Boltzmann Formula Research Articles

ФИЗИКАНЫ ЖАНА АСТРОНОМИЯНЫ ОКУТУУДА ЖАНА ИЗИЛДӨӨДӨ МАТЕМАТИКАНЫН ОРДУ

This scientific article is devoted to the ways of using mathematics in teaching and researching the subjects of physics and astronomy. Newton's first polynomial was used in mathematical modeling of the emission of electron and hole color centers in solid state physics. Shannon's formula was used to determine information entropy, and the logarithmic function of Boltzmann's formula was used to determine statistical entropy. In astronomy, trigonometric functions and stereographic transformations are used to determine celestial coordinates and transform very large angles. This research paper will help students in their study of these disciplines. Particular attention is paid to the close connection between mathematics and astronomy and physics.

Investigation of Angle‐Dependent Shubnikov‐de Haas Oscillations in Topological Insulator Bismuth

The current article investigates the band structure in the presence and absence of spin‐orbit coupling (SOC), examines the Z2 invariants, and investigates the detailed angle‐dependent magneto‐transport of up to 10 T (Tesla) and down to 2 K for the bismuth crystal. The out‐of‐plane field‐dependent magnetoresistance (MR) is positive and is huge to the order of ≈104% at 2 K and 10 T. On the contrary, the longitudinal (in‐plane) field‐dependent MR is relatively small and is negative. The thermal activation energy is also estimated by using the Boltzmann formula from resistivity versus temperature measurement under applied transverse magnetic fields. The topological nature of Bi is confirmed by Z2 invariant calculation using density functional theory (DFT). PBESol bands show trivial but hybrid functional (HSE) bands show non‐trivial topology being present in Bismuth. This article comprehensively studies the dependence of MR oscillations upon the angle between the applied field and the current. The observed oscillations fade away as the angle is increased. This article is an extension of our previous work on bismuth (J. Sup. Novel Mag. 2023, 36, 389), in which a comprehensive analysis of its structural and micro‐structural properties is conducted along with its transport behavior in an applied transverse magnetic field.

Melting entropy of crystals determined by electron-beam-induced configurational disordering.

Upon melting, the molecules in a crystal explore numerous configurations, reflecting an increase in disorder. The molar entropy of disorder can be defined by Boltzmann's formula ΔSd = Rln(Wd), where Wd is the increase in the number of microscopic states, so far inaccessible experimentally. We found that the Arrhenius frequency factor A of the electron diffraction signal decay provides Wd through an experimental equation A = AINTWd, where AINT is an inelastic scattering cross section. The method connects Clausius and Boltzmann experimentally and supplements the Clausius approach, being applicable to a femtogram quantity of thermally unstable and biomolecular crystals. The data also showed that crystal disordering and crystallization of melt are reciprocal, both governed by the entropy change but manifesting in opposite directions.

Multicyclic Norias: A First-Transition Approach to Extreme Values of the Currents

For continuous-time Markov chains we prove that, depending on the notion of effective affinity F, the probability of an edge current to ever become negative is either 1 if F<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F< 0$$\\end{document} else ∼exp-F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim \\exp - F$$\\end{document}. The result generalizes a “noria” formula to multicyclic networks. We give operational insights on the effective affinity and compare several estimators, arguing that stopping problems may be more accurate in assessing the nonequilibrium nature of a system according to a local observer. Finally we elaborate on the similarity with the Boltzmann formula. The results are based on a constructive first-transition approach.

