3.1 Microscopic Configurations
137
microstate energy k (k > j) such that q = (k − j)), then and are related by
=
n j !(n k − 1)!
(n j − 1)!n k !
n j
n k
,
assuming n j , n k 1. From this result, we may deduce that the number of
microstates in the new macrostate increases when an oscillator is moved from a
more populated microstate energy level to a less populated one (i.e., n j > n k ) and
decreases in the opposite case.
Also, we note that if the macroscopic states of the system taking up the energy
are members of the set of statistical equilibrium states for the system of oscillators,
then the occupation numbers are related in each case by Eq. (3.1.3), so that the ratio
// can also be obtained in the form
=
n j
n k
=
n 0 e −ββ j
n 0 e −ββ k
= e
β(( k − j )
= e
βq ,
(3.1.4)
in which q ≡ k − j is the energy added to the system. From this result, we see that
the change in the degeneracy depends only upon the amount of energy that has
been added to the system and is independent of the specific energy levels between
which an oscillator has been moved. The same end result is obtained in the event
that the added energy is distributed amongst several oscillators, rather than simply
taken up by a single oscillator, so that the result in Eq. (3.1.4) also does not depend
upon the number of oscillators that are redistributed.
If we take the natural logarithm of both sides of Eq. (3.1.4), we arrive at the result
ln
= ln
− ln ≡ ln = βq ,
(3.1.5)
which implies that the change in ln is proportional to the amount of energy added
to (q > 0) or removed from(q < 0) the system of oscillators.
Now, let us consider two systems, which we shall label as A and B, with system
A having N A oscillators, total energy E A , and at equilibrium is characterized by
β A ; similarly, system B has N B oscillators, total energy E B , and at equilibrium
is characterized by β B . Let us place systems A and B in thermal contact, thereby
allowing an amount q of energy to be transferred spontaneously from one system
to the other, with N A , N B , and E tot = E A + E B all fixed. Initially, for β A = β B ,
each system is in its most likely macrostate, with A , B microstates, respectively.
Because each of the A microstates of system A can be combined with each of
the B microstates of system B, the total number of microstates accessible to the
composite (combined) system A⊕B is tot = A B . 3
3 We shall refer to systems A and B in thermal contact as subsystems of the (composite/combined)
system A⊕B.
137
microstate energy k (k > j) such that q = (k − j)), then and are related by
=
n j !(n k − 1)!
(n j − 1)!n k !
n j
n k
,
assuming n j , n k 1. From this result, we may deduce that the number of
microstates in the new macrostate increases when an oscillator is moved from a
more populated microstate energy level to a less populated one (i.e., n j > n k ) and
decreases in the opposite case.
Also, we note that if the macroscopic states of the system taking up the energy
are members of the set of statistical equilibrium states for the system of oscillators,
then the occupation numbers are related in each case by Eq. (3.1.3), so that the ratio
// can also be obtained in the form
=
n j
n k
=
n 0 e −ββ j
n 0 e −ββ k
= e
β(( k − j )
= e
βq ,
(3.1.4)
in which q ≡ k − j is the energy added to the system. From this result, we see that
the change in the degeneracy depends only upon the amount of energy that has
been added to the system and is independent of the specific energy levels between
which an oscillator has been moved. The same end result is obtained in the event
that the added energy is distributed amongst several oscillators, rather than simply
taken up by a single oscillator, so that the result in Eq. (3.1.4) also does not depend
upon the number of oscillators that are redistributed.
If we take the natural logarithm of both sides of Eq. (3.1.4), we arrive at the result
ln
= ln
− ln ≡ ln = βq ,
(3.1.5)
which implies that the change in ln is proportional to the amount of energy added
to (q > 0) or removed from(q < 0) the system of oscillators.
Now, let us consider two systems, which we shall label as A and B, with system
A having N A oscillators, total energy E A , and at equilibrium is characterized by
β A ; similarly, system B has N B oscillators, total energy E B , and at equilibrium
is characterized by β B . Let us place systems A and B in thermal contact, thereby
allowing an amount q of energy to be transferred spontaneously from one system
to the other, with N A , N B , and E tot = E A + E B all fixed. Initially, for β A = β B ,
each system is in its most likely macrostate, with A , B microstates, respectively.
Because each of the A microstates of system A can be combined with each of
the B microstates of system B, the total number of microstates accessible to the
composite (combined) system A⊕B is tot = A B . 3
3 We shall refer to systems A and B in thermal contact as subsystems of the (composite/combined)
system A⊕B.