Methods Of Statistical Mechanics Research Articles

Deep‐red Emitting Copper(I) Indenediyltrisphosphine Complexes with Minimized Skeletal Vibrations and Configurational Disorder

Abstract: The full width at half maximum (FWHM) of emission spectra, which plays an important role in determining color purity, may not always receive sufficient attention in the design of emissive materials. Particularly for the red emitter, the traditional focus has been on emission maxima, yet broad FWHM values can significantly change the perceived color. For example, the red color (lem = 616–677 nm) emitted from reported Cu(I) complexes is perceived as orange to yellow if FWHM is large. To reduce FWHM, we incorporated a strained and rigid metalaphosphadicyclopenta[a,f]pentalene motif into Cu(I) complexes using trisphosphine ligands featuring a 1H‐indene‐2,3‐diyl backbone (ITP). Herein, we present the synthesis, structure, and emission properties of ITP‐MX and the congeners, showcasing genuinely deep‐red emission with narrow FWHM values of 56 nm. These materials exhibit color coordinates close to pure red on the CIE diagram, unlike reported broader‐emitting counterparts. Measurement of the entropy of disorder of the emissive crystal by a recently reported statistical mechanical method revealed a quantitative correlation between FWHM and the increase in the number of microstates (the degree of freedom) of the crystals and suggested that mechanical stress can increase the entropy of the crystal, which results in emission broadening.

Deep‐red Emitting Copper(I) Indenediyltrisphosphine Complexes with Minimized Skeletal Vibrations and Configurational Disorder

Abstract: The full width at half maximum (FWHM) of emission spectra, which plays an important role in determining color purity, may not always receive sufficient attention in the design of emissive materials. Particularly for the red emitter, the traditional focus has been on emission maxima, yet broad FWHM values can significantly change the perceived color. For example, the red color (lem = 616–677 nm) emitted from reported Cu(I) complexes is perceived as orange to yellow if FWHM is large. To reduce FWHM, we incorporated a strained and rigid metalaphosphadicyclopenta[a,f]pentalene motif into Cu(I) complexes using trisphosphine ligands featuring a 1H‐indene‐2,3‐diyl backbone (ITP). Herein, we present the synthesis, structure, and emission properties of ITP‐MX and the congeners, showcasing genuinely deep‐red emission with narrow FWHM values of 56 nm. These materials exhibit color coordinates close to pure red on the CIE diagram, unlike reported broader‐emitting counterparts. Measurement of the entropy of disorder of the emissive crystal by a recently reported statistical mechanical method revealed a quantitative correlation between FWHM and the increase in the number of microstates (the degree of freedom) of the crystals and suggested that mechanical stress can increase the entropy of the crystal, which results in emission broadening.

Pattern phase transition of spin particle lattice system

To better understand the pattern phase transition of both physical and biological systems, we investigate a two-dimensional spin particle lattice system using statistical mechanics methods together with XY model governed by Hamiltonian equations of motion. By tweaking the coupling strength and the intensity of the generalization field, we observe phase transitions among four patterns of spin particles, i.e., vortex, ferromagnet, worm and anti-ferromagnet. In addition, we analyze the effect of space boundaries on the formations of vortex and worm. Considering the inherent dynamics of individual particles, we revealed the forming mechanism of such phase transitions, which provides a new perspective for understanding the emergence of phase transition of spin particles systems.

Kinetics in the Two-Level System with Strong Time-De-pendent Coupling of Its States to the Phonon Bath: Spin-Boson Description

Using the methods of nonequilibrium statistical mechanics, the master equation for the density matrix of an open dissipative quantum system is obtained under conditions, when the coupling between the electronic states of the system and the nuclear displacements in it is controlled by the alternating field. A time-dependent polaron transformation is proposed, which made it possible to solve kinetic equations using an expansion in a parameter characterizing transitions between “phonon-dressed” electronic states of the system. As an example, a mechanism is shown that can control the kinetics in a two-level system by applying a periodic force field to electron-phonon coupling.

On the Analysis of Wealth Distribution in the Context of Infectious Diseases.