The Boltzmann distribution function for equilibrium fluctuations

A complex of issues related to the problem of proving the Boltzmann formula for the probability density (distribution) w(η) of the values of a certain set of parameters η of an equilibrium system is investigated. The classical Boltzmann formula expresses this distribution through the non-equilibrium free energy of the system F (η) , which depends on the mentioned parameters. This is possible because after the occurrence of fluctuations, the system finds itself in a non-equilibrium state, which evolves further to equilibrium. The specified parameters are chosen according to the problem under consideration. In the theory of phase transitions, they are called the order parameters. Questions under consideration include definition and construction of the free energy F (η) of a non-equilibrium system and distribution w(η ) for it in the microscopic theory. The approaches of Landau, Leontovich, and Peletminsky are discussed. It is proposed to investigate the results for states in the vicinity of equilibrium. The leading ideas of the research are considering the non-equilibrium state as being realized in the presence of an appropriate external field, using the Gibbs formula for the non-equilibrium system entropy, and applying the Boltzmann formula as a definition of the free energy of the non-equilibrium system. The article is a continuation of the authors' works and sets the task of clarifying some of their statements as well as simplifying and clarifying the calculations. Among other things, the following are discussed: the formula for the expansion of the free energy of the system in powers of the field, the simplification of the distribution w(η ) calculation, the normalization of the approximate expressions for the distribution w(η) , the possibilities of studying the free energy of the equilibrium system using our expression for the effective Landau Hamiltonian, the refinement of the calculation of the non-equilibrium free energy of a spatially inhomogeneous system, the investigation of new types of effective interactions with the Landau Hamiltonian.

Synthetic dimensions in optical cavities and their analogies to two-dimensional materials

A method to produce synthetic dimensions in an optical cavity is presented. The system is obtained by coupling two one-dimensional optical cavities by means of a moving mirror. Its translation generates a new synthetic dimension resulting in a transmittance pattern that corresponds to the energy dispersion of electrons in a two-dimensional lattice: in this case a strained triangular lattice. We elucidate the analogy between these two systems by relating the distribution of transmittances of the optical cavity to the density of states of the two-dimensional lattice and the Bragg diffraction modes to the Van Hove singularities. Our mapping makes the coupled optical cavity a simulator for lattice Hamiltonians found in solid-state physics, providing an easier alternative platform to access some of their properties, as for example, the electronic conductivity, which is found here using the Boltzmann formula. Moreover, it is proved that the truly synthetic behavior appears only when the two cavities have an irrational length ratio as modes resonant on one cavity are never resonant in the other. Mathematically, wave phase differences act independently as they are given by a function that behaves as a pseudorandom number generator. This provides an elegant way to study synthetic dimensions without the need to move the central mirror, i.e., in a single, fixed geometry device. Thus, our work establishes a bridge between the fields of optical cavities, synthetic dimensions, and two-dimensional materials.

A reasonable approach to describe the atom distributions and configurational entropy in high entropy alloys based on site preference

The Boltzmann formula was commonly employed to describe the configurational entropy of high entropy alloys (HEAs) based on the assumption of ideal solid solution, however, the entropy might rather be called composition entropy cautiously due to the site preference on the sublattice was omitted. Here we proposed a reasonable and general approach beyond the Boltzmann formula to describe the site occupying fractions (SOFs) of atoms on sublattices and thus the configurational entropy of HEAs. For this purpose, the thermodynamic function of the chemical reaction of complex HEA involving SOFs synthesized from the corresponding pure elements was computed using the sublattice model, where the density functional perturbation theory (DFPT) was employed to calculate the relevant thermodynamic properties at definite temperature. Three typical high entropy alloys with different phase structures were quantatively calculated and graphically demonstrated. The site preferences exist due to the different atom characters and sublattice environments. For example, Ni atoms only occupy the 1a sublattice, while Co, Cr, and Fe atoms mainly occupy the 3c sublattice in FCC_CoCrFeNi HEA, and the site preferences tend to be weakened considerably with the increase of heat treatment temperature. The configurational entropies were thus re-defined based on the sublattice information and SOFs. The calculated configurational entropies for FCC_CoCrFeNi HEA are 7.95, 9.10, and 9.49 J/mol·K at 473K, 973K, and 1473K, respectively, while those for BCC_AlCoCrFeNi are 11.68, 13.18, and 13.34 J/mol·K, respectively, and for HCP_HfScTiZr HEA are 11.22, 11.39, and 11.46 J/mol·K, respectively, all the configurational entropies are less than the calculated values using Boltzmann formula based on the composition. From the atom distributing configuration based on SOFs, the short range or the local ordering behaviors were explored further by statistically analyzing the coordination number among the same type of atoms, and we found that the coordination number is far less than the coordination number of the corresponding pure metal with FCC, BCC or HCP structure, and the size of the cluster formed from the same type of atoms is considerably small. The quantative and graphical characterizations of HEAs are essential to give a clue for the high resolution experiment at the atomic level, as well as provide the indispensable structural information used to predict the lattice distortion and diverse properties further.

