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New intermolecular potential energy surfaces (PESs) for the quintet, triplet, and singlet states of two rigid oxygen (O2) molecules in their triplet ground electronic states were developed. Quintet interaction energies were obtained for 896 O2–O2 configurations by supermolecular coupled cluster (CC) calculations at levels up to CC with single, double, triple, and perturbative quadruple excitations [CCSDT(Q)] with unrestricted Hartree–Fock (UHF) reference wave functions. Corrections for scalar relativistic effects were calculated as well. Triplet interaction energies were obtained by combining the quintet interaction energies with accurate estimates for the differences between the quintet and triplet energies obtained at the UHF-CCSD(T) level of theory. Here, we exploited the fact that the triplet state is almost identical to the readily accessible "broken-symmetry" state, as shown by Valentin-Rodríguez et al. [J. Chem. Phys. 152, 184304 (2020)]. The singlet interaction energies were estimated from the quintet and triplet interaction energies by employing the Heisenberg Hamiltonian description of the spin splittings. The three PESs are represented analytically by site–site models with five sites per molecule and anisotropic site–site interactions. To validate the PESs, we calculated at temperatures from 55 to 2000 K the second virial coefficient using statistical thermodynamics and the shear viscosity, thermal conductivity, and self-diffusion coefficient in the dilute gas phase using the kinetic theory of molecular gases. The calculated property values are in excellent agreement with the most accurate experimental data from the literature. Therefore, we also propose new reference correlations for the investigated properties based solely on the calculated values. [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]

AbstractBetween the two world wars, a group of Italian mathematicians, statisticians and economists set out to use the tools of statistical mechanics to interpret the Pareto income distribution curve as a probability distribution. Models based on Boltzmann's thermal equilibrium were independently developed to find an economic variable, comparable to the kinetic energy of gas theory, that could explain the heavy tail of the Pareto curve. The various avenues explored eventually led them back to Pareto's much debated explanation, namely that the hyperbolic distribution of income was due to the unequal distribution of talent. [ABSTRACT FROM AUTHOR]

The self-diffusion coefficients of carbonaceous fuels in a supercritical CO2 environment provide transport information that can help us understand the Allam Cycle mechanism at a high pressure of 300 atm. The diffusion coefficients of pure CO2 and binary CO2/CH4 and CO2/C2H6 at high temperatures (500 K~2000 K) and high pressures (100 atm~1000 atm) are determined by molecular dynamics simulations in this study. Increasing the temperature leads to an increase in the diffusion coefficient, and increasing the pressure leads to a decrease in the diffusion coefficients for both methane and ethane. The diffusion coefficient of methane at 300 atm is approximately 0.012 cm2/s at 1000 K and 0.032 cm2/s at 1500 K. The diffusion coefficient of ethane at 300 atm is approximately 0.016 cm2/s at 1000 K and 0.045 cm2/s at 1500 K. The understanding of diffusion coefficients potentially leads to the reduction in fuel consumption and minimization of greenhouse gas emissions in the Allam Cycle. [ABSTRACT FROM AUTHOR]

In the Standard Model of gaseous stars, temperature is primary both in the initiation of thermonuclear reactions to form heavier elements and the emission of radiation. These processes have been described using thermodynamic expressions. However, within any given thermodynamic relation, not only must units balance on each side, but so must thermodynamic character. Temperature, whether or not equilibrium conditions are established, must always be intensive in macroscopic thermodynamics, and mass must be extensive. This ensures that the laws of thermodynamics are respected. The theory of temperatures and nuclear reactions within gaseous stars is constructed from the kinetic theory of an ideal gas, by which temperature is introduced, in combination with gravitational and Coulomb forces. The resulting thermodynamic relations impart a nonintensive character to temperature and a nonextensive character to mass. Consequently, the theory of nuclear reactions in gaseous stars is invalid. Deprived of the only theoretical means by which the Standard Model justifies stellar nuclear reactions, the theory of gaseous stars is not viable. The most reasonable alternative rests in lattice confinement fusion and the recognition that the stars are comprised of condensed matter, namely metallic hydrogen. [ABSTRACT FROM AUTHOR]

Unlike the single species gases, the transport coefficients such as Fick, Soret, Dufour coefficients arise in the hydrodynamic limit of multi-species gas mixtures. To the best of the authors’ knowledge, no multi-component relaxational models is reported that produces all these values correctly. In this paper, we establish the existence of unique stationary mild solutions to the BGK models for gas mixtures which produces the correct Fick coefficients in the Navier–Stokes limit for inert gases (Brull in Eur J Mech B 33:74–86, 2012), and for reactive gases (Brull and Schneider in Commun Math Sci 12(7):1199–1223, 2014) in a unified manner. [ABSTRACT FROM AUTHOR]

