1 Kinetics 1.2 Estimates of ideal gasses and the equipartition theorem 1.4 The velocity and speed distributions

1.3 The Boltzmann factor

(September 1, 2023)

A system has total energy $U$ . If a subsystem within the system has energy $\epsilon$ , the rest of the system has energy $U-\epsilon$ . The subsystem should be small and independent of the rest of the system (except in regard to energy exchange), e.g. a molecule in an ideal gas. Probability the subsystem will have energy $\epsilon$ is proportional to $e^{-\epsilon/{k_{\scriptscriptstyle B}T}}$

P(\epsilon)\propto e^{-\epsilon/{k_{\scriptscriptstyle B}T}}

(1.40)

This is the Boltzmann factor. We can simplify $\beta=k_{B}T$ such as to not have to write so much. If you have a set of microscopically small states $i=1...N$ , the sum of all of these probabilities is 1, as shown below. Since $P(\epsilon)=Ce^{-\beta\epsilon}$ we have

\sum_{i}Ce^{-\beta\epsilon_{i}}=1\,,

(1.41)

which determines the constant $C$ which we call $1/Z$

C=\frac{1}{Z}\quad\mbox{with}\quad Z\equiv\sum_{i}e^{-\beta\epsilon_{i}}

(1.42)

$Z$ is known as the partition function and is important in what follows. Then the probability of finding the subsystem in state $r$ with energy $\epsilon_{r}$ is the following

P(\epsilon)=\frac{e^{-\epsilon/{k_{\scriptscriptstyle B}T}}}{Z}

(1.43)

The partition function is a function of the temperature, $Z(\beta)$ . The derivative of $Z$ with respect to (minus) $\beta$ determine the mean energy via the formula

\left\langle\epsilon\right\rangle=\frac{1}{Z(\beta)}\left(-\frac{\partial Z}{% \partial\beta}\right)

(1.44)

Higher derivatives with respect to minus $\beta$ determine higher moments of the energy, e.g. to find the second moment $\left\langle\epsilon^{2}\right\rangle$ we have

\left\langle\epsilon^{2}\right\rangle=\frac{1}{Z(\beta)}\left(-\frac{\partial}% {\partial\beta}\right)^{2}Z(\beta)=\frac{1}{Z(\beta)}\frac{\partial^{2}Z}{% \partial\beta^{2}}

(1.45)