Fundamental formalism of statistical mechanics and thermodynamics of negative kinetic energy systems

The solutions of a particle’s Dirac equation contains a negative kinetic energy (NKE) branch. Such an energy spectrum has an upper limit but no lower limit, so that the system with this spectrum, called NKE system, is of negative temperature. Fundamental formulas of statistical mechanics and thermodynamics of NKE systems are presented. All the formulas have the same forms of those of positive kinetic energy (PKE) systems. Almost all thermodynamic quantities, except entropy and specific heat, have a contrary sign compared to those of PKE systems. Specially, pressure is negative and its microscopic mechanism is given. Entropy is always positive and Boltzmann entropy formula remains valid. The three laws of thermodynamics remain valid, as long as the thermodynamic quantities have a negative sign. Negative temperature Carnot engine can work between two negative temperatures. Since the NKE levels need not be fully filled, it is argued that the concept of Dirac’s Fermion Sea can be totally abandoned.

Study of entropy-diffusion relation in deterministic Hamiltonian systems through microscopic analysis.

Although an intimate relation between entropy and diffusion has been advocated for many years and even seems to have been verified in theory and experiments, a quantitatively reliable study and any derivation of an algebraic relation between the two do not seem to exist. Here, we explore the nature of this entropy-diffusion relation in three deterministic systems where an accurate estimate of both can be carried out. We study three deterministic model systems: (a) the motion of a single point particle with constant energy in a two-dimensional periodic potential energy landscape, (b) the same in the regular Lorentz gas where a point particle with constant energy moves between collisions with hard disk scatterers, and (c) the motion of a point particle among the boxes with small apertures. These models exhibit diffusive motion in the limit where ergodicity is shown to exist. We estimate the self-diffusion coefficient of the particle by employing computer simulations and entropy by quadrature methods using Boltzmann's formula. We observe an interesting crossover in the diffusion-entropy relation in some specific regions, which is attributed to the emergence of correlated returns. The crossover could herald a breakdown of the Rosenfeld-like exponential scaling between the two, as observed at low temperatures. Later, we modify the exponential relation to account for the correlated motions and present a detailed analysis of the dynamical entropy obtained via the Lyapunov exponent, which is rather an important quantity in the study of deterministic systems.

The transport mechanisms at localized states of thin films of GexAsySe1-x-y chalcogenide glasses under off-equilibrium conditions

The resistance of chalcogenide glass GexAsySe1–x–y film is difficult to be observed due to the electrons pinned near the Fermi level, the large electron binding energy, and the large film resistance. Because the resistance is too large, the test method of Hall effect cannot be used to investigate the electrical properties for chalcogenide glass. In this work, the resistance relaxation states of a series of GexAsySe1–x–y films with different components are observed by reducing the resistance under non-equilibrium temperature conditions from 300 to 440 K, and then the local state electron transport mechanism is investigated. It is interesting that the electrical properties parameters show three relaxation states: 1) The conventional trend is that the conductivity decreases with the increase of temperature. 2) One particular trend is that the resistance presents a wavy shape as the temperature increases. 3) The most special trend is that as the temperature is raised during the conductivity measurements, the conductivity increases before decreases after having reach a certain temperature. Computer calculation results show that the Boltzmann fit formula is more suitable than simply Boltzmann formula for GexAsySe1–x–y chalcogenide film in terms of a low-mobility semiconductor with deep Fermi level and hopping conduction. Through calculations, it is found that three relaxation states are decided by the density of the localized states at Fermi level.

State space of a black hole and soft hair

In this article, we reflect on a problem of a black hole entropy. The main point of the article is that the black hole horizon should be treated as a boundary as well as the boundary at infinity. To make things more concrete, we apply the general ideas to the extremal Oliva-Tempo-Troncoso black hole and construct the corresponding Hilbert space using near-horizon hair. After creating the state space by using the proposed construction, we identify the natural candidates for the microstates responsible for the black hole entropy. The correct value of the black hole entropy is reproduced by counting the number of distinct microstates and applying the Boltzmann formula.