Dissipation processes derived from the kinetic theory of gases (shear viscosity and heat conduction) are employed to examine the solar wind that interacts with planetary ionospheres. The purpose of this study is to estimate the mean free path of wave-particle interactions that produce a continuum response in the plasma behavior. Wave-particle interactions are necessary to support the fluid dynamic interpretation that accounts for the interpretation of various features measured in a solar wind–planet ionosphere region; namely, (i) the transport of solar wind momentum to an upper ionosphere in the presence of a velocity shear, and (ii) plasma heating produced by momentum transport. From measurements conducted in the solar wind interaction with the Venus ionosphere, it is possible to estimate that in general terms, the mean free path of wave-particle interactions reaches λH ≥ 1000 km values that are comparable to the gyration radius of the solar wind particles in their Larmor motion within the local solar wind magnetic field. Similar values are also applicable to conditions measured by the Mars ionosphere and in cometary plasma wakes. Considerations are made in regard to the stochastic trajectories of the plasma particles that have been implied from the measurements made in planetary environments. At the same time, it is as possible that the same phenomenon is applicable to the interaction of stellar winds with the ionosphere of exoplanets, and also in regions where streaming ionized gases reach objects that are subject to rotational motion in other astrophysical problems (galactic flow–plasma interactions, black holes, etc.). [ABSTRACT FROM AUTHOR]

The book "Metaphors in the History of Economic Thought: Crises, Business Cycles and Equilibrium" is a collection of essays that explores the use of metaphors in economics. The book focuses on how economists have used metaphors to discuss economic crises, business cycles, and equilibrium. The authors argue that metaphors play a significant role in shaping economic discourse and can reveal important insights about economic theories. The book covers a range of topics, including the historical development of economic metaphors and their pedagogical and heuristic functions. However, the book does not extensively reference previous literature on the topic, and some chapters are based on previously published articles. Overall, the book provides a thought-provoking exploration of the role of metaphors in economics and encourages further research in this area. [Extracted from the article]

Unlike the case for classical particles, the literature on BGK type models for relativistic gas mixture is extremely limited. There are a few results in which such relativistic BGK models for gas mixture are employed to compute transport coefficients. However, to the best knowledge of authors, relativistic BGK models for gas mixtures with complete presentation of the relaxation operators are missing in the literature. In this paper, we fill this gap by suggesting a BGK model for relativistic gas mixtures for which the existence of each equilibrium coefficients in the relaxation operator is rigorously guaranteed in a way that all the essential physical properties are satisfied such as the conservation laws, the H-theorem, the capturing of the correct equilibrium state, the indifferentiability principle, and the recovery of the classical BGK model in the Newtonian limit. [ABSTRACT FROM AUTHOR]

A gas flow in a square cavity driven by a lid sliding in the direction of its line of contact with the cavity wall is considered. The steady behavior of the gas is numerically investigated based on the linearized Bhatnagar–Gross–Krook kinetic equation and the diffuse reflection boundary condition. When one applies the Stokes equation and the no-slip boundary condition to the system considered here, the flow velocity becomes multivalued at the corner between the lid and the cavity wall, and the shear stress diverges at the corner inversely proportionally to the distance from there, which is known as the so-called corner singularity. In the present work, the behavior of the gas near the corner is examined based on numerical results obtained from the kinetic theory. Although the range of the flow velocity value in the kinetic solution is limited due to the significant velocity slip near the corner, the flow velocity is, nevertheless, multivalued at the corner. The shear stress varies inversely proportionally to the distance from the corner up to the position that is a few tens of mean free paths away from there. The increase in the stress is suppressed at positions closer to the corner and its magnitude remains bounded. Thus, the total forces acting on the lid and the side cavity walls are bounded as well. Due to the distinctive behavior of the stress near the corner, the resulting nondimensional total forces behave with an unconventional rate Kn ln Kn for small Knudsen numbers Kn. [ABSTRACT FROM AUTHOR]

Integral equations, which are defined as "the equation containing an unknown function under the integral sign", have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer theory. A special case of these equations, known as the quadratic Chandrasekhar integral equation, given by x (s) = 1 + λ x (s) ∫ 0 1 s t + s x (t) d t , can be very often encountered in many applications, where x is the function to be determined, λ is a parameter, and t , s ∈ [ 0 , 1 ] . In this paper, using a fixed-point theorem, the existence conditions for the solution of Fredholm integral equations of the form χ (l) = ϱ (l) + χ (l) ∫ p q k (l , z) (V χ) (z) d z are investigated in the space C ω p , q , where χ is the unknown function to be determined, V is a given operator, and ϱ , k are two given functions. Moreover, certain important applications demonstrating the applicability of the existence theorem presented in this paper are provided. [ABSTRACT FROM AUTHOR]

