Boltzmann Probability Distribution

We have gained some understanding of the macroscopic properties of the air in a classroom (say). For instance, we know something about its internal energy and specific heat capacity. How can we obtain information about the statistical properties of the molecules that make up this air? Consider a specific molecule. It constantly collides with its immediate neighbor molecules, and occasionally bounces off the walls of the room. These interactions “inform” it about the macroscopic state of the air, such as its temperature, pressure, and volume. The statistical distribution of the molecule over its own particular internal states must be consistent with this macroscopic state. In other words, if we have a large group of such molecules with similar statistical distributions then they must be equivalent to air with the appropriate macroscopic properties. So, it ought to be possible to calculate the probability distribution of the molecule over its internal states from a knowledge of these macroscopic properties.

We can think of the interaction of a molecule with the air in a classroom as analogous to the interaction of a small system, $A$, in thermal contact with a heat reservoir, $A'$. 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, $P_r$, of finding system $A$ in one particular internal state, $r$, of energy $\epsilon_r$, when it is thermal equilibrium with the heat reservoir, $A'$.

As usual, we assume fairly weak interaction between $A$ and $A'$, so that the energies of these two systems are additive. The energy of $A$ is not known at this stage. In fact, only the total internal energy of the combined system, $A^{(0)} = A + A'$, is known. Suppose that the total internal energy lies in the range $U^{(0)}$ to $U^{(0)} + \delta U$. The overall internal energy is constant in time, because $A^{(0)}$ is assumed to be an isolated system, so

$$\epsilon_r + U' = U^{(0)},$$ (5.321)

where $U'$ denotes the internal energy of the reservoir $A'$. Let ${\mit\Omega}'(U')$ be the number of accessible states of the reservoir when its internal energy lies in the range $U'$ to $U' + \delta U$. Clearly, if system $A$ has an energy $\epsilon_r$ then the reservoir $A'$ must have an energy close to $U'=U^{(0)} - \epsilon_r$. Hence, because $A$ is in one definite state (i.e., state $r$), and the total number of states accessible to $A'$ is ${\mit\Omega}'(U^{(0)} - \epsilon_r)$, it follows that the total number of states accessible to the combined system is simply ${\mit\Omega}'(U^{(0)} - \epsilon_r)$. The principle of equal a priori probabilities tells us the probability of occurrence of a particular situation is proportional to the number of accessible states. Thus,

$$P_r = C' \,{\mit\Omega}'(U^{(0)} - \epsilon_r),$$ (5.322)

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

$$\sum_r P_r = 1,$$ (5.323)

where the sum is over all possible states of system $A$, irrespective of their energy. [See Equation (5.3).]

Let us now make use of the fact that system $A$ is far smaller than system $A'$. It follows that $\epsilon_r\ll U^{(0)}$, so the slowly-varying logarithm of $P_r$ can be Taylor expanded about $U' = U^{(0)}$. Thus,

$$\ln P_r = \ln C' +\ln {\mit\Omega}'(U^{(0)}) -\left[\frac{\partial \ln {\mit\Omega}'} {\partial U'} \right]_0 \epsilon_r +\cdots.$$ (5.324)

Note that we must expand $\ln P_r$, rather than $P_r$ 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 (5.324) can be safely neglected, because $\epsilon_r \ll U^{\,(0)}$. Now, the derivative

$$\left[\frac{\partial \ln {\mit\Omega}'}{\partial U'} \right]_0 \equiv \beta$$ (5.325)

is evaluated at the fixed energy $U' = U^{\,(0)}$, and is, thus, a constant, independent of the energy, $\epsilon_r$, of $A$. In fact, we know, from the previous section, that this derivative is just the temperature parameter $\beta = (k_B\,T)^{-1}$ characterizing the heat reservoir $A'$. Here, $T$ is the absolute temperature of the reservoir. Hence, Equation (5.324) becomes

$$\ln P_r = \ln C' + \ln {\mit\Omega}'(U^{\,(0)}) - \frac{\epsilon_r}{k_B\,T},$$ (5.326)

giving

$$P_r = C \exp\left(-\frac{\epsilon_r}{k_B\,T}\right),$$ (5.327)

where $C$ is a constant independent of $r$. The parameter $C$ is determined by the normalization condition, (5.323), which gives

$$C^{\,-1} = \sum_r \exp\left(-\frac{\epsilon_r}{k_B\,T}\right).$$ (5.328)

We conclude that the probability of a measurement of the energy of some system $A$, that is in thermal equilibrium with a heat reservoir of temperature $T$, yielding the result $\epsilon_r$ is

$$P_r = \frac{\exp(-\epsilon_r/k_B\,T)}{\sum_r \exp(-\epsilon_r/k_B\,T)}.$$ (5.329)

This probability distribution is known as the Boltzmann probability distribution.