Geometric integrator for simulations in the canonical ensemble

In this work we introduce a geometric integrator for molecular dynamics simulations of physical systems in the canonical ensemble. In particular, we consider the equations arising from the so-called density dynamics algorithm with any possible type of thermostat and provide an integrator that preserves the invariant distribution. Our integrator thus constitutes a unified framework that allows the study and comparison of different thermostats and of their influence on the equilibrium and non-equilibrium (thermo-)dynamic properties of the system. To show the validity and the generality of the integrator, we implement it with a second-order, time-reversible method and apply it to the simulation of a Lennard-Jones system with three different thermostats, obtaining good conservation of the geometrical properties and recovering the expected thermodynamic results.Comment: 9 pages, 5 figure

Fine structure of distributions and central limit theorem in diffusive billiards

We investigate deterministic diffusion in periodic billiard models, in terms of the convergence of rescaled distributions to the limiting normal distribution required by the central limit theorem; this is stronger than the usual requirement that the mean square displacement grow asymptotically linearly in time. The main model studied is a chaotic Lorentz gas where the central limit theorem has been rigorously proved. We study one-dimensional position and displacement densities describing the time evolution of statistical ensembles in a channel geometry, using a more refined method than histograms. We find a pronounced oscillatory fine structure, and show that this has its origin in the geometry of the billiard domain. This fine structure prevents the rescaled densities from converging pointwise to gaussian densities; however, demodulating them by the fine structure gives new densities which seem to converge uniformly. We give an analytical estimate of the rate of convergence of the original distributions to the limiting normal distribution, based on the analysis of the fine structure, which agrees well with simulation results. We show that using a Maxwellian (gaussian) distribution of velocities in place of unit speed velocities does not affect the growth of the mean square displacement, but changes the limiting shape of the distributions to a non-gaussian one. Using the same methods, we give numerical evidence that a non-chaotic polygonal channel model also obeys the central limit theorem, but with a slower convergence rate.Comment: 16 pages, 19 figures. Accepted for publication in Physical Review E. Some higher quality figures at http://www.maths.warwick.ac.uk/~dsander

Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

We perform numerical measurements of the moments of the position of a tracer particle in a two-dimensional periodic billiard model (Lorentz gas) with infinite corridors. This model is known to exhibit a weak form of super-diffusion, in the sense that there is a logarithmic correction to the linear growth in time of the mean-squared displacement. We show numerically that this expected asymptotic behavior is easily overwhelmed by the subleading linear growth throughout the time-range accessible to numerical simulations. We compare our simulations to the known analytical results for the variance of the anomalously-rescaled limiting normal distributions.Comment: 10 pages, 4 figure

L\'evy walks on lattices as multi-state processes

Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of L\'evy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which case a regime of superdiffusion may lurk underneath one of normal diffusion.Comment: 27 pages, 4 figure

Transport properties of L\'evy walks: an analysis in terms of multistate processes

Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of L\'evy walks on lattices. By including exponentially-distributed waiting times separating the successive jump events of a walker, we are led to a description of such L\'evy walks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay differential equations. Using simple arguments, we obtain asymptotic solutions to these equations and rederive the scaling laws for the mean squared displacement of such processes. Our calculation includes the computation of all relevant transport coefficients in terms of the parameters of the models.Comment: 5 pages, 2 figures. New references adde

Diffusion coefficients for multi-step persistent random walks on lattices

We calculate the diffusion coefficients of persistent random walks on lattices, where the direction of a walker at a given step depends on the memory of a certain number of previous steps. In particular, we describe a simple method which enables us to obtain explicit expressions for the diffusion coefficients of walks with two-step memory on different classes of one-, two- and higher-dimensional lattices.Comment: 27 pages, 2 figure