A mathematical model is established to investigate the economic effects of infectious diseases. The distribution of wealth among two types of agents in the context of the epidemic is discussed. Using the method of statistical mechanics, the evolution of the entropy weak solutions for the model of the susceptible and the infectious involving wealth density functions is analyzed. We assume that as time tends to infinity, the wealth density function of the infectious is linearly related to the wealth density function of the susceptible individuals. Our results indicate that the spreading of disease significantly affects the wealth distribution. When time tends to infinity, the total wealth density function behaves as an inverse gamma distribution. Utilizing numerical experiments, the distribution of wealth under the epidemic phenomenon and the situation of wealth inequality among agents are discussed.

Dynamical distribution of continuous service time model involving non-Maxwellian collision kernel and value functions

Abstract The distribution of continuous service time in call centers is investigated. A non-Maxwellian collision kernel combining two different value functions in the interaction rule are used to describe the evolution of continuous service time, respectively. Using the statistical mechanical and asymptotic limit methods, Fokker–Planck equations are derived from the corresponding Boltzmann-type equations with non-Maxwellian collision kernels. The steady-state solutions of the Fokker–Planck equation are obtained in exact form. Numerical experiments are provided to support our results under different parameters.

Theoretical investigation of interaction measurements in liquid systems with viscosity distributions

Optical tweezers and colloid probe atomic force microscopies have been used for measurement of the force curve between a sample surface and a probe. In the measurement, the force curve is mainly obtained by using both the spring constant and the displacement of the probe (we call it the general method). However, when there is a viscosity gradient in the interfacial liquid, the general method cannot predict the force curve accurately. In this study, we compared the force curves calculated from the general method and a method considering both the viscosity gradient and the acceleration of the probe. From the comparison, we found a problem in the general method and use of both the viscosity gradient and the acceleration is important for measurement of an accurate force curve. However, the viscosity gradient in the interfacial liquid is unknown in general. Applying statistical mechanics and Brownian motion simulation, we found a fact that the force curve can be accurately determined without information of the viscosity gradient. In addition, we propose an inverse calculation method of the viscosity distribution with use of the force curve obtained from the statistical mechanics method.

A statistical investigation of thermoluminescence processes in YAG:Ce

This work applies the methods of statistical mechanics to two processes relevant to thermoluminescence; (1) the charging of traps, and (2) the discharging of traps. Assuming thermal and diffusive equilibrium, we calculate the trap ionization probabilities over a wide temperature range and for a variety of traps in YAG:Ce. The goal is to investigate whether a statistical approach yields results consistent with experimental thermoluminescence observations. We are able to reproduce with good accuracy the glow curves of several traps observed in YAG:Ce. Our calculated glow curves are shown to be consistent with systems exhibiting second order kinetics. For systems of traps that are at or near thermal and diffusive equilibrium with the conduction electrons, the results indicate that the position of a glow curve peaks is determined by two factors: the trap depths and the density of conduction electrons.

Quantum Dynamics Studies of the Li + Na2 (V = 0, j = 0) → Na + NaLi Reaction on a New Neural Network Potential Energy Surface.

The ultracold reaction offers a unique opportunity to elucidate the intricate microscopic mechanism of chemical reactions, and the Na2Li system serves as a pivotal reaction system in the investigation of ultracold reactions. In this work, a high-precision potential energy surface (PES) of the Na2Li system is constructed based on high-level ab initio energy points and the neural network (NN) method, and a proper asymptotic functional form is adopted for the long-range interaction, which is suitable for the study of cold or ultracold collisions. Based on the new NN PES, the dynamics of the Li + Na2 (v = 0, j = 0) → Na + NaLi reaction are studied in the collision energy range of 10-7 to 80 cm-1. In the high collision energy range of 8 to 80 cm-1, the dynamics of the reaction is studied using the time-dependent wave packet method and the statistical quantum mechanical (SQM) method. Comparing the results of the two methods, it is found that the SQM method provides a rough description of the product ro-vibrational state distribution but overestimates the integral cross-section values. With the decrease of collision energy, the reaction differential cross section gradually changes from forward-backward symmetric scattering to predominant forward scattering. In the low collision energy range from 10-7 to 8 cm-1, the SQM method is used to study the reaction dynamics, and the rate constant in the Wigner threshold region is estimated to be 2.87 × 10-10 cm3/s.

