3.1 Microscopic Configurations
131
3.1.1 Illustration of the Role of the Basic Postulate
Let us construct an ensemble of N systems: if N r have parameter y = E r , then
p r ≡
N r
N
=
r )
(E)
.
(3.1.1)
To illustrate more concretely how the basic postulate of statistical mechanics enters
into these considerations we shall consider as an illustrative example a collection of
simple harmonic oscillators (SHOs), each satisfying the same energy quantization
rule, namely v = (v +
1
2 )hν osc , and localized in space, so that the oscillators are
identical yet distinguishable. We shall proceed by examining three examples [1],
specifically, a set of three quanta of energy to be shared amongst three oscillators,
followed by a set of five quanta of energy to be shared amongst five oscillators, then
by a set of five quanta of energy to be shared amongst ten oscillators. Each of these
examples will be examined in detail in order to help us to understand the role played
by the Fundamental Postulate of Statistical Mechanics enunciated earlier.
(i) Three oscillators sharing three quanta of energy:
There will be three ways in which one of the three oscillators receives all three
quanta of energy, while the other two oscillators receive none. Similarly, there are
six ways in which one oscillator receives two quanta of energy, a second oscillator
receives one quantum of energy, and the third oscillator receives none. Finally, there
is only one way in which each of the three oscillators receives one quantum of
energy. This method of reasoning gives us two pieces of information about the
sharing of three quanta of energy by the three-oscillator system. Firstly, we note that
there will be three generic configurations (arrangements) of the three oscillators,
namely one arrangement in which one oscillator takes up all three quanta of energy,
while the other two oscillators receive none; one arrangement in which one oscillator
receives two quanta of energy, a second oscillator receives one quantum of energy,
and the third oscillator receives none; one arrangement in which each oscillator
receives one quantum of energy. We shall call each such generic configuration a
macrostate of the system, and we shall label it using a Roman numeral. Secondly,
we note that for the first macrostate, I, there are three specific arrangements of
the oscillators, each of which we shall call a microstate of the system. Similarly,
we find that there are six microstates corresponding to macrostate II, and one
microstate corresponding to macrostate III. In general, the number of microstates
corresponding to a given macrostate ‘i’ is typically referred to as its degeneracy
i , {i = I, II, III, . . .}.
A combinatoric expression for the degeneracy of such system macrostates may
be expressed in terms of the number of quanta, n q , available for sharing, the number
of oscillators, N , sharing the n q quanta of energy, and the numbers of oscillators,
N i , that receive i quanta of energy (i = 0, 1, 2, . . . , n q ): this expression for i is
131
3.1.1 Illustration of the Role of the Basic Postulate
Let us construct an ensemble of N systems: if N r have parameter y = E r , then
p r ≡
N r
N
=
r )
(E)
.
(3.1.1)
To illustrate more concretely how the basic postulate of statistical mechanics enters
into these considerations we shall consider as an illustrative example a collection of
simple harmonic oscillators (SHOs), each satisfying the same energy quantization
rule, namely v = (v +
1
2 )hν osc , and localized in space, so that the oscillators are
identical yet distinguishable. We shall proceed by examining three examples [1],
specifically, a set of three quanta of energy to be shared amongst three oscillators,
followed by a set of five quanta of energy to be shared amongst five oscillators, then
by a set of five quanta of energy to be shared amongst ten oscillators. Each of these
examples will be examined in detail in order to help us to understand the role played
by the Fundamental Postulate of Statistical Mechanics enunciated earlier.
(i) Three oscillators sharing three quanta of energy:
There will be three ways in which one of the three oscillators receives all three
quanta of energy, while the other two oscillators receive none. Similarly, there are
six ways in which one oscillator receives two quanta of energy, a second oscillator
receives one quantum of energy, and the third oscillator receives none. Finally, there
is only one way in which each of the three oscillators receives one quantum of
energy. This method of reasoning gives us two pieces of information about the
sharing of three quanta of energy by the three-oscillator system. Firstly, we note that
there will be three generic configurations (arrangements) of the three oscillators,
namely one arrangement in which one oscillator takes up all three quanta of energy,
while the other two oscillators receive none; one arrangement in which one oscillator
receives two quanta of energy, a second oscillator receives one quantum of energy,
and the third oscillator receives none; one arrangement in which each oscillator
receives one quantum of energy. We shall call each such generic configuration a
macrostate of the system, and we shall label it using a Roman numeral. Secondly,
we note that for the first macrostate, I, there are three specific arrangements of
the oscillators, each of which we shall call a microstate of the system. Similarly,
we find that there are six microstates corresponding to macrostate II, and one
microstate corresponding to macrostate III. In general, the number of microstates
corresponding to a given macrostate ‘i’ is typically referred to as its degeneracy
i , {i = I, II, III, . . .}.
A combinatoric expression for the degeneracy of such system macrostates may
be expressed in terms of the number of quanta, n q , available for sharing, the number
of oscillators, N , sharing the n q quanta of energy, and the numbers of oscillators,
N i , that receive i quanta of energy (i = 0, 1, 2, . . . , n q ): this expression for i is