Collisional statistics of the hard-sphere gas

Gaseous diffusion as a correlated random walk

The mean-square displacement per collision of a molecule immersed in a gas at equilibrium is given by its mean-square displacement between two consecutive collisions (mean-square free path) corrected by a prefactor in the form of a series. The nth term of the series is proportional to the mean value of the scalar product r_{1}·r_{n}, where r_{i} is the displacement of the molecule between the (i-1)-th and ith collisions. Simple arguments are used to obtain approximate expressions for each term. The key finding is that the ratio of consecutive terms in the series closely approximates the so-called mean persistence ratio. Exact expressions for the terms in the series are considered and their ratios for several consecutive terms are calculated for the case of hard spheres, showing an excellent agreement with the mean persistence ratio. These theoretical results are confirmed by solving the Boltzmann equation by means of the direct simulation Monte Carlo method. By summing the series, the mean-square displacement and the diffusion coefficient can be determined using only two quantities: the mean-square free path and the mean persistence ratio. A simple and an improved expression for the diffusion coefficient D are considered and compared with the so-called first and second Sonine approximations to D as well as with computer simulations of the Boltzmann equation. It is found that the improved diffusion coefficient shows very good agreement with simulation results over all intruder and molecule mass ranges. When the intruder mass is smaller than that of the gas molecules, the improved formula even outperforms the first Sonine approximation.

Fluctuating diffusivity emerges even in binary gas mixtures

Diffusivity in some soft matter and biological systems changes with time, called the fluctuating diffusivity. In this work, we propose a novel origin for fluctuating diffusivity based on stochastic simulations of binary gas mixtures. In this system, the fraction of one component is significantly small, and the mass of the minor component molecule is different from that of the major component. The minor component exhibits fluctuating diffusivity when its mass is sufficiently smaller than that of the major component. We elucidate that this fluctuating diffusivity is caused by the time scale separation between the relaxation of the velocity direction and the speed of the minor component molecule.

Rotationally-Resolved Two-Dimensional Infrared Spectroscopy of CO2(g): Rotational Wavepackets and Angular Momentum Transfer

Angular momentum transfer and wavepacket dynamics of CO₂(g) were measured on the picosecond time scale using polarization-resolved two-dimensional infrared (2D-IR) spectroscopy. The dynamics of rotational levels up to J_max ≈ 50 are observed simultaneously at room temperature. Rotational wavepackets launched by the pump pulses cause oscillations in the intensity of individual peaks and beating patterns in the 2D-IR spectra. The structure of the rotationally resolved 2D-IR spectrum is explained using nonlinear response function theory. Spectral diffusion of the rotationally resolved 2D-IR peaks reveals information about angular momentum transfer. We demonstrate the ability to directly measure inelastic angular momentum dynamics simultaneously across the ∼50 thermally excited rotational levels over several hundred picoseconds.

Gas Flow at the Ultra-nanoscale: Universal Predictive Model and Validation in Nanochannels of Ångstrom-Level Resolution

Gas transport across nanoscale pores is determinant in molecular exchange in living organisms as well as in a broad spectrum of technologies. Here, we report an unprecedented theoretical and experimental analysis of gas transport in a consistent set of confining nanochannels ranging in size from the ultra-nanoscale to the sub-microscale. A generally applicable theoretical approach quantitatively predicting confined gas flow in the Knudsen and transition regime was developed. Unlike current theories, specifically designed for very simple channel geometries, our approach can be applied to virtually all geometries, for which the probability distribution of path lengths for particle-interface collisions can be computed, either analytically or by numerical simulations. To generate a much needed benchmark experimental model, we manufactured extremely reproducible membranes with two-dimensional nanochannels. Channel sizes ranged from 2.5 to 250 nm, and angstrom level of size control and interface tolerances were achieved using leading-edge nanofabrication techniques. We then measured gas flow in the Knudsen number range from 0.2 to 20. Excellent agreement between theoretical predictions and experimental data was found, demonstrating the validity and potential of our approach.

