6.14 Maxwell-Boltzmann Distribution

The previous sections showed that the thermal statistics of a system of identical bosons is normally dramatically different from that of a system of identical fermions. However, if the temperature is high enough, and the box holding the particles big enough, the differences disappear. These are ideal gas conditions.

Under these conditions the average number of particles per
single-particle state becomes much smaller than one. That average can
then be approximated by the so-called

Figure 6.16 gives a picture of the distribution for noninteracting particles in a box. The energy spectrum to the right shows the average number of particles per state as the relative width of the red region. The wave number space to the left shows a typical system energy eigenfunction; states with a particle in them are in red.

Since the (anti) symmetrization requirements no longer make a
difference, the Maxwell-Boltzmann distribution is often represented as
applicable to distinguishable

particles. But of
course, where are you going to get a macroscopic number of, say,
1d

in

The Maxwell-Boltzmann distribution was already known before quantum
mechanics. The factor

The example of the isothermal atmosphere can be used to illustrate the
idea of intrinsic chemical potential. Think of the entire atmosphere
as build up out of small boxes filled with particles. The walls of
the boxes conduct some heat and they are very slightly porous, to
allow an equilibrium to develop if you are very patient. Now write
the energy of the particles as the sum of their gravitational
potential energy plus an intrinsic energy (which is just their kinetic
energy for the model of noninteracting particles). Similarly write
the chemical potential as the sum of the gravitational potential
energy plus an intrinsic chemical potential:

Since

However, different boxes have different intrinsic chemical potentials. The entire system of boxes has one global temperature and one global chemical potential, since the porous walls make it a single system. But the global chemical potential that is the same in all boxes includes gravity. That makes the intrinsic chemical potential in boxes at different heights different, and with it the number of particles in the boxes.

In particular, boxes at higher altitudes have less molecules. Compare
states with the same intrinsic, kinetic, energy for boxes at different
heights. According to the Maxwell-Boltzmann distribution, the number
of particles in a state with intrinsic energy

Now suppose that you make the particles in one of the boxes hotter.
There will then be a flow of heat out of that box to the neighboring
boxes until a single temperature has been reestablished. On the other
hand, assume that you keep the temperature unchanged, but increase the
chemical potential in one of the boxes. That means that you must put
more particles in the box, because the Maxwell-Boltzmann distribution
has the number of particles per state equal to

While the Maxwell-Boltzmann distribution was already known
classically, quantum mechanics adds the notion of discrete energy
states. If there are more energy states at a given energy, there are
going to be more particles at that energy, because (6.21)
is per state. For example, consider the number of thermally excited
atoms in a thin gas of hydrogen atoms. The number

The final exponential is due to the Maxwell-Boltzmann distribution. The leading factor arises because there are eight electron states at energy

Key Points

- The Maxwell-Boltzmann distribution gives the number of particles per single-particle state for a macroscopic system at a nonzero temperature.

- It assumes that the particle density is low enough, and the temperature high enough, that (anti) symmetrization requirements can be ignored.

- In particular, the average number of particles per single-particle state should be much less than one.

- According to the distribution, the average number of particles in a state decreases exponentially with its energy.

- Systems for which the distribution applies can often be described well by classical physics.

- Differences in chemical potential promote particle diffusion.