In this work, I will discuss the solution of the new in a nonlinear integro-differential equation (NI-DE) form. The NI-DE is established from the nuclear transport equation, which is a linearized derivative of the equation developed by Boltzmann for the kinetic theory of gases. In this study, I will introduce the basic equations and boundary conditions for the formulation of the problem. Then, the NI-DE is established from the planar geometry problem of the neutron transport equation. In addition, I will prove the existence of a unique solution to the problem. In addition, I will use the numerical method to obtain a system of NI-DEs. Therefore, I will prove, under certain conditions, the existence of a unique solution to this system. Finally, I will present two different effective methods to solve the problem numerically, and I will discuss the results. [ABSTRACT FROM AUTHOR]

The modeling and understanding of micro- and nano-scale transport processes have raised increasing attention and extensive investigation during the past decades. In this mini-review, we aim to summarize our recent progress on the non-equilibrium thermodynamics of micro- and nano-scale flow and heat transfer. Special emphasis is put on the entropy generation at the interface, which plays a dominant role at small scale due to the strong non-equilibrium nature of particle-boundary interaction. We also prove the thermodynamic compatibility of both the macroscopic hydrodynamic equation and the non-equilibrium boundary conditions from the perspective of bulk and interfacial entropy generations respectively, as supported by the kinetic theory of microscopic particles. The present review will contribute to a clearer elaboration of thermodynamics at micro/nano-scale and its statistical mechanical demonstration, and thus will promote its further development in the future. [ABSTRACT FROM AUTHOR]

The reliable incorporation of hydrogen as a clean energy source necessitates an in-depth comprehension of the dynamics of flammable clouds resulting from potential leakage incidents. This research introduces an innovative model rooted in the principles of the kinetic theory of gases to determine the dimensions of flammable hydrogen clouds during leaks. The model integrates critical variables such as leak rate, leak direction, wind speed, and wind direction, in addition to surrounding atmospheric conditions, offering a precise forecast of hydrogen dispersion and behaviour during leakage scenarios. This modelling approach relies on two dimensionless groups derived through dimensional analysis, and the correlation between these groups has been established. Furthermore, the model incorporates three constants accounting for hydrogen buoyancy, gas dilution, and mixing, which are fine-tuned through a minimal number of Computational Fluid Dynamics (CFD) simulations. This indispensable information is crucial for the formulation of effective safety protocols, evacuation plans, and risk assessment strategies. Validation of the model against simulated data attests to its accuracy and reliability, underscoring its potential as a valuable tool for steering the secure integration of hydrogen technologies. This research significantly contributes to the advancement of safety measures in the utilisation of hydrogen as a clean energy source, providing a robust foundation for the design and implementation of precautionary measures in real-world applications. • A method for predicting hydrogen flammable cloud size based on dimensional analysis and CFD is proposed. • This method allows an improvement in the calculation efficiency while ensuring its accuracy. • A detailed comparative analysis of three prediction methods was presented in this paper. • This method is suitable for explosion risk analysis during the preliminary design phase. [ABSTRACT FROM AUTHOR]

We study and solve some class of infinite systems of algebraic equations with monotone nonlinearity and Toeplitz-type matrices. Such systems for the specific representations of nonlinearities arise in the discrete problems of dynamic theory of clopen -adic strings for a scalar field of tachyons, the mathematical theory of spatio-temporal spread of an epidemic, radiation transfer theory in inhomogeneous media, and the kinetic theory of gases in the framework of the modified Bhatnagar–Gross–Krook model. The noncompactness of the corresponding operator in the bounded sequence space and the criticality property (the presence of trivial nonphysical solutions) is a distinctive feature of these systems. For these reasons, the use of the well-known classical principles of existence of fixed points for such equations do not lead to the desired results. Constructing some invariant cone segments for the corresponding nonlinear operator, we prove the existence and uniqueness of a nontrivial nonnegative solution in the bounded sequence space. Also, we study the asymptotic behavior of the solution at . In particular, we prove that the limit at of a solution is finite. Also, we show that the difference between this limit and a solution belongs to . By way of illustration, we provide some special applied examples. [ABSTRACT FROM AUTHOR]

