Canonical Probability Distribution

We can think of the interaction of a molecule with the air in a classroom as analogous to the interaction of a small system, , in thermal contact with a heat reservoir, . The air acts like a heat reservoir because its energy fluctuations due to interactions with the molecule are far too small to affect any of its macroscopic parameters. Let us determine the probability, , of finding system in one particular microstate, , of energy , when it is thermal equilibrium with the heat reservoir, .

As usual, we assume fairly weak interaction between and , so that the energies of these two systems are additive. The energy of is not known at this stage. In fact, only the total energy of the combined system, , is known. Suppose that the total energy lies in the range to . The overall energy is constant in time, because is assumed to be an isolated system, so

(7.1) |

where denotes the energy of the reservoir . Let be the number of microstates accessible to the reservoir when its energy lies in the range to . Clearly, if system has an energy then the reservoir must have an energy close to . Hence, because is in one definite state (i.e., state ), and the total number of states accessible to is , it follows that the total number of states accessible to the combined system is simply . The principle of equal a priori probabilities tells us the probability of occurrence of a particular situation is proportional to the number of accessible microstates. Thus,

(7.2) |

where is a constant of proportionality that is independent of . This constant can be determined by the normalization condition

(7.3) |

where the sum is over all possible states of system , irrespective of their energy.

Let us now make use of the fact that system is far smaller than system . It follows that , so the slowly-varying logarithm of can be Taylor expanded about . Thus,

Note that we must expand , rather than itself, because the latter function varies so rapidly with energy that the radius of convergence of its Taylor series is too small for the series to be of any practical use. The higher-order terms in Equation (7.4) can be safely neglected, because . Now, the derivative

(7.5) |

is evaluated at the fixed energy , and is, thus, a constant, independent of the energy, , of . In fact, we know, from Chapter 5, that this derivative is just the temperature parameter characterizing the heat reservoir . Here, is the absolute temperature of the reservoir. Hence, Equation (7.4) becomes

(7.6) |

giving

(7.7) |

where is a constant independent of . The parameter is determined by the normalization condition, which gives

(7.8) |

so that the distribution becomes

(7.9) |

This distribution is known as the

The canonical distribution often causes confusion. People who are familiar with the
principle of equal a priori probabilities, which says that all microstates
are equally probable, are understandably surprised when they come across the
canonical distribution, which says that high energy microstates are markedly less
probable then low energy states. However, there is no need for any
confusion. The principle of equal a priori probabilities applies to
the whole system, whereas the canonical distribution only applies to
a small part of the system. The two results are perfectly consistent.
If the small system is in a microstate with a comparatively high energy,
, then
the remainder of the system (i.e., the reservoir) has a slightly lower energy,
, than
usual (because the overall energy is fixed). The number of accessible microstates
of the reservoir is a very strongly increasing function of its energy. It
follows that if the small system has a
high energy then significantly less states
than usual are accessible to the reservoir, so the number of microstates
accessible
to the overall system is reduced, and, hence, the configuration is comparatively
unlikely. The strong increase in the number of accessible microstates of the
reservoir with increasing
gives rise to the strong (i.e., exponential) decrease
in the likelihood of a state
of the small system with increasing
.
The exponential factor
is called the *Boltzmann factor*.

The canonical distribution gives the probability of finding the small system in one particular state of energy . The probability that has an energy in the small range between and is just the sum of all the probabilities of the states that lie in this range. However, because each of these states has approximately the same Boltzmann factor, this sum can be written

(7.10) |

where is the number of microstates of whose energies lie in the appropriate range. Suppose that system is itself a large system, but still very much smaller than system . For a large system, we expect to be a very rapidly increasing function of energy, so the probability is the product of a rapidly increasing function of , and another rapidly decreasing function (i.e., the Boltzmann factor). This gives a sharp maximum of at some particular value of the energy. As system becomes larger, this maximum becomes sharper. Eventually, the maximum becomes so sharp that the energy of system is almost bound to lie at the most probable energy. As usual, the most probable energy is evaluated by looking for the maximum of , so

(7.11) |

giving

(7.12) |

Of course, this corresponds to the situation in which the temperature of is the same as that of the reservoir. This is a result that we have seen before. (See Chapter 5.) Note, however, that the canonical distribution is applicable no matter how small system is, so it is a far more general result than any that we have previously obtained.