11.5 The Canonical Probability Distribution

The particle energy distribution functions in the previous section were derived assuming that the energy is given. In quantum-mechanical terms, it was assumed that the energy had a definite value. However, that cannot really be right, for one because of the energy-time uncertainty principle.

Assume for a second that a lot of boxes of particles are carefully prepared, all with a system energy as precise as it can be made. And that all these boxes are then stacked together into one big system. In the combined system of stacked boxes, the energy is presumably quite unambiguous, since the random errors are likely to cancel each other, rather than add up systematically. In fact, simplistic statistics would expect the relative error in the energy of the combined system to decrease like the square root of the number of boxes.

But for the carefully prepared individual boxes, the future of their lack of energy uncertainty is much bleaker. Surely a single box in the stack may randomly exchange a bit of energy with the other boxes. Of course, when a box acquires much more energy than the others, the exchange will no longer be random, but almost certainly go from the hotter box to the cooler ones. Still, it seems unavoidable that quite a lot of uncertainty in the energy of the individual boxes would result. The boxes still have a precise temperature, being in thermal equilibrium with the larger system, but no longer a precise energy.

Then the appropriate way to describe the individual boxes is no longer in terms of given energy, but in terms of probabilities. The proper expression for the probabilities is deduced in derivation {D.59}. It turns out that when the temperature $T$, but not the energy of a system is certain, the system energy eigenfunctions $\psi^{\rm S}_q$ can be assigned probabilities of the form

P_q = \frac{1}{Z} e^{-{\vphantom' E}^{\rm S}_q/{k_{\rm B}}T}
$} %
\end{displaymath} (11.4)

where $k_{\rm B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.380,65 10$\POW9,{-23}$ J/K is the Boltzmann constant. This equation for the probabilities is called the Gibbs canonical probability distribution. Feynman [18, p. 1] calls it the summit of statistical mechanics.

The exponential by itself is called the “Boltzmann factor.” The normalization factor $Z$, which makes sure that the probabilities all together sum to one, is called the “partition function.” It equals

Z = \sum_{{\rm all}\;q} e^{-{\vphantom' E}^{\rm S}_q/{k_{\rm B}}T}
$} %
\end{displaymath} (11.5)

You might wonder why a mere normalization factor warrants its own name. It turns out that if an analytical expression for the partition function $Z(T,V,I)$ is available, various quantities of interest may be found from it by taking suitable partial derivatives. Examples will be given in subsequent sections.

The canonical probability distribution conforms to the fundamental assumption of quantum statistics that eigenfunctions of the same energy have the same probability. However, it adds that for system eigenfunctions with different energies, the higher energies are less likely. Massively less likely, to be sure, because the system energy ${\vphantom' E}^{\rm S}_q$ is a macroscopic energy, while the energy ${k_{\rm B}}T$ is a microscopic energy level, roughly the kinetic energy of a single atom in an ideal gas at that temperature. So the Boltzmann factor decays extremely rapidly with energy.

Figure 11.7: Probabilities of shelf-number sets for the simple 64 particle model system if there is uncertainty in energy. More probable shelf-number distributions are shown darker. Left: identical bosons, middle: distinguishable particles, right: identical fermions. The temperature is the same as in the previous two figures.
... \put(344.5,-11){\makebox(0,0)[b]{\small 36\%}}

So, what happens to the simple model system from section 11.3 when the energy is no longer certain, and instead the probabilities are given by the canonical probability distribution? The answer is in the middle graphic of figure 11.7. Note that there is no longer a need to limit the displayed energies; the strong exponential decay of the Boltzmann factor takes care of killing off the high energy eigenfunctions. The rapid growth of the number of eigenfunctions does remain evident at lower energies where the Boltzmann factor has not yet reached enough strength.

There is still an oblique energy line in figure 11.7, but it is no longer limiting energy; it is merely the energy at the most probable shelf occupation numbers. Equivalently, it is the expectation energy of the system, defined following the ideas of chapter 4.4.1 as

\langle E \rangle \equiv \sum_{{\rm all}\;q} P_q {\vphantom' E}^{\rm S}_q \equiv E

because for a macroscopic system size, the most probable and expectation values are the same. That is a direct result of the black blob collapsing towards a single point for increasing system size: in a macroscopic system, essentially all system eigenfunctions have the same macroscopic properties.

In thermodynamics, the expectation energy is called the internal energy and indicated by $E$ or $U$. This book will use $E$, dropping the angular brackets. The difference in notation from the single-particle/shelf/system energies is that the internal energy is plain $E$ with no subscripts or superscripts.

Figure 11.7 also shows the shelf occupation number probabilities if the example 64 particles are not distinguishable, but identical bosons or identical fermions. The most probable shelf numbers are not the same, since bosons and fermions have different numbers of eigenfunctions than distinguishable particles, but as the figure shows, the effects are not dramatic at the shown temperature, ${k_{\rm B}}T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.85 in the arbitrary energy units.