The shock wave and boundary layer interaction (SWBLI) plays an important role in the design of hypersonic vehicles. However, discrepancies between the numerical results of high-temperature gas dynamics and experiment data have not been fully addressed. It is believed that the rarefaction effects are important in SWBLI, but the systematic analysis of the temperature-jump boundary conditions and the role of translational/rotational/vibrational heat conductivities are lacking. In this paper, we derive the three-temperature Navier–Stokes–Fourier (NSF) equations from the gas kinetic theory, with special attention paid to the components of heat conductivity. With proper temperature-jump boundary conditions, we simulate the SWBLI in the double cone experiment. Our numerical results show that, when the three heat conductivities are properly recovered, the NSF equations can capture the position and peak value of the surface heat flux, in both low- and high-enthalpy inflow conditions. Moreover, the separation bubble induced by the separated shock and the reattachment point induced by impact between transmitted shock and boundary layer are found to agree with the experimental measurement. [ABSTRACT FROM AUTHOR]

This study investigates the dynamic modeling of a spherical vapor bubble in an infinite liquid. Initially, the liquid is in a state of saturation, but then, there is a change in ambient pressure. Unlike previous studies that assumed vapor saturation, our model fully considers the metastable fluid state of subcooled vapor and superheated liquid. The theoretical model is based on the state equation of vapor and the kinetic theory of gases, allowing the visualization of the effects of superheating and subcooling on the bubble pressure and the phase transition rate. The accuracy of the numerical solutions for the bubble collapse and growth is confirmed by experimental results using water as the working fluid. Two approaches of heat transfer simplification are validated in the numerical solution, and an adhesion coefficient within the range of 0.1–1 is recommended for the calculation. Additionally, this study provides insights into how the metastable fluid state affects the bubble pressure and phase transition, especially in the early stages of bubble collapse. Furthermore, the conservation of vapor mass inside the bubble is also demonstrated. [ABSTRACT FROM AUTHOR]

Heat and mass transfer across the interface between liquid and vapor is studied by means of molecular dynamics simulation. Two scenarios are considered to access the interface resistivities, specifying either the evaporation rate or the temperature gradient. Spatially resolved profiles of density, temperature, chemical potential, pressure tensor elements, and hydrodynamic velocity are sampled with large-scale molecular dynamics simulations to elucidate the structural and dynamic properties across the interface under non-equilibrium conditions. The employed interaction model is appropriate for simple fluids, like argon, while its thermodynamic properties in bulk phases are fully known. Most of the temperature range from the triple point to the critical point is investigated, varying the heat flux and the particle flux over one to two orders of magnitude. Different approaches are followed to determine the interface resistivities, and their results are compared to literature data and kinetic gas theory. It is found that the interface resistivities are a sole function of the interface temperature and are independent of the chemical potential gradient or the temperature gradient. This also holds for its thickness and surface tension up to the very large gradients that are typically imposed in molecular dynamics simulations. It stands to reason that this is also the case under the presence of gradients with a magnitude that is technically relevant and thus much smaller. [ABSTRACT FROM AUTHOR]

Alternative conceptions become obstacles in physics. However, it is difficult to find instruments that can identify students' alternative conceptions, especially in gases kinetic theory (DIGaKiT). The purpose of this research was to development of diagnostic instrument of DIGaKiT in identifying students' alternative conceptions by Rasch analysis model. The research method used the defining, designing, developing, and disseminating (4D). The samples are 31 students (12 male students and 19 female students, their ages were typically 16 years old) at one of the senior high schools at Belitung. Rasch analysis was used to identify the validity, reliability, and distribution of students' alternative conceptions. The result is that the level of validity and reliability of the instrument is in a good category. Meanwhile, alternative conceptions of the kinetic theory of gases can be identified in all questions, and the questions with the highest alternative conceptions are questions with code Q11 (77%) and the lowest are questions with codes Q1, Q5, and Q6 (4%). Therefore, teachers must design learning processes that can reduce students' alternative conceptions of the kinetic theory of gases material. [ABSTRACT FROM AUTHOR]

