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

$$\fbox{$\displaystyle \mbox{Maxwell-Boltzmann distributio... ...m p}- \mu)/{k_{\rm B}}T}} \qquad \iota^{\rm{d}} \ll 1 $} %$$ (6.21)

Here ${\vphantom' E}^{\rm p}$ is again the single-particle energy, $\mu$ the chemical potential, $T$ the absolute temperature, and $k_{\rm B}$ the Boltzmann constant. Under the given conditions of a low particle number per state, the exponential is big enough that the $\pm1$ found in the Bose-Einstein and Fermi-Dirac distributions (6.9) and (6.19) can be ignored.

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.

Figure 6.16: Particles at high-enough temperature and low-enough particle density.

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, 10$\POW9,{20}$ particles, each of a different type? The imagin­ation boggles. Still, the “d” in $\iota^{\rm {d}}$ refers to distin­guishable.

The Maxwell-Boltzmann distribution was already known before quantum mechanics. The factor $e^{-{\vphantom' E}^{\rm p}/{k_{\rm B}}T}$ in it implies that the number of particles at a given energy decreases exponentially with the energy. A classical example is the decrease of density with height in the atmosphere. In an equilibrium (i.e. isothermal) atmosphere, the number of molecules at a given height $h$ is propor­tional to $e^{-mgh/{k_{\rm B}}T}$ where $mgh$ is the gravita­tional potential energy of the molecules. (It should be noted that normally the atmosphere is not isothermal because of the heating of the earth surface by the sun and other effects.)

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:

$${\vphantom' E}^{\rm p}= mgh + {\vphantom' E}^{\rm p}_{\rm i} \qquad \mu = mgh + \mu_{\rm i}$$

Since ${\vphantom' E}^{\rm p}-\mu$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\vphantom' E}^{\rm p}_{\rm {i}}-\mu_{\rm {i}}$, the Maxwell-Boltzmann distribution is not affected by the switch to intrinsic quantities. But that implies that the relationship between kinetic energy, intrinsic chemical potential, and number of particles in each individual box is the same as if gravity was not there. In each box, the normal ideal gas law applies in terms of intrinsic quantities.

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 ${\vphantom' E}^{\rm p}_{\rm {i}}$ is 1/$e^{({\vphantom' E}^{\rm p}_{\rm {i}}+mgh-\mu)/{k_{\rm B}}T}$. That decreases with height propor­tional to $e^{-mgh/{k_{\rm B}}T}$, just like classical analysis predicts.

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 $e^{\mu/{k_{\rm B}}T}$. The excess particles will slowly leak out through the slightly porous walls until a single chemical potential has been reestablished. Apparently, then, too high a chemical potential promotes particle diffusion away from a site, just like too high a temperature promotes thermal energy diffusion away from a site.

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 $I_2$ of atoms that are thermally excited to energy $E_2$ is in terms of the number $I_1$ with the ground state energy $E_1$:

$$\frac{I_2}{I_1}= \frac{8}{2}e^{-(E_2-E_1)/{k_{\rm B}}T}$$

The final exponential is due to the Maxwell-Boltzmann distribution. The leading factor arises because there are eight electron states at energy $E_2$ and only two at energy $E_1$ in a hydrogen atom. At room temperature ${k_{\rm B}}T$ is about 0.025 eV, while $E_2-E_1$ is 10.2 eV, so there are not going to be any thermally excited atoms at room temperature.

Key Points

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

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

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

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

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

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Differen­ces in chemical potential promote particle diffusion.