Molecular Simulation/Diffusion Coefficients

Einstein Equation

$⟨|r(t)-r(0){|}^2⟩=6Dt$

The mean square displacement as a function of time of an oxygen molecule diffusing water and pentane. The plots were calculated from a molecular dynamics simulation of a periodic simulation cell containing the oxygen molecule and the solvent. These plots can be used to calculate the diffusion coefficient of the solute using the Einstein relation.

File:Liquid Argon MD No PBC.webm

Green-Kubo Theory

The diffusion coefficient of a liquid can be calculated from the velocity autocorrelation function of an MD simulation in the canonical ensemble,

$D=\underset{0}{\overset{\mathrm{\infty }}{\int }}⟨v(t)\cdot v(t+\tau )⟩d\tau$

where v(t) is the velocity of the particle at time t. Note that care must be taken in using an MD simulation with a stochastic thermostat, which introduce an external source of friction and random force.

See Also

Wikipedia:Einstein relation (kinetic theory)

Wikipedia:Diffusion