Acoustic finite-size effects of simulated fluids under periodic boundary conditions

Question

Consider a fluid simulated in a finite box of specific size. An impulse to the fluid element at the center in a given direction is physically expected to propagate at the speed of sound and attenuate during propagation. If the impulse is attenuated before reaching the box, there is no reflection. If the impulse reaches the box, it is reflected back and forth until it is attenuated. In both cases the simulated fluid is expected to behave similarly to the physical fluid.

Now consider a fluid simulated in periodic boundary conditions being a model for an unbounded fluid. If the periodic box is large enough, the impulse could be attenuated before reaching the box and so the finite size of the box would have no influence. However if the impulse reaches the periodic box it will reappear from the other side of the box and keep drifting through the periodic box until it is attenuated. So this should provoke a finite size artifact.

So independently of the modeling framework used, i.e molecular dynamics, Boltzmann equation, Navier-Stokes and so on.., this effect should be known. But searching with terms like "acoustic-finite size effects in periodic boundary conditions" did not give any results. So any hints would be appreciated.

Answer 1

The finite-size effects you mention are quite well-known by practitioners of Molecular dynamics simulations. Unfortunately, a comprehensive discussion in standard textbooks is still missing. They are mentioned and studied in specific studies, usually in the process of assessing the quantitative effects of their presence on dynamic correlation functions or quantities derived from them.

One recent reference where you could find an explicit mention of such periodic boundary condition artifacts in connection with the evaluation of the shear viscosity is Kim, K. S., Kim, C., Karniadakis, G. E., Lee, E. K., & Kozak, J. J. (2019). Density-dependent finite system-size effects in equilibrium molecular dynamics estimation of shear viscosity: Hydrodynamic and configurational study. The Journal of Chemical Physics, 151(10). Moreover, you may find interesting their refs 17, 18 and 24.