Collisional statistics and dynamics of two-dimensional hard-disk systems: From fluid to solid

We perform extensive MD simulations of two-dimensional systems of hard disks, focusing on the collisional statistical properties. We analyze the distribution functions of velocity, free flight time, and free path length for packing fractions ranging from the fluid to the solid phase. The behaviors of the mean free flight time and path length between subsequent collisions are found to drastically change in the coexistence phase. We show that single-particle dynamical properties behave analogously in collisional and continuous-time representations, exhibiting apparent crossovers between the fluid and the solid phases. We find that, both in collisional and continuous-time representation, the mean-squared displacement, velocity autocorrelation functions, intermediate scattering functions, and self-part of the van Hove function (propagator) closely reproduce the same behavior exhibited by the corresponding quantities in granular media, colloids, and supercooled liquids close to the glass or jamming transition.

Collision statistics in sheared inelastic hard spheres

The dynamics of sheared inelastic-hard-sphere systems is studied using nonequilibrium molecular-dynamics simulations and direct simulation Monte Carlo. In the molecular-dynamics simulations Lees-Edwards boundary conditions are used to impose the shear. The dimensions of the simulation box are chosen to ensure that the systems are homogeneous and that the shear is applied uniformly. Various system properties are monitored, including the one-particle velocity distribution, granular temperature, stress tensor, collision rates, and time between collisions. The one-particle velocity distribution is found to agree reasonably well with an anisotropic Gaussian distribution, with only a slight overpopulation of the high-velocity tails. The velocity distribution is strongly anisotropic, especially at lower densities and lower values of the coefficient of restitution, with the largest variance in the direction of shear. The density dependence of the compressibility factor of the sheared inelastic-hard-sphere system is quite similar to that of elastic-hard-sphere fluids. As the systems become more inelastic, the glancing collisions begin to dominate over more direct, head-on collisions. Examination of the distribution of the times between collisions indicates that the collisions experienced by the particles are strongly correlated in the highly inelastic systems. A comparison of the simulation data is made with direct Monte Carlo simulation of the Enskog equation. Results of the kinetic model of Montanero [J. Fluid Mech. 389, 391 (1999)] based on the Enskog equation are also included. In general, good agreement is found for high-density, weakly inelastic systems.

Velocity-correlation distributions in thermal and granular hard-core gases

Collision statistics of hard-core systems (thermal and dissipative) is investigated through the velocity-correlation distributions after n collisions of a tagged hard-core particle: These quantities provide information on the velocity correlations for a given number of collisions. We obtain exact results for arbitrary dimension for the velocity-correlation distribution after the first collision as well as for the velocity-correlation function after an infinite number of collisions. For Gaussian velocity distributions, we show that the decay of the first-collision velocity-correlation distribution for negative argument is always exponential in any dimension; the decay rate is then a function of the mass and the coefficient of restitution. For granular gases, where deviations from Gaussian are relevant, expressions including Sonine corrections are also derived for the velocity-correlation distribution and a comparison with a direct simulation Monte Carlo (DSMC) shows accurate agreement with theoretical results. We emphasize that these quantities can be easily obtained in simulations and most likely also in experiments: therefore they could be an efficient probe of the local environment and of the degree of inelasticity of the collisions.

Exciting hard spheres

We investigate the collision cascade that is generated by a single moving particle in a static and homogeneous hard-sphere gas. We argue that the number of moving particles at time t grows as t;{xi} and the number collisions up to time t grows as t;{eta} , with xi=2d(d+2) , eta=2(d+1)(d+2) , and d the spatial dimension. These growth laws are the same as those from a hydrodynamic theory for the shock wave emanating from an explosion. Our predictions are verified by molecular dynamics simulations in d=1 and 2. For a particle incident on a static gas in a half-space, the resulting backsplatter ultimately contains almost all the initial energy.