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

$$\fbox{$\displaystyle P_q = \frac{1}{Z} e^{-{\vphantom' E}^{\rm S}_q/{k_{\rm B}}T} $} %$$ (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 [17, 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

$$\fbox{$\displaystyle Z = \sum_{{\rm all}\;q} e^{-{\vphantom' E}^{\rm S}_q/{k_{\rm B}}T} $} %$$ (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.

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.