Hard-disk Fluid Research Articles

Lyapunov hydrodynamics in the dilute limit

We study the hydrodynamic Lyapunov exponents of the hard-disk fluid in the dilute limit. These exponents appear in discrete groups and their values depend on the system size. In a previous paper [Phys. Rev. E 64 (2001) 051103], we presented a theory for these exponents, explaining them as the growth rate of collective, organized perturbations in phase space. That theory successfully described the basic features of the hydrodynamic exponents, but it had some difficulties. Many of these difficulties can be removed by considering the dilute limit.

First normal stress difference and crystallization in a dense sheared granular fluid

The first normal stress difference (N1) and the microstructure in a dense sheared granular fluid of smooth inelastic hard-disks are probed using event-driven simulations. While the anisotropy in the second moment of fluctuation velocity, which is a Burnett-order effect, is known to be the progenitor of normal stress differences in dilute granular fluids, we show here that the collisional anisotropies are responsible for the normal stress behavior in the dense limit. As in the elastic hard-sphere fluids, N1 remains positive (if the stress is defined in the compressive sense) for dilute and moderately dense flows, but becomes negative above a critical density, depending on the restitution coefficient. This sign-reversal of N1 occurs due to the microstructural reorganization of the particles, which can be correlated with a preferred value of the average collision angle θav=π/4±π/2 in the direction opposing the shear. We also report on the shear-induced crystal-formation, signaling the onset of fluid–solid coexistence in dense granular fluids. Different approaches to take into account the normal stress differences are discussed in the framework of the relaxation-type rheological models.

Induced freezing and re-entrant melting in the hard-disc fluid; applications of thefundamental measure functional

A fundamental measure functional is used to study the induced freezing andre-entrant melting for the hard-disc fluid in an external periodic potential. Thephase diagram obtained shows good agreement with recent experimental studies.We also use the functional to describe the hard-disc fluid density near a hard walland investigate whether this 2D functional can correctly describe the 0D limit.

Van der Waals-Tonks-type equations of state for hard-disk and hard-sphere fluids.

Using the known virial coefficients of hard-disk and hard-sphere fluids, we develop van der Waals-Tonks-type equations of state for hard-disk and hard-sphere fluids. In the low-density fluid regime, these equations of state are in good agreement with the simulation results and the existing equations of state.

Equation of state of additive hard-disk fluid mixtures: a critical analysis of two recent proposals.

A detailed analysis of two different theoretical equations of state for a binary mixture of additive hard disks [C. Barrio and J. R. Solana, Phys. Rev. E 63, 011201 (2001); A. Santos, S. B. Yuste, and M. López de Haro, Mol. Phys. 96, 1 (1999)], including their comparison with Monte Carlo results, is carried out. It is found that both proposals, which require the equation of state of the single-component system as input, lead to comparable accuracy when the same input is used in both, but that advocated by Santos et al. is simpler and complies with the exact limit in which the small disks are point particles.

Erratum: “Contact pair correlation functions and equation of state for additive hard disk fluid mixtures” [J. Chem. Phys. 115, 7123 (2001)

Abstract

Fundamental measure theory and dimensional interpolation for the hard spheres fluid

This paper presents, a short review of the recent advances in the development of density functional for hard spheres. The fundamental measure theory, as developed originally by Y. Rosenfeld and in its more recent versions, is compared with other density functional approximations. The results for dimensional crossover from the hard spheres fluid in three dimensions to hard rods in one dimension and to zero-dimensional cavities are discussed. New results on the fluid–crystal coexistence and on the crossover to the hard disc fluid in two dimensions are presented.

The Properties of Voronoi Tessellations for 2D Ising Fluid Near the Second-Order Magnetic Phase Transition

The topological and geometrical properties of Voronoi cells generated for 2D fluid of hard disks with Ising-like spins near the second-order phase transition from the paramagnetic to ferromagnetic phase are described for different disk densities. The comparison with Voronoi cells generated for the random hard disk system is given.

