136
3 Ensembles: Systems of Particles
constrained to be the total number of oscillators, N, in the system, i.e.,
i n i = N .
Now, we note that the ratio of the number of oscillators n i in microstate i to the
number of oscillators n 0 in the lowest-lying microstate can be expressed as
n i
n 0
=
n i
n i−1
·
n i−1
n i−2
· · ·
n 2
n 1
·
n 1
n 0
= r
i .
(3.1.2)
As suggested by Schoepf [2], one convenient way of expressing r is as r = e −ββ ,
with β some positive constant that characterizes the equilibrium condition that we
have imposed. The physical behaviour that we would anticipate is clearly satisfied
by this expression for r: for closely-spaced energy levels ( ≈ 0) r approximates
unity, so that all oscillator energy levels are equally populated, while for widelyspaced energy levels ( 1) r approaches zero, so that almost all of the oscillators
will be found in the very lowest energy levels.
Substitution of r = e −ββ into Eq. (3.1.2) allows us to obtain an expression
relating the occupation numbers in our equilibrium system, namely
n i = n 0 e
−βii
(3.1.3)
which, as we shall see shortly, is just the Boltzmann distribution that holds generally
for an equilibrium system of classical particles. We could thus say that having
equally-spaced energy levels is a sufficient condition for the establishment of a
Boltzmann distribution of particles amongst the available energy states. However,
we shall establish more generally that the attainment of a Boltzmann equilibrium
distribution does not require equally-spaced energy levels, so that it is not a
necessary condition.
We may now ask what happens when a system of oscillators occupying a set of
equilibrium-type configurations is allowed to exchange energy with its (external)
surroundings. We shall focus here upon the consequences for a system of oscillators
that takes up energy from its surroundings, with which it is in (thermal) contact.
It will be clear from what has been said thus far that if energy is added to our
system of oscillators, the number of microstates associated with a particular
macrostate of the system will change. Unlike the changes that occur under energy
conservation, where we are restricted to minimally moving two oscillators, one
into a higher-energy microstate, the other into a correspondingly lower-energy
microstate, an influx of energy can be accommodated by changing the energy state
of a single oscillator. If we go from a configuration having macroscopic energy
E (and corresponding degeneracy ) to a new configuration having macroscopic
energy E > E (and degeneracy ), and if we consider the added energy 2 q to
be taken up by a single oscillator, thereby moving it from microstate energy j to
2 The amount of energy q being added is considered here to be small enough that the new
configuration obtained from its uptake remains a member of the set of already existing equilibrium
macrostates.
3 Ensembles: Systems of Particles
constrained to be the total number of oscillators, N, in the system, i.e.,
i n i = N .
Now, we note that the ratio of the number of oscillators n i in microstate i to the
number of oscillators n 0 in the lowest-lying microstate can be expressed as
n i
n 0
=
n i
n i−1
·
n i−1
n i−2
· · ·
n 2
n 1
·
n 1
n 0
= r
i .
(3.1.2)
As suggested by Schoepf [2], one convenient way of expressing r is as r = e −ββ ,
with β some positive constant that characterizes the equilibrium condition that we
have imposed. The physical behaviour that we would anticipate is clearly satisfied
by this expression for r: for closely-spaced energy levels ( ≈ 0) r approximates
unity, so that all oscillator energy levels are equally populated, while for widelyspaced energy levels ( 1) r approaches zero, so that almost all of the oscillators
will be found in the very lowest energy levels.
Substitution of r = e −ββ into Eq. (3.1.2) allows us to obtain an expression
relating the occupation numbers in our equilibrium system, namely
n i = n 0 e
−βii
(3.1.3)
which, as we shall see shortly, is just the Boltzmann distribution that holds generally
for an equilibrium system of classical particles. We could thus say that having
equally-spaced energy levels is a sufficient condition for the establishment of a
Boltzmann distribution of particles amongst the available energy states. However,
we shall establish more generally that the attainment of a Boltzmann equilibrium
distribution does not require equally-spaced energy levels, so that it is not a
necessary condition.
We may now ask what happens when a system of oscillators occupying a set of
equilibrium-type configurations is allowed to exchange energy with its (external)
surroundings. We shall focus here upon the consequences for a system of oscillators
that takes up energy from its surroundings, with which it is in (thermal) contact.
It will be clear from what has been said thus far that if energy is added to our
system of oscillators, the number of microstates associated with a particular
macrostate of the system will change. Unlike the changes that occur under energy
conservation, where we are restricted to minimally moving two oscillators, one
into a higher-energy microstate, the other into a correspondingly lower-energy
microstate, an influx of energy can be accommodated by changing the energy state
of a single oscillator. If we go from a configuration having macroscopic energy
E (and corresponding degeneracy ) to a new configuration having macroscopic
energy E > E (and degeneracy ), and if we consider the added energy 2 q to
be taken up by a single oscillator, thereby moving it from microstate energy j to
2 The amount of energy q being added is considered here to be small enough that the new
configuration obtained from its uptake remains a member of the set of already existing equilibrium
macrostates.