Information Across the Ecological Hierarchy

The ecosystem is a theatre upon which is presented, in various degrees and at differing scales, a drama of constraint and information vs. disorganization and entropy. Concerning biology, most think immediately of genomic information. It strongly constrains the form and behavior of individual species, but its influence upon community structure is indeterminate. At the community level, information acts as a formal cause behind regular patterns of development. Community structure is an amalgam of information and entropy, and the Gibbs–Boltzmann formula departs from the thermodynamic sense of entropy. It measures only the extreme that entropy might reach if the elements of the system were completely independent. A closer analogy to physical entropy in systems with interactions is the conditional entropy—the amount by which the Shannon measure is reduced after the information in the constraints among elements has been subtracted. Finally, at the whole ecosystem level, in communities that inhabit mostly fixed physical environments (e.g., landscapes or seabeds), the distributions of plants and animals appear to be independent both of causal mechanisms and trophic controls, and assume instead forms that maximize the overall entropy of dispersal.

Heat capacity calculation of the CuAlS2 semiconductor compound using the Lambert W function

ABSTRACTIn term of the Lambert W function, simple and reliable analytical expression for the estimation of the specific heat capacity of the CuAlS2 semiconductor is proposed by using the Lambert Boltzmann distribution. The obtained results have been compared with the corresponding experimental results in the 100-300 K temperature range. The estimated results are in good agreement with the experimental results over the entire temperature range. However, accurate values of the heat capacity cannot be calculated using the standard Boltzmann formula.

OPUS‐SSF: A side‐chain‐inclusive scoring function for ranking protein structural models

We introduce a side-chain-inclusive scoring function, named OPUS-SSF, for ranking protein structural models. The method builds a scoring function based on the native distributions of the coordinate components of certain anchoring points in a local molecular system for peptide segments of 5, 7, 9, and 11 residues in length. Differing from our previous OPUS-CSF [Xu et al., Protein Sci. 2018; 27: 286-292], which exclusively uses main chain information, OPUS-SSF employs anchoring points on side chains so that the effect of side chains is taken into account. The performance of OPUS-SSF was tested on 15 decoy sets containing totally 603 proteins, and 571 of them had their native structures recognized from their decoys. Similar to OPUS-CSF, OPUS-SSF does not employ the Boltzmann formula in constructing scoring functions. The results indicate that OPUS-SSF has achieved a significant improvement on decoy recognition and it should be a very useful tool for protein structural prediction and modeling.

Hawking Radiation of Relativistic Particles from Black Strings

We study Hawking radiation of relativistic particles from uncharged and charged black strings in 3+1 dimensions in detail. We use the method of quantum tunneling in the framework of the Hamilton–Jacobi approach. We show that the radial function of the action allows calculating the tunneling rate of the emitted relativistic particles. Using the Boltzmann formula, we derive the Hawking temperatures of uncharged and charged black strings. We discuss the influence of the temporal contribution to the tunneling rate.

Quantum and Thermodynamic Estimation of Mesostate Behaviour of Alkyl Benzoic Acids in Dielectric Medium: Comparative Study

Quantum and thermodynamic estimation of mesostate behaviour of p-n-alkyl benzoic acid with pentyl (PBA), hexyl (HBA) groups has been studied in dielectric medium (benzene). The complete neglect differential overlap (CNDO/2) method has been implemented for the charges, dipole moment and its components analysis. The long-range intermolecular interactions have been examined using the modified Rayleigh–Schrodinger perturbation method coupled with multicentred multipole expansion method. For understanding the short-range interactions, a ‘6-exp’ potential function has been used. The interaction energy values have been used to estimate the probability of each configuration in a dielectric medium (benzene) using the Maxwell–Boltzmann formula at room temperature (300 K) and at nematic–isotropic transition temperatures. Thermodynamic view has been presented based on Helmholtz free energy and entropy. A correlation has been made between quantum statistical and thermodynamic data to estimate the mesostate behaviour of the title molecules.

RNA3DCNN: Local and global quality assessments of RNA 3D structures using 3D deep convolutional neural networks.