Effect of global shrinkage parameter of horseshoe prior in compressed sensing

This study investigates the effect of the global shrinkage parameter τ of a horseshoe prior, one of the global–local shrinkage priors, on linear regression in sparse signal processing. Statistical mechanics methods are employed to examine the accuracy of signal estimation. The phase diagram of the success and failure of signal recovery in noiseless compressed sensing with varying τ is discussed from the viewpoint of dynamic characterization of approximate message passing (AMP) as a solving algorithm and static characterization of the free-energy landscape. It is found that there exists a parameter region where the AMP algorithm can hardly recover the true signal, even though the true signal is locally stable. The analysis of the free-energy landscape also provides important insight into the optimal choice of τ.

Tracing the classical molecular nature of entropy generation

The second law of thermodynamics and the concept of positive entropy generation are analyzed following classical statistical mechanics methods. Previously, using the generalized Boltzmann–Gibbs entropy and its associated general entropy conservation relation, positive entropy generation expressions were obtained in agreement with phenomenological results and the work of Boltzmann and Gibbs. In this study, using the general approach, we formally and explicitly trace the specific entropy generation expressions to truncations in the full N-body description of the entropy state to a lower s-body description. Using higher-order superposition approximations, it is formally shown that the generalized Boltzmann–Gibbs entropy in the s-order state is always less than the corresponding Boltzmann–Gibbs entropy in the lower (s − 1)-order state. Using the general form of the entropy conservation equation, entropy generation is shown to be a required compensatory effect to ensure that all physical variables and physical processes associated with heat, work, temperature, etc., are independent of the particular entropy definition state.

High-pressure high-temperature EXAFS and Debye–Waller factor of platinum

The temperature and pressure effects on the mean-square relative displacement (MSRD) corresponding to the extended X-ray absorption fine structure (EXAFS) Debye–Waller factor of platinum have been studied based on the statistical moment method in quantum statistical mechanics. We implement numerical calculations for platinum up to pressure of 400 GPa. Examining the temperature effects on the EXAFS Debye–Waller factor at ambient pressure we show that theoretical MSRD curve is in good agreement with the previous measurement and calculations up to nearly 800 K. When pressure increases, our pressure-dependent MSRD curve follows very well with the prediction based on the numerical integration along the Hugoniot. Using the theoretical EXAFS Debye–Waller factor, we obtain the Hugoniot temperatures which are highly consistent with recent experimental results. This research advances the correlated Einstein model on the calculation of MSRDs of materials at high temperature and high pressure, specifically, it includes the zero-point vibrations at low temperature and contains the anharmonicity contributions caused by thermal lattice vibrations at high temperature.

Use of <i>q</i>-statistics for study of pulsating aurora

The non-extensive statistical mechanics method of Tsallis (or q-statistics) is first applied to study pulsating auroras, which are regularly observed in the auroral ionosphere during geomagnetic disturbances. For systems with long-range interactions, such as ionized gas or plasma, whose dynamics are primarily determined by long-range electromagnetic forces, one can expect that non-additive and non-extensive thermostatistical principles may characterize their macroscopic behavior. This paper shows that pulsating polar auroras exhibit non-extensive properties and can be described, in part, by q-statistics. It is also demonstrated that the non-extensive parameter q correlates well with the flatness coefficient and scaling index, indicating the applicability of this approach to auroral emissions. Thus, q-statistics can be used to analyze phenomena in the high-latitude region of the Earth.

Quasicrystal Structure Prediction: A Review

AbstractPredicting quasicrystal structures is a multifaceted problem that can involve predicting a previously unknown phase, predicting the structure of an experimentally observed phase, or predicting the thermodynamic stability of a given structure. We survey the history and current state of these prediction efforts with a focus on methods that have improved our understanding of the structure and stability of known metallic quasicrystal phases. Advances in the structural modeling of quasicrystals, along with first principles total energy calculation and statistical mechanical methods that enable the calculation of quasicrystal thermodynamic stability, are illustrated by means of cited examples of recent work.

Isotopic effect in thermodynamic properties of solid neon

Isotopic effects in mean-square displacement, specific heat at constant pressure and linear thermal expansion coefficient of solid neon have been considered using the statistical moment method in statistical mechanics. This approach allows us to study these effects including the anharmonicity of lattice vibrations. We implement calculations for 20Ne and 22Ne isotopes in temperature range from 0 K to the melting temperature (24.6 K) using Lennard-Jones potential to describe the interaction between Ne quantum particles. Our numerical results are compared with those of path-integral Monte Carlo simulations and previous experimental data showing the good agreement. This research presents an effective statistical approach to study isotopic effects in thermodynamic quantities of materials.

