Hard-sphere Gas Research Articles

Hadron resonance gas model with induced surface tension

Here we present a physically transparent generalization of the multicomponent Van der Waals equation of state in the grand canonical ensemble. For the one-component case the third and fourth virial coefficients are calculated analytically. It is shown that an adjustment of a single model parameter allows us to reproduce the third and fourth virial coefficients of the gas of hard spheres with small deviations from their exact values. A thorough comparison of the compressibility factor and speed of sound of the developed model with the one and two component Carnahan-Starling equation of state is made. It is shown that the model with the induced surface tension is able to reproduce the results of the Carnahan-Starling equation of state up to the packing fractions 0.2-0.22 at which the usual Van der Waals equation of state is inapplicable. At higher packing fractions the developed equation of state is softer than the gas of hard spheres and, hence, it breaks causality in the domain where the hadronic description is expected to be inapplicable. Using this equation of state we develop an entirely new hadron resonance gas model and apply it to a description of the hadron yield ratios measured at AGS, SPS, RHIC and ALICE energies of nuclear collisions. The achieved quality of the fit per degree of freedom is about 1.08. We confirm that the strangeness enhancement factor has a peak at low AGS energies, while at and above the highest SPS energy of collisions the chemical equilibrium of strangeness is observed. We argue that the chemical equilibrium of strangeness, i.e. $\gamma_s \simeq 1$, observed above the center of mass collision energy 4.3 GeV may be related to the hadronization of quark gluon bags which have the Hagedorn mass spectrum, and, hence, it may be a new signal for the onset of deconfinement.

Impact of roughness on the instability of a free-cooling granular gas.

A linear stability analysis of the hydrodynamic equations with respect to the homogeneous cooling state is carried out to identify the conditions for stability of a granular gas of rough hard spheres. The description is based on the results for the transport coefficients derived from the Boltzmann equation for inelastic rough hard spheres [Phys. Rev. E 90, 022205 (2014)PLEEE81539-375510.1103/PhysRevE.90.022205], which take into account the complete nonlinear dependence of the transport coefficients and the cooling rate on the coefficients of normal and tangential restitution. As expected, linear stability analysis shows that a doubly degenerate transversal (shear) mode and a longitudinal ("heat") mode are unstable with respect to long enough wavelength excitations. The instability is driven by the shear mode above a certain inelasticity threshold; at larger inelasticity, however, the instability is driven by the heat mode for an inelasticity-dependent range of medium roughness. Comparison with the case of a granular gas of inelastic smooth spheres confirms previous simulation results about the dual role played by surface friction: while small and large levels of roughness make the system less unstable than the frictionless system, the opposite happens at medium roughness. On the other hand, such an intermediate window of roughness values shrinks as inelasticity increases and eventually disappears at a certain value, beyond which the rough-sphere gas is always less unstable than the smooth-sphere gas. A comparison with some preliminary simulation results shows a very good agreement for conditions of practical interest.

The Propagation of Chaos for a Rarefied Gas of Hard Spheres in the Whole Space

We discuss old and new results on the mathematical justification of Boltzmann's equation. The classical result along these lines is a theorem which was proven by Lanford in the 1970s. This paper is naturally divided into three parts. I. Classical. We give new proofs of both the uniform bounds required for Lanford's theorem, as well as the related bounds due to Illner & Pulvirenti for a perturbation of vacuum. The proofs use a duality argument and differential inequalities, instead of a fixed point iteration. II. Strong chaos. We introduce a new notion of propagation of chaos. Our notion of chaos provides for uniform error estimates on a very precise set of points; this set is closely related to the notion of strong (one-sided) chaos and the emergence of irreversibility. III. Supplemental. We announce and provide a proof (in Appendix A) of propagation of partial factorization at some phase-points where complete factorization is impossible.

Microscopic solutions of the Boltzmann-Enskog equation in the series representation

The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation.

Higher-order moment theories for dilute granular gases of smooth hard spheres

Grad’s method of moments is employed to develop higher-order Grad moment equations – up to the first 26 moments – for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff’s law while the other higher-order moments decay on a faster time scale. The nonlinear terms of the fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have a negligible effect on Haff’s law. Furthermore, an even larger Grad moment system, which includes the fully contracted sixth moment, is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of the scalar sixth-order moment into the Grad moment system has a negligible effect on Haff’s law. The constitutive relations for the stress and heat flux (i.e. the Navier–Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via the Chapman–Enskog expansion and via computer simulations. The linear stability of the homogeneous cooling state is analysed through the Grad 26-moment system and various subsystems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution, while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber, in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment theory is in excellent agreement with that obtained through existing theories.

