7.1 Introduction
365
The numerator of Eq. (7.1.14) can be simplified if we note that the derivative of
W a (x) in the integrand will be appreciable only for x − a ≤ δ due to the nature
of the potential term W a (x): this means that we may approximate xe −βV ext (x) by
ae −βV ext (a) and extend the upper limit of integration from b to ∞, thereby giving
b
a
x
dW a
dx
e
−β[V ext (x)+W a (x)+W b (x)] dx ae
−βV ext (a)
∞
a
dW a
dx
e
−βW a (x) dx
for the numerator. Parameter differentiation then gives
b
a
x
dW a
dx
e
−β[V ext (x)+W a (x)+W b (x)] dx = ae
−βV ext (a)
−
1
β
∞
a
∂e −βW a (x)
∂x
dx .
Finally, the integral on the right-hand side of this expression may be evaluated
directly, to give
b
a
x
dW a
dx
e
−β[V ext (x)+W a (x)+W b (x)] dx = −k B T a e
−βV ext (a) .
For the integral in the denominator of Eq. (1.2.12), we note that the exponential
e −βV tot (x) only differs from e −βV ext (x) in regions that extend only by approximately
δ from the walls and, so long as δ is much smaller than b − a, we obtain
b
a
e
−βV tot (x) dx
b
a
e
−βV ext (x) dx .
A similar treatment of the term involving W b (x) on the left-hand side gives the same
denominator, and the numerator as
b
a
x
dW b
dx
e
−β[V ext (x)+W a (x)+W b (x)] dx k B T b e
−βV ext (b) .
Upon collecting terms, we thus obtain the result
x
dV ext
dx
b
a
= k B T
⎡
⎢
⎢
⎣ 1 −
be
−βV ext (b)
− ae
−βV ext (a)
b
a
e
−βV ext (x) dx
⎤
⎥
⎥
⎦ .
(7.1.15)
This result shows explicitly that the introduction of a ‘wall-potential’ that has a
range that is short in comparison with typical system dimensions allows us to
obtain the generalized equipartition principle (7.1.13) that does not depend upon
the detailed form for the wall-potential and, when an external potential acts on the
system, to incorporate a correction due to the presence of the boundaries.