it with some kinds of weight. In this paper, we reveal the relation between these Riemann metrics and the Euclidean metric, by showing that they are locally equivalent. Moreover, various Riemann metrics have this property provided the corresponding weight satisfies a bounded condition. Based on this, we prove that the two-side Łojasiewicz inequality holds near the Gibbs distribution with Łojasiewicz exponent 1 2 1 2. Then we use it to prove the solution of the discrete Fokker-Planck equation converges to the Gibbs distribution with exponential rate. As a corollary of Łojasiewicz inequality, we show that the two-side Talagrand-type inequality holds under different Riemann metrics. [ABSTRACT FROM AUTHOR]

We present a simple algorithm that perfectly samples configurations from the unique Gibbs measure of a spin system on a potentially infinite graph G. The sampling algorithm assumes strong spatial mixing together with subexponential growth of G. It produces a finite window onto a perfect sample from the Gibbs distribution. The run-time is linear in the size of the window, in particular it is constant for each vertex. [ABSTRACT FROM AUTHOR]