Kinetic Theory Of Dilute Gases Research Articles

Is the Boltzmann Equation Reversible? A Large Deviation Perspective on the Irreversibility Paradox

We consider the kinetic theory of dilute gases in the Boltzmann–Grad limit. We propose a new perspective based on a large deviation estimate for the probability of the empirical distribution dynamics. Assuming Boltzmann molecular chaos hypothesis (Stosszahlansatz), we derive a large deviation rate function, or action, that describes the stochastic process for the empirical distribution. The quasipotential for this action is the negative of the entropy when the conservation laws are verified, as should be expected. While the Boltzmann equation appears as the most probable evolution, corresponding to a law of large numbers, the action describes a genuine reversible stochastic process for the empirical distribution, in agreement with the microscopic reversibility. As a consequence, this large deviation perspective gives the expected meaning to the Boltzmann equation and explains its irreversibility as the natural consequence of limiting the physical description to the most probable evolution. More interestingly, it also quantifies the probability of any dynamical evolution departing from solutions of the Boltzmann equation. This picture is fully compatible with the heuristic classical view of irreversibility, but makes it much more precise in various ways. We also explain that this large deviation action provides a natural gradient structure for the Boltzmann equation.

Determination of the potential energy surfaces of some hydrocarbon refrigerants and their gas transport properties via semi-empirically based assessment

Abstract Effective and isotropic potentials of three hydrocarbon refrigerants were determined from an iterative inversion procedure, including ethylene (R1150), isobutane (R600a), and neopentane (R601b). By employing the inverted potentials, predictions of transport properties were performed with the aid of the kinetic theory of dilute gases. The viscosity coefficient, self-diffusion coefficient, and thermal diffusion factor of pure R1150, R600a, and R601b were presented over the temperature range from 273.15 K to 973.15 K at low density. Comparisons of the theoretically calculated values with the literature data were discussed. It is shown that the potentials obtained by the inversion scheme were able to predict accurate transport properties for these slightly polar ozone-safe hydrocarbon refrigerants.

Accounting for rotational non-equilibrium effects in subsonic DSMC boundary conditions

From study of the kinetic theory of dilute gases, it becomes clear that the subsonic, implicit boundary conditions which are commonly used in direct simulation Monte Carlo simulations of micro-scale channel flows do not take any form of rotational non-equilibrium into account. The consequences of not considering non-equilibration of the rotational mode are discussed and a new form of the boundary conditions which attempt to take this effect into account are proposed. It is shown that for a highly compressible test case involving these implicit boundaries in the literature that a degree of rotational non-equilibrium is present. The changes in the recovered macroscopic properties when the new boundary is applied in the same case include an improved rotational temperature distribution and a slightly lower velocity at the channel exit. A further test case exhibiting no rotational non-equilibrium was performed and it is shown that the macroscopic properties which are recovered are essentially the same for both boundaries. The results indicate that the newly proposed outlet boundary condition increases the range of applicability of these types of boundary conditions, without adding any significant computational expense.

Calculations of the thermophysical properties of binary mixtures of noble gases at low density from ab initio potentials

Interaction potentials determined ab initio from quantum mechanics were used to calculate the thermophysical properties of six binary mixtures of noble gases: helium–neon, helium–argon, helium–krypton, neon–argon, neon–krypton and argon–krypton. From the ab initio potentials, the second pressure virial coefficient, the binary diffusion coefficient and the thermal diffusion factor were computed for the considered binary mixtures at low density in the temperature range 100–5000 K employing well-established formulas in statistical thermodynamics and the kinetic theory of dilute gases. In all cases, while not as good as that for pure noble gases, close agreement between experimental data and the theoretically calculated values shows that ab initio potentials are reliable for the calculation of accurate thermophysical properties of binary mixtures of noble gases over a wide temperature range.