The wave kinetic equation predicts the averaged temporal evolution of a continuous spectral density of waves either randomly interacting or scattered by the fine structure of a medium. In a wide range of systems, the wave kinetic equation is derived from a fundamental equation of wave motion, which is symmetric through time reversal. By contrast, the corresponding wave kinetic equations is time irreversible: Its solutions monotonically increase an entropy-like quantity. A similar paradox appears whenever one makes a mesoscopic description of the evolution of a very large number of microscopic degrees of freedom, the paradigmatic example being the kinetic theory of dilute gas molecules leading to the Boltzmann equation. Since Boltzmann, it has been understood that a probabilistic understanding solves the apparent paradox. More recently, it has been understood that the kinetic description itself, at a mesoscopic level, should not break time reversal symmetry (Bouchet in J Stat Phys 181(2):515-550, 2020). The time reversal symmetry remains a fundamental property of the mesoscopic stochastic process: Without external forcing, the path probabilities obey a detailed balance relation with respect to an equilibrium quasipotential. The proper theoretical or mathematical tool to derive fully this mesoscopic time reversal stochastic process is large deviation theory: A large deviation principle uncovers a time reversible field theory, characterized by a large deviation Hamiltonian, for which the deterministic wave kinetic equation appears as the most probable evolution. Its irreversibility appears as a consequence of an incomplete description, rather than as a consequence of the kinetic limit itself, or some related chaotic hypothesis. This paper follows Bouchet (J Stat Phys 181(2):515-550, 2020) and a series of other works that derive the large deviation Hamiltonians of the main classical kinetic theories, for instance, Guioth et al. (J Stat Phys 189(20):20, 2022) for homogeneous wave kinetics. We propose here a derivation of the large deviation principle in an inhomogeneous situation, for the linear scattering of waves by a weak random potential. This problem involves microscopic scales corresponding to the typical wavelengths and periods of the waves and mesoscopic ones which are the scales of spatial inhomogeneities in the spectral density of both the random scatterers and the wave spectrum, and the time needed for the random scatterers to alter the wave spectrum. The main assumption of the kinetic regime is a large separation of these microscopic and mesoscopic scales. For the sake of simplicity, we consider a generic model of wave scattering by weak disorder: the Schrödinger equation with a random potential. We derive the path large deviation principle for the local spectral density and discuss its main properties. We show that the mesoscopic process obeys a time reversal symmetry at the level of large deviations. [ABSTRACT FROM AUTHOR]

Based on gas kinetic theory, a linear stability analysis method for low-speed rarefied flows was developed by Zou et al. ["A new linear stability analysis approach for microchannel flow based on the Boltzmann Bhatnagar–Gross–Krook equation," Phys. Fluids 34, 124114 (2022) and "A novel linear stability analysis method for plane Couette flow considering rarefaction effects," J. Fluid Mech. 963, A33 (2023)]. In the present study, we extended the method to high-speed rarefied flows using the Bhatnagar–Gross–Krook model. The Chebyshev spectral method is employed to discretize physical space, and the Gauss–Hermite and fourth-order Newton–Cotes quadrature methods are used to discretize velocity space. The fourth-order Newton–Cotes quadrature method was found to have sufficient accuracy for the stability analysis, laying the foundation for future research on hypersonic flows. The stability analysis of compressible rarefied Couette flow showed that acoustic modes are reflected between the wall and the relative sonic line, and the variation in their phase speed and growth rate with the wavenumber is not affected by the Mach number (Ma) and the Knudsen number (Kn). Increasing Kn has a stabilizing effect on both the acoustic and viscous modes, but as Ma increases, the attenuation rate of each mode's growth rate gradually decreases. In subsonic and sonic flows, the least stable viscous mode dominates in the case of small numbers. As Kn increases, the viscous mode gradually dominates over all wavenumber ranges considered in subsonic flow. In sonic flow, mode 1 is dominant in the region beyond the range of small wavenumbers. In supersonic flow, mode 2 is the least stable in the large wavenumber ranges, while mode 1 is the least stable in other wavenumber ranges. At a fixed wavenumber, as Kn increases, the decay rate of the growth rate of mode 2 is the highest. Additionally, under different Knudsen numbers, the growth rates of mode 1, mode 2, and the least stable viscous mode monotonically increase with an increase in Ma, with mode 2 showing the most significant increase. [ABSTRACT FROM AUTHOR]

The study of gas mixture flows through micro- and macro-channels remains a very attractive area for theorists and experimentalists worldwide, mainly due to their great practical applicability in several aspects of science and industry. The present work includes a comparative study between the linear (McCormack model) and the nonlinear (DSMC method) kinetic theories of binary gas mixture flows through channels over a wide range of the involved parameters. The results show that the McCormack model is a reliable kinetic model for predicting the gas mixture flow behavior. Specific criteria with respect to the applicability range of the linear kinetic theory of short and long capillaries are proposed. The analysis shows that the separation phenomenon remains strong even in the case of small pressure drops. The applicability range of the linear kinetic theory for binary gas mixture flows driven by large pressure drops is obtained to be smaller compared to that of single gases but still wide enough covering a wide range of pressure ratios. Furthermore, the present work shows that the long capillary theory remains a very powerful tool for studying the gas mixture flow behavior under weak and strong nonequilibrium conditions. This work provides a kinetic database of the linear kinetic data (as supplementary material), which until today is still missing from the literature, and as it is shown throughout this work, it can be used far beyond the restrictions defined by the linear kinetic theory. [ABSTRACT FROM AUTHOR]

