10.2 Statistical Basis of Thermodynamics
423
waves for the possible alternative states can constructively or destructively interfere
with each other, unless the phases of the component waves have a random character.
In the classical limit, averages over each distinct quantum state become averages
over positions and momenta. 4
With our probabilistic interpretation of thermodynamics, a macrostate in thermodynamic equilibrium has a distribution of its particles amongst the most likely
distributions. The more likely distributions are those that can be created in many
ways microscopically, just as the distribution of heads and tails of 100 tossed pennies
will be near half-and-half because there are lots of ways for the coins to land
that way, and only a few if most of the coins end up heads (or tails). At a finite
temperature, the occupation of many of the microstates in the system will change,
but if there are lots of microstates available which make a given macroscopic state,
then the system will spend more time among these states than those which have only
a few ways to be generated.
To see which distributions are more likely, we imagine all particles in a system
occupying possible quantized energy states, with n i particles in the energy state i .
The range of the index i may be finite or extend to infinity. As we expressed before,
the total energy of the system is then
E =
i
n i i .
(10.1)
In the following argument, we take the system as isolated, so that E is fixed. In
addition, we assume that the total number of particles N is fixed, 5 with
N =
i
n i .
(10.2)
The number of ways W of putting all N of the particles of a system into the
possible quantum energy levels i will now be considered. The fact that particles
such as electrons, atoms, and molecules are indistinguishable is a critical realization
in finding W . 6 Unlike billiard balls, according to quantum theory, we cannot tell one
4 Nicely, quantum descriptions over discrete values are simpler to express than over a continuum.
An even more extreme extension of discreteness would propose that all natural processes involve
a finite number of logical relationships among integers. In such a universe, no continuum would
exist. Evolution of this universe would be definite, and all information initially present would
be preserved. We mortals, however, may lose the ability to keep track of that information, and in
these circumstances introduce entropy as the measure of information which has become practically
inaccessible.
5 If the system is a ‘box of photons’ in equilibrium with the walls of the box, then N is no longer
fixed, so this case is handled separately.
6 Boltzmann’s calculation of W assumed distinguishable particles. With this assumption, the
entropy change when two ideal gases are allowed to mix is not zero. But if both are the same
gas, there should be no entropy of mixing. This is referred to as the Willard Gibbs ‘paradox’,
which was also resolved by Gibbs. In 1875, Gibbs, considering the entropy of mixing of gases,
423
waves for the possible alternative states can constructively or destructively interfere
with each other, unless the phases of the component waves have a random character.
In the classical limit, averages over each distinct quantum state become averages
over positions and momenta. 4
With our probabilistic interpretation of thermodynamics, a macrostate in thermodynamic equilibrium has a distribution of its particles amongst the most likely
distributions. The more likely distributions are those that can be created in many
ways microscopically, just as the distribution of heads and tails of 100 tossed pennies
will be near half-and-half because there are lots of ways for the coins to land
that way, and only a few if most of the coins end up heads (or tails). At a finite
temperature, the occupation of many of the microstates in the system will change,
but if there are lots of microstates available which make a given macroscopic state,
then the system will spend more time among these states than those which have only
a few ways to be generated.
To see which distributions are more likely, we imagine all particles in a system
occupying possible quantized energy states, with n i particles in the energy state i .
The range of the index i may be finite or extend to infinity. As we expressed before,
the total energy of the system is then
E =
i
n i i .
(10.1)
In the following argument, we take the system as isolated, so that E is fixed. In
addition, we assume that the total number of particles N is fixed, 5 with
N =
i
n i .
(10.2)
The number of ways W of putting all N of the particles of a system into the
possible quantum energy levels i will now be considered. The fact that particles
such as electrons, atoms, and molecules are indistinguishable is a critical realization
in finding W . 6 Unlike billiard balls, according to quantum theory, we cannot tell one
4 Nicely, quantum descriptions over discrete values are simpler to express than over a continuum.
An even more extreme extension of discreteness would propose that all natural processes involve
a finite number of logical relationships among integers. In such a universe, no continuum would
exist. Evolution of this universe would be definite, and all information initially present would
be preserved. We mortals, however, may lose the ability to keep track of that information, and in
these circumstances introduce entropy as the measure of information which has become practically
inaccessible.
5 If the system is a ‘box of photons’ in equilibrium with the walls of the box, then N is no longer
fixed, so this case is handled separately.
6 Boltzmann’s calculation of W assumed distinguishable particles. With this assumption, the
entropy change when two ideal gases are allowed to mix is not zero. But if both are the same
gas, there should be no entropy of mixing. This is referred to as the Willard Gibbs ‘paradox’,
which was also resolved by Gibbs. In 1875, Gibbs, considering the entropy of mixing of gases,