3.2 The Canonical Ensemble: Closed Systems
139
temperatures T A and T B to our two objects (systems A and B, respectively), we
may say that
(a) if T A > T B , then the combined (or composite) system is not at equilibrium, and
energy therefore transfers spontaneously from system A into system B;
(b) if T A = T B , then the combined system is at equilibrium.
Comparison between these two statements for spontaneous energy transfer from
system A into system B, namely β A < β B , or equivalently, T A > T B , indicates that
β and T must be inversely proportional to one another. We cannot at this stage be
any more specific about the constant of proportionality other than to say that it must
be dimensioned so that β represents a reciprocal energy that characterizes such an
(equilibrium) system.
3.2 The Canonical Ensemble: Closed Systems
Let us now consider in greater detail what happens when two systems can exchange
energy between them, i.e., they are not individually isolated systems. Can we make
any quantitative statements about such systems? The answer to this question turns
out to be ‘Yes’, provided that one of the systems is small in comparison to the other
one. For example, if one takes a piece of hot metal and drops it into a lake, the
metal can be expected to cool down, and the lake to take up the heat lost by the
metal—however, one does not expect the temperature of the lake to be changed by
very much.
In our quest let us call the object of interest ‘the system A’ and its environment
‘the heat reservoir A ’. Place A in contact with A , and wait until equilibrium has
been established. We now ask ‘What is the probability p r of finding the system A
in any one particular state r of energy E r ?’
We shall denote by (E ) the number of states accessible to the reservoir A
when its energy is equal to E , i.e., when the energy lies between E and E + δE ,
with δE very small compared with the separation between the energy levels of A,
but still large enough to contain many possible states of A . We shall denote by A ∗
the total system, consisting of the system A plus its surroundings A , as illustrated
in Fig. 3.1.
We may treat the whole system A ∗ = A ⊕ A as an isolated system, so that E ∗
must be constant. If we now associate the energy E r with A as A ←→ E r , then the
energies E ∗ , E , E r are related via
E
∗
= E
+ E r
E
= E
∗
− E r ,
(3.2.1)
E r = E
∗
− E
.
139
temperatures T A and T B to our two objects (systems A and B, respectively), we
may say that
(a) if T A > T B , then the combined (or composite) system is not at equilibrium, and
energy therefore transfers spontaneously from system A into system B;
(b) if T A = T B , then the combined system is at equilibrium.
Comparison between these two statements for spontaneous energy transfer from
system A into system B, namely β A < β B , or equivalently, T A > T B , indicates that
β and T must be inversely proportional to one another. We cannot at this stage be
any more specific about the constant of proportionality other than to say that it must
be dimensioned so that β represents a reciprocal energy that characterizes such an
(equilibrium) system.
3.2 The Canonical Ensemble: Closed Systems
Let us now consider in greater detail what happens when two systems can exchange
energy between them, i.e., they are not individually isolated systems. Can we make
any quantitative statements about such systems? The answer to this question turns
out to be ‘Yes’, provided that one of the systems is small in comparison to the other
one. For example, if one takes a piece of hot metal and drops it into a lake, the
metal can be expected to cool down, and the lake to take up the heat lost by the
metal—however, one does not expect the temperature of the lake to be changed by
very much.
In our quest let us call the object of interest ‘the system A’ and its environment
‘the heat reservoir A ’. Place A in contact with A , and wait until equilibrium has
been established. We now ask ‘What is the probability p r of finding the system A
in any one particular state r of energy E r ?’
We shall denote by (E ) the number of states accessible to the reservoir A
when its energy is equal to E , i.e., when the energy lies between E and E + δE ,
with δE very small compared with the separation between the energy levels of A,
but still large enough to contain many possible states of A . We shall denote by A ∗
the total system, consisting of the system A plus its surroundings A , as illustrated
in Fig. 3.1.
We may treat the whole system A ∗ = A ⊕ A as an isolated system, so that E ∗
must be constant. If we now associate the energy E r with A as A ←→ E r , then the
energies E ∗ , E , E r are related via
E
∗
= E
+ E r
E
= E
∗
− E r ,
(3.2.1)
E r = E
∗
− E
.