For a system of nonlinear integral equations on the semiaxis, we study a boundary value problem whose matrix kernel has unit spectral radius. This boundary value problem has applications in various areas of physics and biology. In particular, such problems arise in the dynamical theory of -adic strings for the scalar field of tachyons, in the mathematical theory of spread of epidemic diseases, in the kinetic theory of gases, and in the theory of radiative transfer. The questions of the existence, absence, and uniqueness of a nontrivial solution of this boundary value problem are discussed. In particular, it is proved that a boundary value problem with a zero boundary conditions at infinity has only a trivial solution in the class of nonnegative and bounded functions. It is also proved that if at least one of the values at infinity is positive, then this problem has a convex nontrivial nonnegative bounded and continuous solution. At the end of this paper, examples of the matrix kernel and nonlinearity are provided that satisfy all the conditions of the proved theorems. [ABSTRACT FROM AUTHOR]

We show that the kinetic theory of gases developed by Clausius, Maxwell, and Boltzmann can be understood as stochastic thermodynamics. This comprehension is based on the recognition that the dynamics used by Maxwell to demonstrate his distribution of velocities and by Boltzmann to derive his fundamental equation is a stochastic dynamics. An essential feature of the transition rates devised by Maxwell and used by Boltzmann is the preservation of energy and momentum. Since the dynamics is stochastic and not deterministic, we may say retrospectively that the objections to the kinetic theory raised by the paradox of irreversibility is resolved or is immaterial. Using these transition rates we obtain the Kolmogorov equation associated with the stochastic dynamics We show that it reduces to the Boltzmann equation if the velocities of distinct molecules can be considered to be statistically independent. We also show how the Boltzmann H-theorem is related to the production of entropy.

In teaching mathematics to first-year undergraduates, and thus in the appropriate calculus textbooks, the task of calculating an integral that satisfies a specific first-order or second-order recurrence relation often appears. These relations are obtained mainly by applying the method of integration by parts. Calculating such integrals is usually tedious, especially for an integer n > 2, time-consuming, and presents the possibility of making a large number of errors when computing involves multiple iterative steps. In [1], it is shown that in two cases (Theorems 2.1. and 2.3), the process of calculating integrals satisfying first-order recurrence relations can be performed quickly using easily memorised closed-form formulas for corresponding primitive functions. The question can rightly be asked whether there is a faster way to calculate other integrals of this type. In this paper, our goal is to give an affirmative answer to such a question, though without convering all situations. Since each recurrence relation is equivalent to a difference equation of the same order, the calculation of integrals mentioned above can be reduced to solving the corresponding difference equations. Since every first-order or second-order linear difference equation is solvable, it follows that for every integral which can be reduced to a first- order or second-order recurrence formula, it is possible to find corresponding primitive functions directly. Sometimes such a procedure is much faster than iterative solving of the integral. Closed-form formulas for the integrals discussed in the following sections are not unknown (see [2]). However, here our goal is to present the idea of computing indefinite integrals using difference equations. We will discuss it in more detail in Section 2. In Section 3, we discuss the application of the results obtained to calculate several improper integrals and the application of some of them in different sciences. An exciting example of such an application is the integral , which in the case n = 1 is used in the kinetic theory of gases, particularly in the Maxwell-Boltzmann distribution of gas molecules by energies (see Remark 4). Also, we compare the formulas obtained by the method of difference equations with the formulas obtained using Wolfram Alpha software (see Remark 5). [ABSTRACT FROM AUTHOR]

Some factors, such as pressure and temperature, affect the rate of chemical reactions. In addition, the activation energy barrier must be overcome for the reaction to be initiated. It can be preferred to overcome this barrier by using catalysts and preheating. The catalyst ensures that it obtains the energy to react quickly by transferring it to the reactants. Similarly, the translational, vibrational, and rotational energy levels of reactants can be increased by preheating. According to the kinetic molecular theory of gases, preheating increases the kinetic energies of the gases and the speed of their collision, so the reaction takes place faster. This study theoretically investigates possible reactions of methane that can occur with the effect of only vibrational energy levels. The vibrational excitation of the molecules affects the reaction rates, and the activation barrier is overcome with lower energies. Using laser-based techniques makes the excitation of well-defined vibrational modes possible. This study investigated inelastic collisions of a methane molecule with well-characterized energy levels in infrared spectroscopy with some gases and the vibrational energy transfers that occur in these collisions. The methane molecule is the simplest form of a molecular structure consisting of more than three atoms of hydrogen atoms, which play an essential role in combustion chemistry. It shows that C⸺H stretch excitation increases the reaction rate of methane (CH4) molecules. [ABSTRACT FROM AUTHOR]

