3.1 Microscopic Configurations
135
We have just made a case for the tendency of a system to evolve in time towards
the macrostate having the maximal number of microstates. For our small example
system, there will likely be only one macrostate having a clearly defined maximal
value for (here VI = 840 vs. IV = V = 360). Let us note, however, that
for a system with a large number of oscillators (10 3 , 10 6 , 10 9 , etc.), there may well
be many macrostates that have similarly large numbers of microstates, in which
case once the much larger number of macrostates with markedly smaller values of
have evolved into the members of this set of maximal macrostates, they will fluctuate
amongst one another due to the continuing interactions between the particles making
up the system.
The long-time, equilibrium, behaviour of a system of oscillators is described
by its occupation of a set of macrostates characterized by similarly large numbers
of microstates, so that the ratios of their macrostate degeneracies are close to
unity. If, following Schoepf [2], we consider an energy-conserving change from a
macrostate with degeneracy to a nearby macrostate with degeneracy via the
interaction between two oscillators in a particular energy microstate such that one
oscillator goes into the next-lowest-energy microstate while the other oscillator goes
into the next-highest-energy microstate, the equilibrium requirement that // ≈ 1
can be restated in terms of microstate occupation numbers n k as
=
n i (n i − 1)
(n i+1 + 1)(n i−1 + 1)
≈ 1 .
For a realistic system of oscillators (even a ‘nano-system’) the occupation numbers
n i would be sufficiently large that we could meaningfully make the approximation
n i ± 1 ≈ n i , in which case, our equilibrium criterion becomes
n 2
i
n i+1 n i−1
1 .
Upon carrying out a slight rearrangement of this result, this criterion may be reexpressed as
n i
n i−1
n i+1
n i
.
Thus, for the special case that we have been considering, which consists of
oscillators possessing equally-spaced energy levels, the ratio, r, of the numbers of
oscillators in two adjacent energy levels at equilibrium is the same. Mathematically,
this means that
n i+1
n i
=
n i
n i−1
= · · · =
n 2
n 1
=
n 1
n 0
≡ r ,
so that the occupation numbers n i (i = 0, 1, 2, . . .) form a geometric progression.
For the present case, we will have r < 1, as the sum over the occupation numbers is
135
We have just made a case for the tendency of a system to evolve in time towards
the macrostate having the maximal number of microstates. For our small example
system, there will likely be only one macrostate having a clearly defined maximal
value for (here VI = 840 vs. IV = V = 360). Let us note, however, that
for a system with a large number of oscillators (10 3 , 10 6 , 10 9 , etc.), there may well
be many macrostates that have similarly large numbers of microstates, in which
case once the much larger number of macrostates with markedly smaller values of
have evolved into the members of this set of maximal macrostates, they will fluctuate
amongst one another due to the continuing interactions between the particles making
up the system.
The long-time, equilibrium, behaviour of a system of oscillators is described
by its occupation of a set of macrostates characterized by similarly large numbers
of microstates, so that the ratios of their macrostate degeneracies are close to
unity. If, following Schoepf [2], we consider an energy-conserving change from a
macrostate with degeneracy to a nearby macrostate with degeneracy via the
interaction between two oscillators in a particular energy microstate such that one
oscillator goes into the next-lowest-energy microstate while the other oscillator goes
into the next-highest-energy microstate, the equilibrium requirement that // ≈ 1
can be restated in terms of microstate occupation numbers n k as
=
n i (n i − 1)
(n i+1 + 1)(n i−1 + 1)
≈ 1 .
For a realistic system of oscillators (even a ‘nano-system’) the occupation numbers
n i would be sufficiently large that we could meaningfully make the approximation
n i ± 1 ≈ n i , in which case, our equilibrium criterion becomes
n 2
i
n i+1 n i−1
1 .
Upon carrying out a slight rearrangement of this result, this criterion may be reexpressed as
n i
n i−1
n i+1
n i
.
Thus, for the special case that we have been considering, which consists of
oscillators possessing equally-spaced energy levels, the ratio, r, of the numbers of
oscillators in two adjacent energy levels at equilibrium is the same. Mathematically,
this means that
n i+1
n i
=
n i
n i−1
= · · · =
n 2
n 1
=
n 1
n 0
≡ r ,
so that the occupation numbers n i (i = 0, 1, 2, . . .) form a geometric progression.
For the present case, we will have r < 1, as the sum over the occupation numbers is