4.1 Canonical Ensemble: Closed Systems
173
so that
C V =
1
k B T 2
∂ 2 ln Z N
∂β 2
V
.
(4.1.5)
Thus, upon comparison of this expression with Eq. (4.1.4c), we see that
k B T
2 C V = =((E)
2
,
(4.1.6)
thereby connecting energy fluctuations to the heat capacity at constant volume.
4.1.1 Thermodynamics from the Canonical Ensemble
Let us consider a system of interest A, together with its associated set of energy
states E r , in thermal equilibrium with a bath (or reservoir) at temperature T . We
shall characterize the system, for the moment, by a single external parameter x (such
as the volume V , for example). Let us now change x slightly, bring it to a new
value x + dx, and ask what energy changes occur in our system of interest. The
energy levels will be functions of the external parameter, so that we should write
E r = E r (x). We can see that the energy levels do depend upon such parameters
in a straightforward way if we consider the example of the particle-in-a-box. The
particle-in-a-box has energy given by
n x n y n z =
h 2
8m
n 2
x
a 2 +
n 2
y
b 2 +
n 2
z
c 2
and clearly depends upon the volume V (through a, b, c) and upon the mass m of
the particle. If we change x, then there will be a corresponding change x E r in E r ,
which can be expressed as follows:
x E r =
∂E r
∂x
dx .
(4.1.7)
We should now compare this result with the usual definition of ‘work’ in
mechanics. The basic concept can be expressed as it was by Newton: ‘work’ =
‘force’בdisplacement’. We can also generalize this concept to say: If there is
something (call it X) that changes when something else (i.e., x) changes, then
the system tends to move away from the state of energy E r at which it was in
equilibrium. However, if the change is produced very slowly, then the system will
do work to remain in that state, and
δw = dE r =
∂E r
∂x
dx ≡ −X r dx ,
(4.1.8)
173
so that
C V =
1
k B T 2
∂ 2 ln Z N
∂β 2
V
.
(4.1.5)
Thus, upon comparison of this expression with Eq. (4.1.4c), we see that
k B T
2 C V = =((E)
2
,
(4.1.6)
thereby connecting energy fluctuations to the heat capacity at constant volume.
4.1.1 Thermodynamics from the Canonical Ensemble
Let us consider a system of interest A, together with its associated set of energy
states E r , in thermal equilibrium with a bath (or reservoir) at temperature T . We
shall characterize the system, for the moment, by a single external parameter x (such
as the volume V , for example). Let us now change x slightly, bring it to a new
value x + dx, and ask what energy changes occur in our system of interest. The
energy levels will be functions of the external parameter, so that we should write
E r = E r (x). We can see that the energy levels do depend upon such parameters
in a straightforward way if we consider the example of the particle-in-a-box. The
particle-in-a-box has energy given by
n x n y n z =
h 2
8m
n 2
x
a 2 +
n 2
y
b 2 +
n 2
z
c 2
and clearly depends upon the volume V (through a, b, c) and upon the mass m of
the particle. If we change x, then there will be a corresponding change x E r in E r ,
which can be expressed as follows:
x E r =
∂E r
∂x
dx .
(4.1.7)
We should now compare this result with the usual definition of ‘work’ in
mechanics. The basic concept can be expressed as it was by Newton: ‘work’ =
‘force’בdisplacement’. We can also generalize this concept to say: If there is
something (call it X) that changes when something else (i.e., x) changes, then
the system tends to move away from the state of energy E r at which it was in
equilibrium. However, if the change is produced very slowly, then the system will
do work to remain in that state, and
δw = dE r =
∂E r
∂x
dx ≡ −X r dx ,
(4.1.8)