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There are two possible thermal transport mechanisms at liquid–gas interfaces, namely, evaporation/condensation (i.e., heat transfer by liquid–vapor phase change at liquid surfaces) and heat conduction (i.e., heat exchange by collisions between gas molecules and liquid surfaces). Using molecular dynamics (MD) simulations, we study thermal transport across the liquid–vapor interface of a model n-dodecane (C12H26) under various driving force conditions. In each MD simulation, we restrict the thermal energy to be transferred across the liquid–vapor interface by only one mechanism. In spite of the complex intramolecular interactions in n-dodecane molecules, our modeling results indicate that the Schrage relationships, which were shown to give accurate predictions of evaporation and condensation rates of monatomic fluids, are also valid in the prediction of evaporation and condensation rates of n-dodecane. In the case of heat conduction at the liquid–vapor interface of n-dodecane, the interfacial thermal conductance obtained from MD simulations is consistent with the prediction from the kinetic theory of gases. The fundamental understanding of thermal transport mechanisms at liquid–gas interfaces will allow us to formulate appropriate boundary conditions for continuum modeling of heating and evaporation of small fuel droplets. [ABSTRACT FROM AUTHOR]

This contribution presents a comprehensive overview of lattice Boltzmann models for non-ideal fluids, covering both theoretical concepts at both kinetic and macroscopic levels and more practical discussion of numerical nature. In that context, elements of kinetic theory of ideal gases are presented and discussed at length. Then a detailed discussion of the lattice Boltzmann method for ideal gases from discretization to Galilean invariance issues and different collision models along with their effect on stability and consistency at the hydrodynamic level is presented. Extension to non-ideal fluids is then introduced in the context of the kinetic theory of gases along with the corresponding thermodynamics at the macroscopic level, i.e. the van der Waals fluid, followed by an overview of different lattice Boltzmann based models for non-ideal fluids. After an in-depth discussion of different well-known issues and artifacts and corresponding solutions, the article finishes with a brief discussion on most recent applications of such models and extensions proposed in the literature towards non-isothermal and multi-component flows. [ABSTRACT FROM AUTHOR]

The ES-BGK model is a generalized version of the BGK model of the Boltzmann equation designed to reproduce the correct Prandtl number in the Navier–Stokes limit. We prove the existence and uniqueness of mild solutions to the ES-BGK model for polyatomic molecules when the total energy of the initial data is not necessarily finite. [ABSTRACT FROM AUTHOR]

The results of viscosity measurements at moderate densities on the two gaseous mixtures carbon dioxide–nitrogen and ethane–methane including the pure gases between 253.15 K and 473.15 K, originally performed by Humberg et al. at Ruhr University Bochum, Germany, using a rotating-cylinder viscometer between 0.1 MPa and 2.0 MPa, were employed to determine the interaction viscosity, η 12 (0) , and the product of molar density and diffusion coefficient, (ρ D 12) (0) , each in the limit of zero density. The isothermal viscosity data were evaluated by those authors with density series restricted to the second order at most to derive the zero-density viscosities and initial density viscosity coefficients, η mix (0) and η mix (1) , for the mixtures, as well as, η i (0) and η i (1) ( i = 1 , 2 ), respectively, for the pure gases. Humberg et al. have already compared their η mix (0) and η i (0) data for carbon dioxide–nitrogen and ethane–methane with corresponding viscosity values theoretically computed for the nonspherical potentials of the intermolecular interaction. Now we employed η mix (0) and η mix (1) as well as η i (0) and η i (1) in two procedures to derive η 12 (0) values. For this, we needed A 12 ∗ values (ratio between effective cross-sections of viscosity and diffusion). But the second procedure applying the initial density viscosity coefficients η mix (1) and η i (1) failed to yield reasonable η 12 (0) values. The first procedure should provide the best results when it is possible to use A 12 ∗ values computed for the nonspherical potential. The effect is comparatively small if η 12 (0) is determined. But if (ρ D 12) (0) is calculated from η 12 (0) using A 12 ∗ values for the nonspherical potential, the impact is several percent. Moreover, the experimentally based η 12 (0) and (ρ D 12) (0) data agree with theoretically calculated values for the nonspherical potentials. [ABSTRACT FROM AUTHOR]

This paper analyzes the mixing of gases in a plane channel at rarefied conditions. The direct simulation Monte Carlo method is employed to simulate gas mixing in parallel mixers working at different Knudsen numbers and having different values of wall accommodation coefficient. Results show that the normal-to-wall component of the mole fraction gradient may have the same sign as the corresponding component of the diffusive mass flux vector near the diffuse solid walls in contrast to the predictions of Fick's law for continuum conditions. This non-continuum behavior, which is called "inversion" in the present study, will become more pronounced at higher Knudsen numbers, whereas it will become less evident for smaller wall accommodation coefficients. To confirm that the observed phenomenon is consistent with the basic physical laws governing the rarefied gas dynamics and it is not an artifact of the numerical method, a new analytical model based on the kinetic theory of gases is developed for the parallel mixers that have diffuse walls and are working in the free-molecular regime. Excellent agreement is observed between the analytical and direct simulation Monte Carlo results in the free molecular flow regime. Both methods predict the occurrence of inversion near the diffuse walls at highly rarefied flow conditions. [ABSTRACT FROM AUTHOR]

