$($a$)$ Define what is understood by 'mean square displacement' $($MSD$)$ in a

molecular dynamics simulation and describe its calculation.

$($b$)($i$)$ Illustrate how, in homogeneous systems, the MSD is related to the

diffusion coefficient $D$ in a molecular simulation.

$($ii$)$ Given that the diffusion coefficient for a Li$+$ tracer diffusing within a

defective LiAl alloy is $2\mathrm{.}1$ and $3\mathrm{.}1{10}^{-6}c\frac{m^2}{sec}$ at $410$ and $660K,$

respectively, evaluate the distance $($in $pm)$ travelled after $1ns$ by a $L{i}^{+}$

tracer at each temperature.

$($iii$)$ Given that the Boltzman constant $k_B=8\mathrm{.}6173{10}^{-5}e\frac{V}{K},$ evaluate the

activation barrier for $L{i}^{+}$tracer diffusion $($in $e\frac{V}{?}$ molecule$)[$using the data

from $($b$)($ii$)].$