Two-phase thermodynamic model for computing entropies of liquids reanalyzed.

The two-phase thermodynamic (2PT) model [S.-T. Lin et al., J. Chem. Phys. 119, 11792-11805 (2003)] provides a promising paradigm to efficiently determine the ionic entropies of liquids from molecular dynamics. In this model, the vibrational density of states (VDoS) of a liquid is decomposed into a diffusive gas-like component and a vibrational solid-like component. By treating the diffusive component as hard sphere (HS) gas and the vibrational component as harmonic oscillators, the ionic entropy of the liquid is determined. Here we examine three issues crucial for practical implementations of the 2PT model: (i) the mismatch between the VDoS of the liquid system and that of the HS gas; (ii) the excess entropy of the HS gas; (iii) the partition of the gas-like and solid-like components. Some of these issues have not been addressed before, yet they profoundly change the entropy predicted from the model. Based on these findings, a revised 2PT formalism is proposed and successfully tested in systems with Lennard-Jones potentials as well as many-atom potentials of liquid metals. Aside from being capable of performing quick entropy estimations for a wide range of systems, the formalism also supports fine-tuning to accurately determine entropies at specific thermal states.

Energy nonequipartition in gas mixtures of inelastic rough hard spheres: The tracer limit.

The dynamical properties of a tracer or impurity particle immersed in a host gas of inelastic and rough hard spheres in the homogeneous cooling state is studied. Specifically, the breakdown of energy equipartition as characterized by the tracer/host ratios of translational and rotational temperatures is analyzed by exploring a wide spectrum of values of the control parameters of the system (masses, moments of inertia, sizes, and coefficients of restitution). Three complementary approaches are considered. On the theoretical side, the Boltzmann and Boltzmann-Lorentz equations (both assuming the molecular chaos ansatz) are solved by means of a multitemperature Maxwellian approximation for the velocity distribution functions. This allows us to obtain explicit analytical expressions for the temperature ratios. On the computational side, two different techniques are used. First, the kinetic equations are numerically solved by the direct simulation Monte Carlo (DSMC) method. Second, molecular dynamics simulations for dilute gases are performed. Comparison between theory and simulations shows a general good agreement. This means that (i) the impact of the molecular chaos ansatz on the temperature ratios is not significant (except at high inelasticities and/or big impurities) and (ii) the simple Maxwellian approximation yields quite reliable predictions.

The explicit expression of the fugacity for weakly interacting Bose and Fermi gases

In this paper, we calculate the explicit expression for the fugacity for two- and three-dimensional weakly interacting Bose and Fermi gases from their equations of state in isochoric and isobaric processes, respectively, based on the mathematical result of the boundary problem of analytic functions --- the homogeneous Riemann-Hilbert problem. We also discuss the Bose-Einstein condensation phase transition of three-dimensional hard-sphere Bose gases.

Boltzmann kinetic equation for a strongly confined gas of hard spheres.

A Boltzmann-like kinetic equation for a quasi-two-dimensional gas of hard spheres is derived. The system is confined between two parallel hard plates separated a distance between one and two particle diameters. An entropy Lyapunov function for the equation is identified. In addition to the usual Boltzmann expression, it contains a contribution associated to the confinement of the particles. The steady properties of the system agree with equilibrium statistical mechanics results. Equations describing the energy transfer between the degrees of freedom parallel and perpendicular to the confining plates are obtained for some simple initial configurations. The theoretical predictions are compared with molecular dynamics simulation data and a good agreement is found.

Flow of a monatomic rarefied gas over a circular cylinder: Calculations based on the ab initio potential method

Two-dimensional flows of argon and helium over a circular cylinder are calculated by the Direct Simulation Monte Carlo (DSMC) method in the range of the free stream Mach number Ma∞ from 0.5 to 10 in nearly free molecular, transitional, and nearly continuum flows. In DSMC simulations, inter-particle collisions are calculated with the ab initio (AI) potential method based on the interatomic interaction potentials established in quantum mechanical calculations. It is shown that the AI potential method enables computationally efficient simulations of multidimensional rarefied gas flows without introducing semi-empirical models of collision cross sections. The calculated values of the drag CD and heat flux CQ coefficients of the cylinder for Ar and He at the same values of Ma∞, rarefaction parameter, and surface-to-free-stream temperature ratio are found to be different in less than 1%, ensuring small sensitivity of CD and CQ to the species of a monatomic gas. The simulation results obtained with the AI potential method are systematically compared with results obtained with the hard sphere (HS) molecular model. It is found that the choice of the HS diameter based on the condition of the identical viscosity of the real and HS gases at the free stream temperature ensures calculations of CD and CQ in sub- and supersonic flows at Ma∞⩽2 with errors less than 3% and 6.5%, correspondingly. For hypersonic flows, this choice of the HS diameter is unsatisfactory and results in the errors up to 7% in CD and 28% in CQ. A semi-empirical rule that defines an optimum HS diameter in super- and hypersonic flows is suggested. With the use of this rule, the HS model is capable of predicting CD and CQ with errors less than 1% and 3%, correspondingly, and also provides a good accuracy in calculations of local flow parameters.