The evolution of a gas can be described by different mathematical models depending on the scale of observation. A natural question, raised by Hilbert in his sixth problem, is whether these models provide mutually consistent predictions. In particular, for rarefied gases, it is expected that the equations of the kinetic theory of gases can be obtained from molecular dynamics governed by the fundamental principles of mechanics. In the case of hard sphere gases, Lanford (1975) has shown that the Boltzmann equation does indeed appear as a law of large numbers in the low density limit, at least for very short times. The aim of this paper is to present recent advances in the understanding of this limiting process. [ABSTRACT FROM AUTHOR]

A class of nonlinear singular integral equations on the whole line is considered. This class of equations for specific representations of nonlinearity and kernel arises in dynamic theory of p-adic strings, in mathematical theory of the pandemic spread within the framework of modified Diekmann-Kapper model, in the theory of radiative transfer in nonhomogeneous medium and kinetic theory of gases. In this paper we will investigate and reveal new qualitative properties of the solution of the above mentioned equations. Relying on these properties as well as some geometric inequalities for convex downward functions we will prove the uniqueness of the solution in the class of bounded functions whose infimum is a positive number. At the end we will present examples of such equations, satisfying all restrictions of the proven statements. [ABSTRACT FROM AUTHOR]

Extensive research has been carried out in the past to estimate the bulk viscosity of dense monatomic fluids; however, little attention has been paid to estimate the same in the dilute gas regime. Here, the term 'dilute' is used in reference to kinetic theory of gases and refers to low density conditions where average inter-molecular spacing is much greater than the size of the molecules. In this work, we have performed molecular dynamics simulations and used the Green–Kubo method to make precise estimations for the bulk viscosity of argon gas at atmospheric conditions. The investigated temperature and pressure range is 300 to 750 K and 0.5 to 1.5 bar respectively. It is observed that the estimated bulk viscosity is O (1 0 − 10) Pa s, which is several orders of magnitude smaller than that of other diatomic and polyatomic gases, but nonetheless, not an absolute zero as typically assumed. It implies that Stokes' hypothesis is true for dilute monatomic gases only in an approximate rather than absolute sense. The variation of bulk viscosity with pressures at constant temperatures has also been studied and is found to be of quadratic nature. The obtained bulk viscosity values are also reported in the reduced Lennard-Jones units to enable extension of the present results to other noble gases as well. It has also been observed that the bulk viscosity of Lennard-Jones monatomic gases becomes temperature independent at very low densities. [ABSTRACT FROM AUTHOR]

The Luria–Delbrück mutation model is a cornerstone of evolution theory and has been mathematically formulated in a number of ways. In this paper, we illustrate how this model of mutation rates can be derived by means of classical statistical mechanics tools—in particular, by modeling the phenomenon resorting to methodologies borrowed from classical kinetic theory of rarefied gases. The aim is to construct a linear kinetic model that can reproduce the Luria–Delbrück distribution starting from the elementary interactions that qualitatively and quantitatively describe the variations in mutated cells. The kinetic description is easily adaptable to different situations and makes it possible to clearly identify the differences between the elementary variations, leading to the Luria–Delbrück, Lea–Coulson, and Kendall formulations, respectively. The kinetic approach additionally emphasizes basic principles which not only help to unify existing results but also allow for useful extensions. [ABSTRACT FROM AUTHOR]

This Review concerns recent results on the quantitative study of convergence toward the stationary state for spatially inhomogeneous kinetic equations. We focus on analytical results obtained by means of certain probabilistic techniques from the ergodic theory of Markov processes. These techniques are sometimes referred to as Harris-type theorems. They provide constructive proofs for convergence results in the L1 (or total variation) setting for a large class of initial data. The convergence rates can be made explicit (for both geometric and sub-geometric rates) by tracking the constants appearing in the hypotheses. Harris-type theorems are particularly well-adapted for equations exhibiting non-explicit and non-equilibrium steady states since they do not require prior information on the existence of stationary states. This allows for significant improvements of some already-existing results by relaxing assumptions and providing explicit convergence rates. We aim to present Harris-type theorems, providing a guideline on how to apply these techniques to kinetic equations at hand. We discuss recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, giving some perspectives on potential extensions to nonlinear equations. [ABSTRACT FROM AUTHOR]