Heat conduction in relativistic neutral gases revisited

The kinetic theory of dilute gases to first order in the gradients yields linear relations between forces and fluxes. The heat flux for the relativistic gas has been shown to be related not only to the temperature gradient but also to the density gradient in the representation where number density, temperature and hydrodynamic velocity are the independent state variables. In this work we show the calculation of the corresponding transport coefficients from the full Boltzmann equation and compare the magnitude of the relativistic correction.

Reference Viscosity of Argon at Low Density in the Temperature Range from 290 K to 680 K

An all-quartz oscillating-disk viscometer of very high precision was used to measure the temperature dependence of the viscosity of argon in the limit of zero density. The measurements were based on a single calibration value at 298.15 K, which was calculated theoretically using an accurate ab initio pair potential for argon and the kinetic theory of dilute gases. The uncertainty of the experimental data is conservatively estimated to be 0.15 % temperature increasing to 0.2 % 680 K. The new data, as well as viscosities determined in 2007 at the National Institute of Standards and Technology in the range from 200 K to 400 K, agree excellently within ±0.1 % theoretically calculated viscosity of argon at zero density. On the contrary, the widely accepted viscosity data recommended by Kestin et al. (J Chem Phys 56:4119, 1972) deviate by as much as 0.9 %

Properties of Helium, Nitrogen and He-N2 Binary Gas Mixtures

An extensive review is conducted and a comprehensive database is compiled for the transport and thermodynamic properties of helium and nitrogen and their binary mixtures at pressures up to 40 MPa and temperatures from 200 to 1500 K. These gases are attractive working fluids in a closed Brayton cycle for energy conversion in space reactor power systems and terrestrial power plants. Semi-empirical correlations are developed for calculating the properties of the He-N 2 binary mixtures based on the Chapman-Enskog kinetic theory for dilute gases and the application of the law of corresponding states. The correlations accurately account for the effects of pressure and temperature and are in excellent agreement with the compiled properties database.

Properties of noble gases and binary mixtures for closed Brayton Cycle applications

A review is conducted of the properties of the noble gases, helium, neon, argon, krypton and xenon, and their binary mixtures at pressures from 0.1 to 20 MPa and temperatures up to 1400 K. An extensive database of experimental measurements is compiled and used to develop semi-empirical properties correlations. The correlations accurately account for the effects of pressure and temperature on the thermodynamic and transport properties of these gases for potential uses in space (∼2 MPa and up to 1400 K) and terrestrial (∼7.0 MPa and up to 1200 K) applications of Closed Brayton Cycle (CBC). The developed correlations are based on the Chapman–Enskog kinetic theory for dilute gases, and on the application of the law of corresponding states to account for the dependence of properties on pressure. The correlations use the critical temperature and density of the gases as scaling parameters, and their predictions are compared with the compiled database. At temperatures ⩾400 K and pressures ⩽2 MPa in CBC space power systems, He and Ne, and the binary mixtures of He–Xe and He–Kr with molecular weights ⩽40 g/mole behave essentially like a perfect gas, and the error of neglecting the effect of pressure on their compressibility factor, specific heats and transport properties is ⩽1%. At a typical operating pressure of 7.0 MPa and up to 1200 K in terrestrial CBC power plants, neglecting the effect of pressure can result in ∼4% error in the properties of noble gases and the binary mixtures of He–Xe and He–Kr with molecular weights ⩽40 g/mole, and as much as 20% error for pure argon. Therefore, when operating at pressures >2.0 MPa and/or using noble gases or binary mixtures with molecular weights > 40 g/mole, the present correlations should be used to accurately predict the thermodynamic and transport properties.

Integral collision kernel for the growth of aerosol particles

Integral collision kernel is elucidated using experimental results for titania, silica and alumina nanoparticles synthesized by FCVD process, and titania submicron particles synthesized in a tube furnace reactor. The integral collision kernel was obtained from a particle number balance equation by the integration of collision rates from the kinetic theory of dilute gases for the free-molecule regime, from the Smoluchowski theory for the continuum regime, and by a semi-empirical interpolation for the transition regime between the two limiting regimes. Comparisons have been made on particle size and the integral collision kernel, showing that the predicted integral collision kernel agreed well with the experimental results in Knudsen number range from about 1.5 to 20.