Lithium Intercalation Mechanisms in Wadsley-Roth Phases

Wadsley-Roth crystallographic shear phases are complex inorganic compounds that intercalate alkali ions at a rapid rate, allowing for fast charging lithium-ion battery electrodes with high power densities. Wadsley-Roth crystallographic shear phases consist of mxn size blocks of corner-sharing octahedra. These mxn blocks of octahedra share edges with other blocks of corner sharing octahedra, thereby forming what are referred to as crystallographic shear planes. A variety of Wadsley-Roth phases have been synthesized with differing structural features including the size of the blocks of corner-sharing transition metal octahedra, transition metal chemistry, and disorder among the transition metal sites. Using first-principles statistical mechanics methods, we have predicted the electrochemical properties of PNb9O25, TiNb2O7, and Nb2O5 (Figure 1), three important Wadsley-Roth shear phases, to explore the interplay between crystal structure, transition metal chemistry, Li site preferences, and the voltage profile. The computational approach relied on a triptych of ab initio density functional theory calculations, cluster expansion methods and finite temperature Monte Carlo methods. The calculations provide insights about lithium site preferences, strain evolution[1], and electronic structure [2, 3] as a function of state of charge in representative Wadsley-Roth phases.For the PNb9O25 compound, we find that lithium prefers to initially fill pyramidal sites at the crystallographic shear planes. It is not until higher lithium concentrations that Li begins to fill window sites at the center of the 3 by 3 octahedral blocks. This inversion in Li-Va ordering is paired with an increase of crystallographic strain and a decrease in octahedral distortions. In contrast to PNb9O25, the TiNb2O7 compound, a commercialized Wadsley-Roth phase, is characterized by Ti-Nb disorder on the transition metal sublattice. We find that the Nb, due to its higher oxidation state, prefers octahedral sites that share fewer edges with neighboring octahedra, relegating the lower oxidation state Ti cations to octahedra at the block edges. First-principles calculations and Monte Carlo simulations predict that the lithium site stability is paired to the local transition metal chemistry and ordering, with Li preferring to fill sites coordinated by TiO6 octahedra. With the use of symmetry adapted collective displacement order parameters we uncover changes in octahedral distortions that differ between the TiO6 and NbO6 octahedra upon intercalation of lithium. Furthermore, we find a similar evolution in strain upon intercalation as is predicted for PNb9O25. To study the effect of octahedral block size on electrochemical properties of Wadsley-Roth phases, we performed an additional in-depth study on Nb2O5, a Wadsley-Roth structure with 4 by 4 corner-sharing octahedral blocks. Predictions for this phase indicate that the dilute limit Li site preference changes when increasing block sizes. Unlike TiNb2O7 and PNb9O25, which have 3 by 3 octahedral block structures, lithium window sites at the center of the block are stable in Nb2O5 at the dilute limit. We attribute this to differences in octahedral distortions in the 4 by 4 and 3 by 3 blocks. Our results indicate that there are a wide variety of structural and chemical levers with which to modify and further optimize the electrochemical properties of Wadsley-Roth phases. References Saber, M., Preefer, M.B., Kolli, S.K., Zhang, W., Laurita, G., Dunn, B., Seshadri, R. and Van der Ven, A., 2021. Role of Electronic Structure in Li Ordering and Chemical Strain in the Fast Charging Wadsley–Roth Phase PNb9O25. Chemistry of Materials, 33(19), pp.7755-7766.Preefer, M.B., Saber, M., Wei, Q., Bashian, N.H., Bocarsly, J.D., Zhang, W., Lee, G., Milam-Guerrero, J., Howard, E.S., Vincent, R.C. and Melot, B.C., 2020. Multielectron redox and insulator-to-metal transition upon lithium insertion in the fast-charging, Wadsley-Roth phase PNb9O25. Chemistry of Materials, 32(11), pp.4553-4563.Baek, S.W., Preefer, M.B., Saber, M., Zhai, K., Frajnkovič, M., Zhou, Y., Dunn, B.S., Van der Ven, A., Seshadri, R. and Pilon, L., 2022. Potentiometric entropy and operando calorimetric measurements reveal fast charging mechanisms in PNb9O25. Journal of Power Sources, 520, p.230776. Figure 1

