University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Mobility of atoms and diffusion.
Einstein relation.
∑
=
−
≡
Δ
≡
N
1
i
2
i
i
2
))
0
(
r
)
t
(
r
(
N
1
)
t
(
r
MSD
r
r
r
•
This expression is called Einstein relation since it was first derived by Albert
Einstein in his Ph.D. thesis in 1905 (see note below)
•
This expression relates macroscopic transport coefficient D with microscopic
information on the mean square distance of molecular migration
•
The 6 in this formula becomes 4 for a twodimensional system and 2 for a one
dimensional system (see next page)
•
This equation is suitable for calculation of D in MD simulation only for
sufficiently high temperatures, when D > 10
12
m
2
/s
•
Time t cannot be too large for a finite system (D drops to 0 when MSD
approaches the size of the system)
•
For periodic boundaries “true” atomic displacements should be used
•
Derivation of this equation is given on pages 7879 of the textbook by D.
Frenkel and B. Smit
In MD simulation we can describe the mobility of atoms through the
mean square displacement that can be calculated as
The MSD contains information on the diffusion coefficient D,
ns
fluctuatio
Dt
6
A
)
t
(
r
MSD
2
+
+
=
Δ
=
r
Historic note:
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 Fall '11
 Zhigilei
 Vector field, diffusion equation, Leonid Zhigilei, Einstein Relation

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