Chapter 6 - Boltzmann Statistics
Section 6.1 - The Boltzmann Factor
Consider some system coupled to a reservoir. Then, we can associate each microstate of the system with some energy level \(E\).
Definition. We say an energy level is degenerate if it corresponds to more than one microstate.
Now, consider two microstates \(s_1\) and \(s_2\), with energies \(E(s_1)\) and \(E(s_2)\) and probabilities \(\mathcal{P}(s_1)\) and \(\mathcal{P}(s_2)\) respectively. While in an isolated system, the fundamental assumption of statistical mechanics tells us that each microstate is equally probable, this is not the case in a closed or open system.
We can now consider the reservoir when the system is in state \(s_1\). There are multiple microstates the reservoir can be in under these constrains, so let us define \(\Omega_R(s_1)\) as the multiplicity of the macrostate of the reservoir while the system is in state \(s_1\).
This then lets us derive the relation
We can then see that as \(S = k ln \Omega\), we can write \(\Omega_R = \exp(S_R/k)\), allowing us to write
If the system is small compared to the reservoir, such as if the system is an atom, we can write \(S_R(s_2) - S_R(s_1) = dS_R\), and invoke the thermodynamic identity. We further can assume that \(P dV_R\), while nonzero, is negligible compared to \(dU_R\). Additionally, we assume \(dN = 0\) for closed systems. Thus,
We can substitute this into the ratio of probabilities to see that
We can rearrange this.
However, for this to be true for all states, the expression must equal a constant, which we denote as \(1/Z\). Thus,
Definition. Here, \(Z\) is the partition function. This is a constant that only depends on temperature, and equals the sum of all unweighted probabilities.