Reference Correlation of the Viscosity of Propane

A new representation of the viscosity of propane includes a zero-density correlation and an initial-density dependence correlation based on the kinetic theory of dilute gases and on the Rainwater–Friend theory. The higher density contributions of the residual viscosity in the representation are formed by a combination of double polynomials in density and reciprocal temperature, and a free-volume term with a temperature-dependent close-packed density. The full surface correlation is based on a set of primary experimental data selected as a result of a critical assessment of the available information from 37 original viscosity studies. The review refers to 96 citations altogether. The validity of the representation extends from the triple point to 600 K and 100 MPa in accordance with the modified Benedict–Webb–Rubin equation of state. The uncertainty of the representation varies from ±0.4% for the viscosity of the dilute gas phase between room temperature and 600 K, to about ±2.5% for the range 100–475 K up to about 30 MPa, and to about ±4% for points outside this range. Tables of the viscosity according to the representative equations at selected temperatures and pressures and along the saturation line provide easy reference as well as the validation of computer codes.

On the causality of a dilute gas as a dissipative relativistic fluid theory of divergence type

The dissipative relativistic fluid theories of divergence type are the simplest theories which are physically consistent and have a well posed-hyperbolic-initial value formulation, since they can be constructed from a single scalar function chi and a dissipation-source tensor Iab, both of them functions of fluid variables. In this work we find the expression for this generating function for the case of a dilute gas using only the knowledge of an equilibrium fluid state, which is known from the kinetic theory of dilute gases. The generating function is obtained by imposing some conditions on the divergence theory, related to the symmetry and trace of the tensor of the fluxes. These conditions come naturally from kinetic theory, and are needed to correctly describe a dilute gas. We prove that in the neighbourhood of the equilibrium states, these divergence type equations for a dilute gas are causal for Boltzmann, Fermi or Bose equilibrium distribution functions.

Correlational kinetics I. General theory

A kinetic theory for dilute gases with short and long range interactions is derived. Not only instantaneous but also retarded interactions are considered. To treat correlations in an appropriate manner a 2-particle Boltzmann density is chosen as the basic quantity of the theory. Thus, the presented theory is the starting point for the derivation of a correlational thermodynamics.

Fifty years of kinetic theory

Abstract While the kinetic theory of dilute gases based on Boltzmann's 1872 equation was essentially completed by Chapman and Enskog around 1915, a concentrated effort to generalize the Boltzmann equation systematically to higher densities to obtain expansions of the transport coefficients in powers of the density was only started about 1945. Around 1965 it was realized, however, that this was impossible and that non-analytic terms in the density would occur. This ended the classical period of kinetic theory and inaugurated the modern era at about the same time as this occurred in the theory of phase transitions. New phenomena for which a kinetic, i.e., molecular, theory were given, include: the long time (tail) correlations in equilibrium time correlation functions related to the transport coefficients and later the long spatial correlations of density fluctuations in non-equilibrium stationary states. In addition a kinetic theory has been developed for the transport properties of dense gases and liquids. The theory is based on identifying the relevant collision processes as suggested by computer simulations of dense hard sphere fluids and by neutron scattering experiments of atomic liquids: ring and cage collisions. The theory has been applied to dense simple liquids as well as to concentrated colloidal suspensions. Finally, some open problems and current investigations are discussed.

A Low‐Pressure Extension of the Stokes‐Einstein Relationship

AbstractAn empirical relation between diffusion coefficients and viscosities is proposed which interpolates between the kinetic theory of dilute gases and the Stokes‐Einstein equation of liquids. This extended Stokes‐Einstein‐relationship is tested with experimental examples. It is useful for studies of reaction processes over large density ranges.

