Einstein Equation

${\displaystyle \langle |r(t)-r(0)|^{2}\rangle =6Dt}$

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.

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,

${\displaystyle D=\int _{0}^{\infty }\langle v(t)\cdot v(t+\tau )\rangle d\tau }$

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