Biomedical Engineering Reference
In-Depth Information
Finally, we take a time average over all the particles in the box, which is assumed to be in
an equilibrium state:
N
N
2
m
N
1
2
m
2
d
d
t
1
−
r
i
.
F
i
=−
r
i
.
v
i
+
v
i
(8.28)
i
=
1
i
=
1
i
=
1
The second term on the right-hand side is obviously the mean kinetic energy of all the
particles in the box. This must be
2
Nk
B
T
, according to the equipartition of energy principle.
Whatever the value of the first average quantity in brackets on the right-hand side it
cannot vary with time because we are dealing with an equilibrium state and so the first time
derivative must vanish:
N
m
2
d
d
t
−
r
i
.
v
i
=
0
i
=
1
Thus we have
N
2
m
N
1
2
1
−
r
i
.
F
i
=
v
i
(8.29)
i
=
1
i
=
1
The summation term on the left hand side
N
1
2
−
r
i
.
F
i
i
=
1
involving the forces and coordinates is often referred to as the
virial of Clausius
. According
to the equipartition of energy principle, it is equal to
2
Nk
B
T
.
