## Classical Limit When the number of particles becomes very large ($N \gg 1$), all probabilities take on a Guassian distribution. $N \gg 1 \hspace{1cm} \Rightarrow \hspace{1cm} \bar{n}_{s} \ll 1 \hspace{0.5cm} , \hspace{0.5cm} e^{\beta H} \gg 1$ ## Equipartition Energy The principle of equipartition states, *"In a system in thermal equilibrium, on the average, an equal amount of energy will be associated with each degree of freedom."* ## Ensembles ### Microcanonical To determine the number of microstates ($\Omega$) in a system, we have to use a variety of counting methods. Many of them involve using the *combinations* and *permutations* formulas. $\ce{_{r}C_{n}} = \begin{pmatrix} n \\ r \end{pmatrix} = \frac{n!}{r! \, (n - r)!} \hspace{2cm} \ce{_{r}P_{n}} = \frac{n!}{(n - r)!}$ The probability distribution function (PDF) is then: $ p(\mu) = \frac{1}{\Omega} \; \begin{cases} 1 & \text{if } E = E_{\mu} \\ 0 & \text{otherwise} \end{cases} $ ### Maxwell-Boltzmann *(Also known as the **Canonical Distribution**)* **Maxwell-Boltzmann Statistics** describes a system containing a large number of identical non-interacting, non-relativistic classical particles in thermodynamic equilibrium. *(Only valid when the temperature is high enough or the particle density is low enough to render quantum effects negligible.)* **Single Partition Function:**$Z_{1} = e^{- \beta H} \quad\hspace{1cm} \text{where} \hspace{1cm} \beta \equiv \frac{1}{k_{\rm B} T}$ **Partition Function:**$Z = \frac{1}{N!} \left( Z_{1} \right)^{N} = \sum_{\mu}^{N} e^{- \beta H_{\mu}}$ **Probability Distribution Function (PDF):**$p(\mu) = \frac{1}{Z} e^{- \beta H_{\mu}} \hspace{1cm} \Rightarrow \hspace{1cm} p(v) = \left[ \frac{m}{2 \pi k_{\rm B} T} \right]^{3/2} \exp\left( - \frac{m v^{2}}{2 k_{\rm B} T} \right)$ **Average Number Density:**$\bar{n}_{s} = \sum n_{s} p(\mu_{s}) = - \frac{1}{\beta} \frac{\partial \ln(Z)}{\partial E_{s}} \hspace{1cm} \Rightarrow \hspace{1cm} \bar{n}_{\rm s} = g_{\rm s} \left(\frac{m_{\rm s} k_{\rm B} T}{2 \pi \hbar^{2}}\right)^{3/2} \exp \left( - \frac{m_{\rm s} c^{2}}{k_{\rm B} T} \right)$ ### Fermi-Dirac & Bose-Einstein *(Also known as the **Quantum Canonical Distribution**)* The **Fermi-Dirac Statistics** ($+$) should be used when dealing with non-interacting, identical particles that obey the Pauli exclusion principle. Specifically, this applies to fermions - particles with odd, half-integer spins. The **Bose-Einstein Statistics** ($-$) should be used when dealing with non-interacting, identical particles that do not obey the Pauli exclusion principle. Specifically, this applies to bosons - particles with integer spins. > [!note] > In the following equations, the *Fermi-Dirac* distributions are represented by choosing the **+** signs. Similarly, the *Bose-Einstein* distributions are represented by choosing the **-** signs. **Single Partition Function:**$Z_{1} = \left( 1 \pm e^{- \beta H} \right)^{\pm 1}$ **Partition Function:**$Z = \prod_{i}^{N} Z_{1}$ **Average Number Density:**$\bar{n}_{s} = \left( e^{\beta H} \pm 1 \right)^{-1}$ ## Partition Function Relationships Where $\Omega$ is the number of microstates, $Z$ is the partition function, and $\mathcal{Z}$ is the Gibb's partition function... **Entropy using Microcanonical Ensemble:**$S = k_{\rm B} \ln \Omega$ **Helmholtz Free Energy:**$F = - k_{\rm B} T \, \ln(Z)$ **Gibbs Free Energy:**$G = - k_{\rm B} T \, \ln(\mathcal{Z})$ ## Expectation Values and Canonical Pairs ### Canonical Pairs For any set of canonical pairs $\left\{ \, (T ,S) \, , \, (J, x) \, , \, (\mu, N) \, , \, \dots \, (\mathcal{O}_{1}, \mathcal{O}_{2}) \, \right\}$, we can calculate the averages of any of the powers by following the pattern... $ \begin{aligned} \langle \mathcal{O}_{1} \rangle_{c} &= - \frac{1}{\beta} \frac{\partial \ln(Z)}{\partial \mathcal{O}_{2}} \\ \langle \mathcal{O}_{1}^{2} \rangle_{c} &= - \frac{1}{\beta} \frac{\partial \langle \mathcal{O}_{1} \rangle_{c}}{\partial \mathcal{O}_{2}} \\ \langle \mathcal{O}_{1}^{3} \rangle_{c} &= - \frac{1}{\beta} \frac{\partial \langle \mathcal{O}_{1}^{2} \rangle_{c}}{\partial \mathcal{O}_{2}} \\ &\dots \end{aligned} \hspace{1cm} \underbrace{\Leftrightarrow}_{\text{flip the negative}} \hspace{1cm} \begin{aligned} \langle \mathcal{O}_{2} \rangle_{c} &= - \frac{1}{\beta} \frac{\partial \ln(Z)}{\partial \mathcal{O}_{1}} \\ \langle \mathcal{O}_{2}^{2} \rangle_{c} &= - \frac{1}{\beta} \frac{\partial \langle \mathcal{O}_{2} \rangle_{c}}{\partial \mathcal{O}_{1}} \\ \langle \mathcal{O}_{2}^{3} \rangle_{c} &= - \frac{1}{\beta} \frac{\partial \langle \mathcal{O}_{2}^{2} \rangle_{c}}{\partial \mathcal{O}_{1}} \\ &\dots \end{aligned} $ ### Energy For the average energy to the $n^{th}$ power... $\langle E^{n} \rangle_{c} = (-1)^{n} \frac{\partial ^{n} \ln (Z)}{\partial \beta^{n}}$ If dealing with a... $ \begin{alignat}{6} &\textbf{Monatomic Gas:} \hspace{1cm}& 3 \text{ trans d.o.f.} &+ 0 \text{ rot d.o.f.} &= 3 \text{ d.o.f.} &\hspace{1cm} \Rightarrow \hspace{1cm} \langle E \rangle &= 3 \, (\tfrac{1}{2} k_{\rm B} T) = \tfrac{3}{2} k_{\rm B} T \\ &\textbf{Diatomic Gas:} \hspace{1cm}& 3 \text{ trans d.o.f.} &+ 2 \text{ rot d.o.f.} &= 5 \text{ d.o.f.} &\hspace{1cm} \Rightarrow \hspace{1cm} \langle E \rangle &= 5 \, (\tfrac{1}{2} k_{\rm B} T) = \tfrac{5}{2} k_{\rm B} T \\ &\textbf{Polyatomic Gas:} \hspace{1cm}& 3 \text{ trans d.o.f.} &+ 3 \text{ rot d.o.f.} &= 5 \text{ d.o.f.} &\hspace{1cm} \Rightarrow \hspace{1cm} \langle E \rangle &= 6 \, (\tfrac{1}{2} k_{\rm B} T) = 3 k_{\rm B} T \\ \end{alignat} $