Transient hydrodynamic finite-size effects in simulations under periodic boundary conditions

Abstract

We use Lattice-Boltzmann and analytical calculations to investigate transient hydrodynamic finite size effects induced by the use of periodic boundary conditions in simulations at the molecular, mesoscopic or continuum levels of description. We analyze the transient response to a local perturbation in the fluid and obtain via linear response theory the local velocity correlation function. This new approach is validated by comparing the finite size effects on the steady-state velocity with the known results for the diffusion coefficient. We next investigate the full time-dependence of the local velocity auto-correlation function. We find at long times a cross-over between the expected t −3/2 hydrodynamic tail and an oscillatory exponential decay, and study the scaling with the system size of the cross-over time, exponential rate and amplitude, and oscillation frequency. We interpret these results from the analytic solution of the compressible Navier-Stokes equation for the slowest modes, which are set by the system size. The present work not only provides a comprehensive analysis of hydrodynamic finite size effects in bulk fluids, but also establishes the Lattice-Boltzmann method as a suitable tool to investigate such effects in general.It is by now well established that hydrodynamic finite size effects arise in simulations due to the use of periodic boundary conditions (PBC). These effects can be understood as the result of spurious hydrodynamic interactions between particles and their periodic images. Following Dünweg and Kremer [1], Yeh and Hummer [2] proposed a complete analysis of the finite size effect on the diffusion coefficient of fluid particles in a cubic box based on the mobility tensor T: . More recently, the extension to anisotropic boxes was also investigated [8,9] and interpreted in terms of the same hydrodynamic arguments [10,11]. The distortion of the flow field due to the finite size of the system (and the associated use of PBC) does not only affect the diffusion coefficient D of particles, but in principle all dynamical properties. In particular, hydrodynamic flows in an unbounded fluid result in long-time tails of correlations functions, e.g. as t −3/2 for the velocity autocorrelation function (VACF) in three dimensions [12,13]. Such long time tails have been reported in molecular simulations for the VACF since the pioneering work of Ref. 14 (see e.g.[15]) as well as in purely hydrodynamic lattice simulations for the VACF or other correlation functions [16][17][18][19]. Such slow hydrodynamic modes also manifest themselves in the non-Markovian dynamics of solutes, which includes a deterministic component of the force exerted by the suspending fluid, well described for colloidal spheres by the Basset-Boussinesq force [20,21]. Simulations displaying such a hydrodynamic memory, either on a coarse-grained [22] or molecular [23] scale, may therefore suffer from artefacts associated with the use of PBC, at least on long time scales. This was already recognized by Alder and Wainwright in their semi...

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