Error estimation on the box size correction formula of diffusion coefficient in molecular simulations

Correcting the diffusion coefficient for simulation box size is an essential step in extrapolating small simulations to real systems. The current common approach is a correction formula based on fluid dynamics theory, introduced by Yeh and Hummer [J. Phys. Chem. B 108, 15873 (2004)]. While this formula has been used by some authors for theoretical correction, others have found some difficulties in applying the formula. In this paper, we have reviewed the error issues in diffusion coefficient calculations in molecular simulations and discussed why significant uncertainties arise when applying the Yeh-Hummer correction formula. Furthermore, based on mathematical statistics theory, we suggest using a new correction formula that converges more quickly to the particle number, aiming to achieve better correction results. Using molecular dynamics simulation data of Lennard-Jones fluids, the dependence of the diffusion coefficient on different powers of the number of particles is investigated. It shows that our formula is slightly better than Yeh-Hummer's formula in agreement with the simulation results.

Molecular dynamics simulations were carried out in the NVT ensemble using parallel simulator-LAMMPS. The simulations were run in the reduced unit. The interaction potential was chosen to be Lennard-Jones 12-6 model. The Nose-Hoover thermostat was used to realize a thermal equilibrium state. A cutoff distance was taken as 2.5 times the Lennard-Jones interaction diameter. The time steps were all taken as 0.005 in reduced units. A typical total number of simulation steps was 3.5 × 10⁶, of which 5 × 10⁵ steps were used to bring the system to equilibrium, and the remaining 3 × 10⁶ steps were used to dump the trajectories of particles and other quantities every 1000 steps. The diffusion coefficient is calculated using a separate program.