Quality assessment is essential for the computational prediction and design of RNA tertiary structures. To date, several knowledge-based statistical potentials have been proposed and proved to be effective in identifying native and near-native RNA structures. All these potentials are based on the inverse Boltzmann formula, while differing in the choice of the geometrical descriptor, reference state, and training dataset. Via an approach that diverges completely from the conventional statistical potentials, our work explored the power of a 3D convolutional neural network (CNN)-based approach as a quality evaluator for RNA 3D structures, which used a 3D grid representation of the structure as input without extracting features manually. The RNA structures were evaluated by examining each nucleotide, so our method can also provide local quality assessment. Two sets of training samples were built. The first one included 1 million samples generated by high-temperature molecular dynamics (MD) simulations and the second one included 1 million samples generated by Monte Carlo (MC) structure prediction. Both MD and MC procedures were performed for a non-redundant set of 414 RNAs. For two training datasets (one including only MD training samples and the other including both MD and MC training samples), we trained two neural networks, named RNA3DCNN_MD and RNA3DCNN_MDMC, respectively. The former is suitable for assessing near-native structures, while the latter is suitable for assessing structures covering large structural space. We tested the performance of our method and made comparisons with four other traditional scoring functions. On two of three test datasets, our method performed similarly to the state-of-the-art traditional scoring function, and on the third test dataset, our method was far superior to other scoring functions. Our method can be downloaded from https://github.com/lijunRNA/RNA3DCNN.

Heat Capacity Calculation of Some Semiconductor Compounds Using Lambert W Function

Based on the Lambert-Boltzmann distribution, a simple and efficient expression for the determination of the heat capacity of some semiconductor compounds is proposed. The new expression in exact and algebraic closed form allows to calculate the heat capacity of any semiconductor compound. Its high precision is proved by comparing the calculated results against the experimental results reported in the literature for some semiconductor compounds such as Mg2Pb, SeSn, SeSn2,GaAs, InAs, and ZnO in the 260-360 K temperature range. However, the heat capacity can not be deduced by using the standard Boltzmann formula.

Implicit Temporal Discretization and Exact Energy Conservation for Particle Methods Applied to the Poisson–Boltzmann Equation

We report on a new multiscale method approach for the study of systems with wide separation of short-range forces acting on short time scales and long-range forces acting on much slower scales. We consider the case of the Poisson–Boltzmann equation that describes the long-range forces using the Boltzmann formula (i.e., we assume the medium to be in quasi local thermal equilibrium). We develop a new approach where fields and particle information (mediated by the equations for their moments) are solved self-consistently. The new approach is implicit and numerically stable, providing exact energy conservation. We test different implementations that all lead to exact energy conservation. The new method requires the solution of a large set of non-linear equations. We consider three solution strategies: Jacobian Free Newton Krylov, an alternative, called field hiding which is based on hiding part of the residual calculation and replacing them with direct solutions and a Direct Newton Schwarz solver that considers a simplified, single, particle-based Jacobian. The field hiding strategy proves to be the most efficient approach.

Modulation of membrane potential and ion concentration of isolate ellipsoidal cell exposed to static electric field

Based on the model setting is carried out to describe the interaction between static electric field and cell and the calculation of the transmembrane potential, and the Nernst formula and the Boltzmann formula, the method which to analyze the influence of the static field on the quantity across ellipsoid-cell membrane is investigated by using the theorem of electromagnetic induction. Furthermore, numerical studies were used to detect and verify the transition of ion exchange across the membrane. Some interesting results are approached as follows. The exchange of ions across the membrane and fluctuation of membrane potential are dependent on the position, orientation of electric field applied on the membrane of cell and the cell-shape. With the increasing of the field angle (between the direction of applied field and the direction of ellipsoid semi-major axis) and the semi-axis ratio ρ , it is found that the maximal ratio of ion concentration on the side of the ellipsoid-cell will reduce when external field is applied on the cell membrane, and its positions θ max will increase. Under a certain cell superficial area or volume, θ max is increased as ρ goes up. The larger field intensity for the electrostatic field with the noise, the smaller signal to noise ratio, as a result, the rate of relative change of the ion concentration will increase. The mechanism for transition in transmembrane ions could be that media or cell can be polarized when appropriate electric field is applied on ellipsoid cell even noise is considered.