A simple kinetic theory model of reactive collisions of rigid nonspherical molecules

The classical kinetic theory for dilute gases of rigid convex molecules, as developed by Hoffman (1969), is now applied to the calculation on the bimolecular rate coefficient, the energy-dependent reaction cross section σR(E), and the orientation-dependent differential cross section, for general diatom–diatom reactions. Incorporated in the theory are the angular momentum and the convex shape of the colliding molecules, as well as the dependence of the barrier height upon mutual orientation. Several simple collision systems are considered, including that of two reactive ellipsoidal molecules. For atom–diatom scattering, it is found that, in the post-threshold region (E≳E0), σR(E) has quadratic and higher-order terms in E−E0 but no linear term. Like σR, the differential cross section depends sensitively upon the shape of the colliding molecules, as well as upon the angle-dependent threshold energy. For the near-spherical case, one obtains simple formulas that display explicitly the dependence of the cross sections and the rate coefficient upon the pair-anisotropy parameter λ. In the spherical limit (λ=0), the results reduce to those for the steric model of Levine and Bernstein (1984) and of Smith (1982).

Collision integrals for soft, nonspherical molecules in terms of spherical Ω* integrals

A new formulation of the classical kinetic theory of dilute gases composed of molecules with rotational structure is given which closely parallels that for structureless molecules. The matrix elements of the collision operator are expressed as integral moments of a generalized collision cross section 𝒥. It is shown that a suitable approximation for the momentum dependence of 𝒥 leads to expressions for the matrix elements which can be written in terms of the familiar spherical Ω* integrals.

An experimental test of the Boltzmann equation: Argon

Recent theoretical work has shown that the kinetic theory of dilute gases, as expressed in the Boltzmann equation and the Chapman-Enskog method of solution, is not rigorously based in the sense that the Boltzmann equation has not been shown rigorously to follow from the laws of classical or quantum mechanics. The widely-held view that the theory is justified by agreement with experiment is not firmly based, since no stringent test has ever been made. This paper describes such a test, in which an independently and precisely determined potential function is used to calculate transport properties for argon, which are found to be in excellent agreement with the best modern measurements.

On Symmetric Multicomponent Diffusion Coefficients

The symmetric set of isotropic multicomponent diffusion coefficients introduced by Curtiss in the context of the kinetic theory of dilute gases are given a general definition and shown to possess general symmetry properties by virtue of Onsager reciprocal relations. Their relationships to Onsager coefficients, Curtiss–Hirschfelder diffusion coefficients, and Bearman–Kirkwood multicomponent friction coefficients are established and employed in stability considerations to deduce results which include the vanishing of Curtiss–Hirschfelder thermal diffusion coefficients in dilute solution as a consequence of the second law of thermodynamics.

Kinetic Theory of Polyatomic Gases

A classical-mechanical kinetic theory of dilute gases is developed for molecules with arbitrary degrees of freedom and interacting according to an arbitrary potential. The Boltzmann equation for the truncated singlet distribution function is derived by application of a general form of the divergence theorem, and the equilibrium solution is obtained from an H theorem for the fluid. The solution of the Boltzmann equation by a procedure of the Chapman—Enskog type is described for the case of molecules having translational and rotational degrees of freedom. Formulas for the transport properties of the gas are obtained in the form of multidimensional integral functionals of the intermolecular potential function.

Kinetic Theory of Nuclear Spin Relaxation in Gases

A theory of nuclear spin relaxation in linear molecules is developed for dilute gases and mixtures. The main assumptions used are those often employed in the kinetic theory of dilute gases: (1) binary collisions, (2) no correlation between the effects of successive binary collisions, and (3) classical mechanics for the molecular translational and rotational motion. The calculations result in expressions relating the spin relaxation times to the rotational angular momentum transfered by collisions. Using the results, it is possible to calculate the spin relaxation times for any given intermolecular potential which includes angle-dependent terms.