Normal diffusion in crystal structures and higher-dimensional billiard models with gaps (original) (raw)

Diffusion in a periodic Lorentz gas

Journal of Statistical Physics, 1987

We use a constant "driving force" F a together with a Gaussian thermostatting "constraint force" F,. to simulate a nonequilibrium steady-state current (particle velocity) in a periodic, two-dimensional, classical Lorentz gas. The ratio of the average particle velocity to the driving force (field strength) is the Lorentz-gas conductivity. A regular "Galton-board" lattice of fixed particles is arranged in a dense triangular-lattice structure. The moving scatterer particle travels through the lattice at constant kinetic energy, making elastic hard-disk collisions with the fixed particles. At low field strengths the nonequilibrium conductivity is statistically indistinguishable from the equilibrium Green-Kubo estimate of Machta and Zwanzig. The low-field conductivity varies smoothly, but in a complicated way, with field strength. For moderate fields the conductivity generally decreases nearly linearly with field, but is nearly discontinuous at certain values where interesting stable cycles of collisions occur. As the field is increased, the phase-space probability density drops in apparent fractal dimensionality from 3 to 1. We compare the nonlinear conductivity with similar zero-density results from the two-particle Boltzmann equation. We also tabulate the variation of the kinetic pressure as a function of the field strength.

Many-particle diffusion in continuum: Influence of a periodic surface potential

The Journal of Chemical Physics, 2002

We study the diffusion of Brownian particles with a short-range repulsion on a surface with a periodic potential through molecular dynamics simulations and theoretical arguments. We concentrate on the behavior of the tracer and collective diffusion coefficients D T () and D C (), respectively, as a function of the surface coverage . In the high friction regime we find that both coefficients are well approximated by the Langmuir lattice-gas results for up to Ϸ0.7 in the limit of a strongly binding surface potential. In particular, the static compressibility factor within D C () is very accurately given by the Langmuir formula for 0рр1. For higher densities, both D T () and D C ()show an intermediate maximum which increases with the strength of the potential amplitude. In the low friction regime we find that long jumps enhance blocking and D T () decreases more rapidly for submonolayer coverages. However, for higher densities D T ()/D T (0) is almost independent of friction as long jumps are effectively suppressed by frequent interparticle collisions. We also study the role of memory effects for many-particle diffusion.

Density-dependent diffusion in the periodic Lorentz gas

2000

We study the deterministic diffusion coefficient of the two-dimensional periodic Lorentz gas as a function of the density of scatterers. Based on computer simulations, and by applying straightforward analytical arguments, we systematically improve the Machta–Zwanzig random walk approximation [Phys. Rev. Lett. 50: 1959 (1983)] by including microscopic correlations. We furthermore, show that, on a fine scale, the diffusion coefficient is a non-trivial function of the density.

Diffusion in lattice Lorentz gases with a percolation threshold

Physical Review E, 1999

A mean-field approximation for the diffusion coefficient in lattice Lorentz gases with an arbitrary mixture of pointlike stochastic scatterers in the low-density limit is proposed. In this approximation, the diffusion coefficient is directly related to the first return probability of the moving particle in the corresponding Cayley tree through an effective ring operator. A renormalization scheme for the approximate determination of the first return probability is constructed. The predictions of this mean-field theory and those of the repeated ring approximation ͑RRA͒ are compared with computer simulation results for models in which a fraction x B of the scatterers are deterministic backscatterers, so that the diffusion coefficient vanishes beyond a certain percolation threshold x B c . The approximation proposed in this paper is seen to be in good agreement with the simulation results, in contrast to the RRA, which already fails to give the correct percolation threshold.

Irregular diffusion in the bouncing ball billiard

Physica D-nonlinear Phenomena, 2004

We call a system bouncing ball billiard if it consists of a particle that is subject to a constant vertical force and bounces inelastically on a one-dimensional vibrating periodically corrugated floor. Here we choose circular scatterers that are very shallow, hence this billiard is a deterministic diffusive version of the well-known bouncing ball problem on a flat vibrating plate. Computer simulations show that the diffusion coefficient of this system is a highly irregular function of the vibration frequency exhibiting pronounced maxima whenever there are resonances between the vibration frequency and the average time of flight of a particle. In addition, there exist irregularities on finer scales that are due to higher-order dynamical correlations pointing towards a fractal structure of this curve. We analyze the diffusive dynamics by classifying the attracting sets and by working out a simple random walk approximation for diffusion, which is systematically refined by using a Green–Kubo formula.

Confined diffusion in a random Lorentz gas environment

Physical Review E

We study the diffusive behavior of biased Brownian particles in a two dimensional confined geometry filled with the freezing obstacles. The transport properties of these particles are investigated for various values of the obstacle density η and the scaling parameter f , which is the ratio of work done to the particles to available thermal energy. We show that, when the thermal fluctuations dominate over the external force, i.e., small f regime, particles get trapped in the given environment when the system percolates at the critical obstacle density η c ≈ 1.2. However, as f increases, we observe that particle trapping occurs prior to η c. In particular, we find a relation between η and f which provides an estimate of the minimum η up to a critical scaling parameter f c beyond which the Fick-Jacobs description is invalid. Prominent transport features like nonmonotonic behavior of the nonlinear mobility, anomalous diffusion, and greatly enhanced effective diffusion coefficient are explained for various strengths of f and η. Also, it is interesting to observe that particles exhibit different kinds of diffusive behaviors, i.e., subdiffusion, normal diffusion, and superdiffusion. These findings, which are genuine to the confined and random Lorentz gas environment, can be useful to understand the transport of small particles or molecules in systems such as molecular sieves and porous media, which have a complex heterogeneous environment of the freezing obstacles.

Diffusion anomaly in a three-dimensional lattice gas

Physica A: Statistical Mechanics and its Applications, 2007

We investigate the relation between thermodynamic and dynamic properties of an associating lattice gas (ALG) model. The ALG combines a three dimensional lattice gas with particles interacting through a soft core potential and orientational degrees of freedom. From the competition between the directional attractive forces and the soft core potential results two liquid phases, double criticality and density anomaly. We study the mobility of the molecules in this model by calculating the diffusion constant at a constant temperature, D. We show that D has a maximum at a density ρ max and a minimum at a density ρ min < ρ max. Between these densities the diffusivity differs from the one expected for normal liquids. We also show that in the pressure-temperature phase-diagram the line of extrema in diffusivity is close to the liquid-liquid critical point and it is partially inside the temperature of maximum density (TMD) line.