Temperature effects on thermodynamic and mechanical properties of the InP, InAs and InSb compounds

The thermodynamic and mechanical properties of the zinc-blende indium pnictides InP, InAs and InSb compounds have been investigated thanks to the statistical moment method in statistical mechanics. We have derived the analytical expressions of thermal induced atomic displacement, lattice constant, elastic moduli (Young’s modulus, bulk modulus and shear modulus) and elastic constants of the zinc-blende compounds. The difference of temperature effects on mechanical properties of InSb comparing to InP and InAs compounds has been pointed out. We show that InSb is less affected by temperature while InP changes its mechanical properties from hardness to softness quickly when the temperature increases. The advancement of this method is that it has included the anharmonic effects of thermal lattice vibrations by taking into account the higher-order atomic displacement terms.

Temperature effects on thermodynamic properties of rare-earth ytterbium

Thermodynamic quantities (including Debye temperature and frequency, atomic mean-square displacement, specific heats at constant volume and constant pressure, linear thermal expansion coefficient, and total entropy) of face-centered cubic ytterbium metal have been studied by means of the statistical moment method in statistical mechanics. This theoretical approach allows us to calculate these thermodynamic quantities taking into account the anharmonicity of lattice vibrations. Numerical calculations for ytterbium have been conducted in temperature range from 0 K to 1100 K using Lennard-Jones potential to describe the interaction between ytterbium atoms. Our derived results are compared with those of mean-field potential approach as well as previous experimental data showing the good agreement. We have shown that the atomic mean-square displacement contains the zero-point vibration (quantum effect) at low temperature. At high temperature it changes linearly with respect to temperature. Furthermore, our theoretical calculations have also pointed out the anharmonic contribution to specific heat at constant volume of ytterbium metal is negative with temperature. This research presents an effective statistical approach to study temperature effects on thermodynamic quantities of materials.

Quantifying the cohesive strength of rock materials by roughness analysis using a domain based multifractal framework

Cohesive strength or intrinsic tensile strength as well as cohesive length are two important unknowns for cohesive modelling of fracture and failure analysis of quasi-brittle materials including rocks. There is no direct method for measuring these parameters and their quantification is always challenging and controversial. In this study, a novel multifractal framework is employed to quantify the cohesive length of four different rock types including sandstone, marble, fine-grained granite and coarse-grained granite by analysing the roughness of their fracture surfaces in a wide range of length scales. On the one hand, microstructural heterogeneities of rock material at small enough length scales will cause multifractality of the roughness of its fractured surface. On the other hand, this intrinsic heterogeneity together with extrinsic features including loading and environmental conditions as well as geometrical features including shape and size of a quasi-brittle specimen or structure are forming a fracture process zone (FPZ) in front of any stress concentrators before crack propagation. Therefore, it is proposed that locating the transition from multifractality to mono-fractality of a rough rock fractured surface using the employed statistical mechanics method leads to quantifying the effective length of FPZ of a sharp crack or the cohesive length. This length is quantified for the studied rocks ranging from 0.4 to 1.1 mm. Moreover, by employing the theory of critical distances, the cohesive strength σc of these rocks is also quantified ranging from 15 to 36 MPa.

Effect of Polydispersity on the Structural and Magnetic Properties of a Magnetopolymer Composite.

When using magnetopolymer composites in high-precision industrial and biomedical technologies, the problem of predicting their properties in an external magnetic field arises. In this work, we study theoretically the influence of the polydispersity of a magnetic filler on a composite's equilibrium magnetization and on the orientational texturing of magnetic particles formed during polymerization. The results are obtained using rigorous methods of statistical mechanics and Monte Carlo computer simulations in the framework the bidisperse approximation. It is shown that by adjusting the dispersione composition of the magnetic filler and the intensity of the magnetic field at which the sample's polymerization occurs, it is possible to control the composite's structure and magnetization. The derived analytical expressions determine these regularities. The developed theory takes into account dipole-dipole interparticle interactions and therefore can be applied to predict the properties of concentrated composites. The obtained results are a theoretical basis for the synthesis of magnetopolymer composites with a predetermined structure and magnetic properties.