Modified kinetic theory applied to the shear flows of granular materials

Granular materials are characterized by large collections of discrete particles, where the particle-particle interactions are significantly more important than the particle-fluid interactions. The current kinetic theory captures fairly accurately the granular flow behavior in the dilute case, when only binary interactions are significant, but is not accurate at all in the dense flow regime, where multi-particle interactions and contacts must be modeled. To improve the kinetic theory results for granular flows in the dense flow regime, we propose a Modified Kinetic Theory (MKT) model that utilizes the contact duration or cutoff time to account for the complex particle-particle interactions in the dense regime. The contact duration model, also called TC model, was originally proposed by Luding and McNamara [“How to handle the inelastic collapse of a dissipative hard-sphere gas with the TC model,” Granular Matter 1, 113 (1998)] to solve the inelastic collapse issue existing in the inelastic hard sphere model. This model defines a cutoff time tc such that dissipation is not counted if the time between two consecutive contacts is less than tc. As shown in their study, the use of a cutoff time tc can also reduce the dissipation during multi-particle contacts. In this paper we relate the TC model with the Discrete Element Method (DEM) by choosing the cutoff time tc to be the duration of contact calculated from the linear-spring-dashpot soft-sphere model of the DEM. We examine two types of granular flows, simple shear flow and the plane shear flow, and compare the results of the classical kinetic theory model, the present MKT model, and the DEM model. We show that the MKT model entails a significant improvement over the kinetic theory model for simple shear flows at inertial regimes. With the MKT model the calculations are close to the DEM results at solid fractions as high as 0.57. Even for the plane shear flows, where shear rate and solid fraction are inhomogeneous, the results of the MKT model agree very well with the DEM results.

Kinetic Theory of a Confined Quasi-Two-Dimensional Gas of Hard Spheres

The dynamics of a system of hard spheres enclosed between two parallel plates separated a distance smaller than two particle diameters is described at the level of kinetic theory. The interest focuses on the behavior of the quasi-two-dimensional fluid seen when looking at the system from above or below. In the first part, a collisional model for the effective two-dimensional dynamics is analyzed. Although it is able to describe quite well the homogeneous evolution observed in the experiments, it is shown that it fails to predict the existence of non-equilibrium phase transitions, and in particular, the bimodal regime exhibited by the real system. A critical revision analysis of the model is presented , and as a starting point to get a more accurate description, the Boltzmann equation for the quasi-two-dimensional gas has been derived. In the elastic case, the solutions of the equation verify an H-theorem implying a monotonic tendency to a non-uniform steady state. As an example of application of the kinetic equation, here the evolution equations for the vertical and horizontal temperatures of the system are derived in the homogeneous approximation, and the results compared with molecular dynamics simulation results.

Nuclear and hadron matter equation of state within the induced surface tension approach

We present a novel equation of state which is based on the virial expansion for the multicomponent mixtures with hard core repulsion. The suggested equation of state explicitly contains the surface tension which is induced by particle interaction. At high densities such a surface tension vanishes and in this way it switches the excluded volume treatment of hard core repulsion to its eigen volume treatment. The great advantage of the developed model is that the number of equations to be solved is two and it does not depend on the number of independent hard-core radii. Using the suggested equation of state we obtained a high quality fit of the hadron multiplicities measured at AGS, SPS, RHIC and ALICE energies and studied the properties of the nuclear matter phase diagram. It is shown the developed equation of state is softer than the gas of hard spheres and remains causal up to the several normal nuclear densities. Therefore, it could be applied to the neutron star interior modeling.

Thermal properties of an impurity immersed in a granular gas of rough hard spheres

We study in this work the dynamics of a granular impurity immersed in a low-density granular gas of identical particles. For description of the kinetics of the granular gas and the impurity particles we use the rough hard sphere collisional model. We take into account the effects of non-conservation of energy upon particle collision. We find an (approximate) analytical solution of the pertinent kinetic equations for the single-particle velocity distribution functions that reproduces reasonably well the properties of translational/rotational energy non-equipartition. We assess the accuracy of the theoretical solution by comparing with computer simulations. For this, we use two independent computer data sets, from molecular dynamics (MD) and from Direct Simulation Monte Carlo method (DSMC). Both approach well, with different degrees, the kinetic theory within a reasonable range of parameter values.

