3.1 Microscopic Configurations
133
are, however, major differences in the corresponding degeneracies associated with
the seven resulting macrostates. We obtain degeneracies
I = 10, , II = III = 90, , IV = V = 360, , VI = 840, , VII = 252,
giving a total of 2002 microstates and relative probabilities p I 0.0050, p II =
p III 0.0450, p IV = p V 0.1798, p VI 0.4196, p VII 0.1259, for the seven
macrostates.
A comparison between these last two simple cases, in which five quanta of
energy are shared firstly amongst five and then amongst ten oscillators, shows that
a doubling of the number of oscillators available to take up the five quanta of
energy leads to an approximately 16-fold increase in the total number of microstates
available to the system. By continuing with this type of argument we would obtain
a result that is characteristic of any assembly with a large number N of units. More
specifically, the outcome would be that for very large assemblies by far the largest
number of microstates are associated with a relatively small number of macrostates
that are not only very similar to the predominant macrostate but also give rise to
properties that are essentially indistinguishable on a macroscopic scale.
How does the fundamental postulate enter into this? Well, were it not for this
postulate, we would be unable to conclude that the macrostate containing the largest
number of microstates is the most probable macrostate: this association of ‘largest
number’ of microstates with the ‘most probable’ macrostate is directly based upon
the inherent assumption that every microstate has precisely the same probability
of occurrence. Otherwise, a very large number of microstates occurring in one
macrostate with an associated very small probability factor could be equivalent to
a much smaller number of microstates in another macrostate, but associated with a
proportionally larger probability of occurrence.
Our considerations thus far have been based upon the assumption that the
oscillators in a given macrostate do not interact with one another. In reality, this is
never the case. Hence, let us examine what effect interactions between the oscillators
will have on the potential evolution [2] of a system of particles. 1 To begin with, we
shall assume that any interactions amongst oscillators in a given macrostate will
require them to be simultaneously located in close proximity to one another, so
that two-body interactions will clearly dominate over three-body and other manybody interactions. As a consequence of such an assumption, N -body interactions
will be increasingly less probable the higher the value of N : for the following
discussion, we shall treat any interactions that require more than two oscillators
to be simultaneously in close proximity as improbable. We shall therefore ignore
any outcome that would require simultaneous interaction between more than two
oscillators in order to occur.
As a concrete illustration of our expectations, let us examine transitions that
can take place if we start from macrostate III obtained from sharing five quanta
1 The remainder of this subsection utilizes a line of reasoning introduced in reference [2].
133
are, however, major differences in the corresponding degeneracies associated with
the seven resulting macrostates. We obtain degeneracies
I = 10, , II = III = 90, , IV = V = 360, , VI = 840, , VII = 252,
giving a total of 2002 microstates and relative probabilities p I 0.0050, p II =
p III 0.0450, p IV = p V 0.1798, p VI 0.4196, p VII 0.1259, for the seven
macrostates.
A comparison between these last two simple cases, in which five quanta of
energy are shared firstly amongst five and then amongst ten oscillators, shows that
a doubling of the number of oscillators available to take up the five quanta of
energy leads to an approximately 16-fold increase in the total number of microstates
available to the system. By continuing with this type of argument we would obtain
a result that is characteristic of any assembly with a large number N of units. More
specifically, the outcome would be that for very large assemblies by far the largest
number of microstates are associated with a relatively small number of macrostates
that are not only very similar to the predominant macrostate but also give rise to
properties that are essentially indistinguishable on a macroscopic scale.
How does the fundamental postulate enter into this? Well, were it not for this
postulate, we would be unable to conclude that the macrostate containing the largest
number of microstates is the most probable macrostate: this association of ‘largest
number’ of microstates with the ‘most probable’ macrostate is directly based upon
the inherent assumption that every microstate has precisely the same probability
of occurrence. Otherwise, a very large number of microstates occurring in one
macrostate with an associated very small probability factor could be equivalent to
a much smaller number of microstates in another macrostate, but associated with a
proportionally larger probability of occurrence.
Our considerations thus far have been based upon the assumption that the
oscillators in a given macrostate do not interact with one another. In reality, this is
never the case. Hence, let us examine what effect interactions between the oscillators
will have on the potential evolution [2] of a system of particles. 1 To begin with, we
shall assume that any interactions amongst oscillators in a given macrostate will
require them to be simultaneously located in close proximity to one another, so
that two-body interactions will clearly dominate over three-body and other manybody interactions. As a consequence of such an assumption, N -body interactions
will be increasingly less probable the higher the value of N : for the following
discussion, we shall treat any interactions that require more than two oscillators
to be simultaneously in close proximity as improbable. We shall therefore ignore
any outcome that would require simultaneous interaction between more than two
oscillators in order to occur.
As a concrete illustration of our expectations, let us examine transitions that
can take place if we start from macrostate III obtained from sharing five quanta
1 The remainder of this subsection utilizes a line of reasoning introduced in reference [2].