Hard discs in circular cavities: density functional theory versus simulation

Three different density functional approaches, namely the Rickayzen, Takamiya-Nakanishi and Rosenfeld approximations, have been employed to study the equilibrium particle density distributions of hard-disc fluids in hard circular cavities. Also, Monte Carlo simulations have been performed to test the theoretical results. The comparison with the simulation data shows that the Rosenfeld approximation, which is based on the fundamental geometrical measures of the particles, is better than the Takamiya-Nakanishi and Rickayzen approximations and yields good agreement with the computer simulation data even for higher densities.

Contact pair correlation functions and equation of state for additive hard disk fluid mixtures

The contact pair correlation functions and the equation of state for a binary mixture of additive hard disks is obtained using a procedure similar to that leading to the Boublı́k–Mansoori–Carnahan–Starling–Leland equation of state for mixtures of additive hard spheres. The results from the derived equations are tested against new Monte Carlo data obtained for several diameter ratios and mole fractions. The overall agreement is excellent. The equation of state reproduces exactly the second virial coefficient of the mixture and the third with great accuracy. Predicted values of the fourth and fifth virial coefficients are also in very good agreement with numerical data.

Precollisional velocity correlations in a hard-disk fluid with dissipative collisions.

Velocity correlations are studied in granular fluids, modeled by the inelastic hard sphere gas. Making a density expansion of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for the evolution of the reduced distributions, we predict the presence of precollisional velocity correlations. They are created by the propagation through correlated sequences of collisions (ring events) of the velocity correlations generated after dissipative collisions. The correlations have their origin in the dissipative character of collisions, being always present in granular fluids. The correlations, that manifest microscopically as an alignment of the velocities of a colliding pair produce modifications of collisional averages, in particular, the virial pressure. The pressure shows a reduction with respect to the elastic case as a consequence of the velocity alignment. Good qualitative agreement is obtained for the comparison of the numerical evaluations of the obtained analytical expressions and molecular dynamics results that showed evidence of precollisional velocity correlations [R. Soto and M. Mareschal, Phys. Rev. E 63, 041303 (2001)].

Size selectivity in a confined polydisperse hard-disk fluid

The Rosenfeld density functional approximation has been considered to investigate the local size segregation and selectivity in the adsorption of a polydisperse hard-disk fluid, which has a continuous distribution of the particle diameter. The calculated results have shown that the preferred species in a circular cavity depend strongly on the size ratio of a polydisperse hard-disk fluid as well as the cavity size and bulk packing fraction. The pore average size distribution for small particles decreases roughly linearly with increasing the cavity size, and the pore average size distribution for an average diameter of particle 〈σ〉 is almost the same as the bulk packing fraction. The local relative concentration oscillates with a spatial period close to the diameter of the large particle as well as the equilibrium density distribution does. These results show that a polydisperse hard-disk fluid confined in a circular cavity rivals similar structural properties compared with those of a multicomponent hard-disk fluid.

Theory and computer simulation for the equation of state of additive hard-disk fluid mixtures.

A procedure previously developed by the authors to obtain the equation of state for a mixture of additive hard spheres on the basis of a pure fluid equation of state is applied here to a binary mixture of additive hard disks in two dimensions. The equation of state depends on two parameters which are determined from the second and third virial coefficients for the mixture, which are known exactly. Results are compared with Monte Carlo calculations which are also reported. The agreement between theory and simulation is very good. For the fourth and fifth virial coefficients of the mixture, the equation of state gives results which are also in close agreement with exact numerical values reported in the literature.

Cahn-hilliard theory for unstable granular fluids

A Cahn-Hilliard-type theory for hydrodynamic fluctuations is proposed that gives a quantitative description of the slowly evolving spatial correlations and structures in density and flow fields in the early stages of evolution of freely cooling granular fluids. Two mechanisms for pattern selection and structure formation are identified: unstable modes leading to density clustering (mechanismlike spinodal decomposition, or "uplifting" in structural geology), and selective noise reduction (mechanismlike peneplanation in structural geology) leading to vortex patterns. As time increases, the structure factor for the density field develops a maximum, which shifts to smaller wave numbers. This corresponds to an approximately diffusively growing length scale for density clusters. Analytic expressions are derived for spatial correlation functions and structure factors that agree well with molecular dynamics simulations of a fluid of inelastic hard disks.