Kinetic equation and nonequilibrium entropy for a quasi-two-dimensional gas.

A kinetic equation for a dilute gas of hard spheres confined between two parallel plates separated a distance smaller than two particle diameters is derived. It is a Boltzmann-like equation, which incorporates the effect of the confinement on the particle collisions. A function S(t) is constructed by adding to the Boltzmann expression a confinement contribution. Then it is shown that for the solutions of the kinetic equation, S(t) increases monotonically in time, until the system reaches a stationary inhomogeneous state, when S becomes the equilibrium entropy of the confined system as derived from equilibrium statistical mechanics. From the entropy, other equilibrium properties are obtained, and molecular dynamics simulations are used to verify some of the theoretical predictions.

Transient motion of and heat transfer in a rarefied gas between plane parallel walls with different surface properties

Transient motion of and heat transfer in a rarefied gas between plane parallel walls with different surface properties are studied based on kinetic theory. It is assumed that one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary, and the transient behavior of the gas caused by a sudden heating of one of the walls is studied. The linearized Boltzmann equation for a hard-sphere molecular gas is numerically studied using the modified hybrid scheme of the characteristic coordinate and finite difference methods, to correctly describe the discontinuities in the velocity distribution function. The transient motion of the gas from an early time stage to the final time-independent state is studied over a wide range of the mean free path and the accommodation coefficient of the boundary. Between the two transient flows caused by the heating of the respective walls, the values of the heat flow on the heated wall are different, whereas those on the unheated wall coincide identically. This property, which is a consequence of the symmetric relation of the linearized Boltzmann equation, is numerically confirmed over a wide range of the mean free path. The long time behavior of the heat flow on the walls is quite similar to that of the shear stress in the Couette flow problem, whereas a distinct wavy behavior is observed in an early time stage.

Dimensional Reduction for Generalized Continuum Polymers

The Brydges-Imbrie dimensional reduction formula relates the pressure of a $d$-dimensional gas of hard spheres to a model of $(d+2)$-dimensional branched polymers. Brydges and Imbrie's proof was non-constructive and relied on a supersymmetric localization lemma. The main result of this article is a constructive proof of a more general dimensional reduction formula that contains the Brydges--Imbrie formula as a special case. Central to the proof are invariance lemmas, which were first introduced by Kenyon and Winkler for branched polymers. The new dimensional reduction formulas rely on invariance lemmas for central hyperplane arrangements that are due to M\'esz\'aros and Postnikov. Several applications are presented, notably dimensional reduction formulas for (i) non-spherical bodies and (ii) for corrections to the pressure due to symmetry effects.

Fermi energy for hard-sphere gas in porous media and potential fields

In this paper, we introduce a method for analytically calculating the Fermi energy. As an example, we study the hard-sphere (HS) gas, the gas in porous media and the gas in an external potential field or curved background. These examples are valuable for studying the spectral structure of a real gas.

Recovering Navier–Stokes Equations from Asymptotic Limits of the Boltzmann Gas Mixture Equation

This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles. Specifically the hydrodynamics limit is performed by employing different time and space scalings. The paper shows that, depending on the magnitude of the parameters which define the scaling, the macroscopic quantities (number density, mean velocity and local temperature) are solutions of the acoustic equation, the linear incompressible Euler equation and the incompressible Navier–Stokes equation. The derivation is formally tackled by the recent moment method proposed by [C. Bardos, et al., J. Stat. Phys. 63 (1991) 323] and the results generalize the analysis performed in [C. Bianca, et al., Commun. Nonlinear Sci. Numer. Simulat. 29 (2015) 240].

Thermodynamics of the low-density excluded-volume hadron gas

We consider the thermodynamics of excluded-volume particles at finite temperature and chemical potential in the low-density approximation. We assume Boltzmann statistics and study the influence of the excluded volume on an ideal gas thermodynamics at the same temperature, pressure, and number of particles. We show that considering the change of the free enthalpy due to the excluded volume, and using the Maxwell identities, one can derive relevant thermodynamic functions and parameters of multicomponent gases. The derivation is quite general, because particles may have different sizes and shapes which can also depend on their momenta. Besides its simplicity and generality, our approach has the advantage of eliminating the transcendental equations occurring in earlier studies. A representative example of the excluded-volume thermodynamics is the single-component gas of hard spheres. For this case, using a virial expansion, the validity limits of the low-density approximation are also discussed.