Lyapunov instability for a hard-disk fluid in equilibrium and nonequilibrium thermostated by deterministic scattering

We compute the full Lyapunov spectra for a hard-disk fluid under temperature gradient and under shear. The Lyapunov exponents are calculated using a recently developed formalism for systems with elastic hard collisions. The system is thermalized by deterministic and time-reversible scattering at the boundary, whereas the bulk dynamics remains Hamiltonian. This thermostating mechanism allows for energy fluctuations around a mean value which is reflected by only two vanishing Lyapunov exponents in equilibrium and nonequilibrium. In nonequilibrium steady states the phase-space volume is contracted on average, leading to a negative sum of the Lyapunov exponents. Since the system is driven inhomogeneously we do not expect the conjugate pairing rule to hold, which is indeed shown to be the case. Finally, the Kaplan–Yorke dimension and the Kolmogorov–Sinai entropy are calculated from the Lyapunov spectra.

Molecular Fluids at High Dimensionality

We extend the analysis of orientational first-order phase transitions in anisotropic molecular fluids at high spatial dimensionality to hard-disk fluids, and then to mixture of hard disks and hard spheres. The effect of hard-sphere admixture depends sensitively on the relative sizes of the two geometrical objects, and large spheres completely quench the disk transition. An introductory study is made of spatially ordered states.

Fluctuations in the number of particles of the ideal gas: A simple example of explicit finite-size effects

The fluctuations in the number of particles of the ideal gas are calculated using the canonical and the grand canonical ensembles. The two results differ by a factor which accounts for the relative size of the total volume and the subvolume where the canonical ensemble fluctuations are calculated. This factor gives a simple example of explicit finite-size effects, because it arises from considering a fixed number of particles in the canonical ensemble. Our simulation results for the isothermal compressibility of the hard disk fluid with a finite number of particles are improved by applying this correction.

Thermostating by deterministic scattering: heat and shear flow.

We apply a recently proposed thermostating mechanism to an interacting many-particle system where the bulk particles are moving according to Hamiltonian dynamics. At the boundaries the system is thermalized by deterministic and time-reversible scattering. We first show how this scattering mechanism can be related to stochastic boundary conditions. We subsequently simulate thermal conduction and shear flow for a hard disk fluid. By comparing the transport coefficients obtained from computer simulations to theoretical results we find that this thermostating mechanism yields well-defined nonequilibrium steady states in the range of linear response. Furthermore, the conjectured identity between thermodynamic entropy production and exponential phase-space contraction rates is investigated from the standpoint of our formalism. We find that, in general, these quantities do not agree.

Microcanonical density of states of a hard-particle fluid under gravity

A simple approximate expression is proposed for the energy dependence of the microcanonical density of states of a classical hard-particle fluid column of infinite height in a uniform gravitational field. This expression is based on the exact result in one dimension and depends on an effective parameter that can be related to the mean kinetic energy in the fluid. The behavior of this parameter is investigated by means of molecular dynamics simulations, showing an asymptotically linear regime at high energies. From the density of states an approximate expression is obtained for the microcanonical single-particle momentum modulus distribution function. By comparing the results of this approximation with simulation data for a hard-disk fluid it is found that the energy dependence proposed for the density of states is a very good approximation for a wide range of values of the characteristic parameters of the system.

The structure of two-dimensional dimerizing fluids from Wertheim’s Ornstein-Zernike equation

A dimerizing fluid of hard discs is studied using two-dimensional (2D) Wertheim’s Ornstein-Zernike (WOZ) equation and associative Percus-Yevick (APY) closure. Dimerization takes place due to site-site associative interactions. The dependences of the association constant on disc density at different association energies are obtained. We calculate the compressibility and the virial equations of state (EOS) using the solution of the WOZ equation. Theoretical structure and thermodynamics is compared with Monte Carlo computer simulation data. Extension of our solution for polymerizing models is of special interest for the development of EOS for 2D chain fluids.