$\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}$
Definition 1.1.1 Temperature denoted $T$ is the macroscopically measurable state function that is only equal when two systems are in thermodynamic equilibrium.
Definition 1.1.2 Work denoted $W$ is energy transferred to a system by macroscopic forces.
Definition 1.1.3 Heat denoted $Q$ is energy transferred to a system by microscopic forces.
Definition 1.1.4 Quasi-static processes are slow transformations where the macroscopic properties remain well defined.
Definition 1.1.5 Diathermic walls are walls that allow heat transfer.
Definition 1.1.6 Adiabatic walls are walls that don't allow heat transfer.
Law 1.1.7 The 1st Law of Thermodynamics states that the exact differential energy $dE$ of a system is the sum of the inexact differential heat into the system $\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\dj Q$ and the inexact differential work done on the system $\dj W$.
\[dE = \dj Q + \dj W\]
Definition 1.1.8 Heat engine efficiency $\eta = \frac{W}{Q_H} = \frac{Q_H-Q_C}{Q_H}$ is the amount of work extracted per unit heat consumed.
Definition 1.1.9 Refrigerator coefficient of performance $\omega = \frac{Q_C}{W} = \frac{Q_C}{Q_H-Q_C}$ is the amount of heat extracted per unit work used.
Result 1.1.10 The Carnot engine $\eta_{CE} = \frac{T_H-T_C}{T_H} = 1-\frac{T_C}{T_H}$ is the most efficient possible heat engine.
Result 1.1.11 The Carnot refrigerator $\omega_{CR} = \frac{T_H-T_C}{T_C} = \frac{T_H}{T_C}$ is the most efficient possible refrigerator.
Definition 1.2.1 The entropy of a system is the state function $S$ such that the following holds:
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}dS = \frac{dE}{T} + \frac{\mathbf{P}\cdot d\mathbf{V}}{T}\]
where $\mathbf{P}$ are the generalized pressures of the system and $\mathbf{V}$ are the generalized volumes of the system.
Theorem 1.2.2 Clausius's Theorem states that for an arbitrary cyclic process in phase space $\mathbf{\lambda}$, the integral of differential heat over temperature is non-positive.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\oint\frac{\dj Q(\mathbf{\lambda})}{T(\lambda)} d\mathbf{\lambda}\leq 0\]
Law 1.2.3 The 2nd Law of Thermodynamics states that the entropy of a closed system is non-decreasing.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}} dS \geq \frac{\dj Q}{T}\]
Definition 1.2.4 Reversible processes are processes where $\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}} dS = \frac{\dj Q}{T}$ during the entire process.
Definition 1.2.5 Irreversible processes are processes where $\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}} dS > \frac{\dj Q}{T}$ at some point during the process.
Result 1.2.6 Reversible processes are quasi-static, path independent and do not change the entropy of the system.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\oint\frac{\dj Q(\mathbf{\lambda})}{T(\lambda)}\partial\mathbf{\lambda} = S_f - S_i = 0\quad\text{ (for reversible processes)}\]
Result 1.2.7 Irreversible processes result in a net increase in entropy.
\[\oint\frac{\dj Q(\mathbf{\lambda})}{T(\lambda)}\partial\mathbf{\lambda} > S_f - S_i > 0\quad\text{ (for irreversible processes)}\]
Definition 1.3.1 The partial pressures $\{P_i\}$ are the pressures of the system exerted by each type of particle in the system.
Definition 1.3.2 The volumes $\{V_i\}$ are the amounts of space occupied by each type of particle in the system.
Definition 1.3.3 The generalized forces $\{J_i\}$ are the forces acting on the system.
Definition 1.3.4 The generalized displacements $\{x_i\}$ are the displacements of the system for each of the generalized forces acting on the system.
Definition 1.3.5 The physical work $W_{phy}$ is the work done on the system by generalized forces or partial pressures of the system.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\dj W_{phy} = \sum_i{J_i dx_i} = \mathbf{J}\cdot d\mathbf{x} = \sum_i{-P_i dV} = -\mathbf{P}\cdot d\mathbf{V}\]
Definition 1.3.6 The chemical work $W_{chem}$ is the work done on the system by the generalized chemical potentials $\{\mu_\alpha\}$ and the generalized numbers of particles $\{N_\alpha\}$ which is the work done by a change in particle number.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\dj W_{chem} = \sum_\alpha{\mu_\alpha dN_\alpha} = \mathbf{\mu}\cdot d\mathbf{N}\]
Definition 1.3.7 A closed system is a system that cannot exchange particles with the environment, that is $d\mathbf{N} = 0$.
Definition 1.3.8 An open system is a system that can exchange particles with the environment.
Result 1.3.9 The total work $W$ done on a system is the sum of the physical work $W_{phy}$ and the chemical work $W_{chem}$.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\dj W = \dj W_{phy},\quad \quad \text{Closed System}\]\[\dj W = \dj W_{phy} + \dj W_{chem},\quad \text{Open System}\]
Result 1.3.10 Differential energy states that the following holds for all thermodynamic systems.
\[\partial E = T\partial S - \mathbf{P}\cdot\partial\mathbf{V} + \mathbf{\mu}\cdot d\mathbf{N}\]\[T = \left( \frac{\partial E}{\partial S} \right)_{\mathbf{V},\mathbf{N}},\quad -P_i=\left( \frac{\partial E}{\partial V_i} \right)_{S,V_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial E}{\partial N_\alpha} \right)_{S,\mathbf{V},N_{\beta\neq \alpha}}\]
Result 1.3.11 Differential entropy states that the following holds for all thermodynamic systems.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\partial S = \frac{\partial E}{T} + \frac{\mathbf{P}\cdot\partial\mathbf{V}}{T} - \frac{\mathbf{\mu}\cdot d\mathbf{N}}{T}\]\[\frac{1}{T} = \left( \frac{\partial S}{\partial E} \right)_{\mathbf{V},\mathbf{N}},\quad \frac{P_i}{T} = \left( \frac{\partial S}{\partial V_i} \right)_{E,V_{j\neq i},\mathbf{N}},\quad \frac{\mu_\alpha}{T}=\left( \frac{\partial S}{\partial N_\alpha} \right)_{E,\mathbf{V},N_{\beta\neq \alpha}}\]
Result 1.3.12 The Gibbs-Duhem Relation states that $T$, $\mathbf{P}$ and $\mathbf{\mu}$ are related by the following differential equation.
\[SdT - \mathbf{V}\cdot d\mathbf{P} + \mathbf{N}\cdot d\mathbf{\mu} = 0\]
1.4.1 The enthalpy is a state function $H$ defined by
\[H = E+\mathbf{P}\cdot\mathbf{V}\]
Result 1.4.2 For adiabatic systems with constant external forces, the enthalpy is minimized.
\[dH\leq 0\]
Result 1.4.3 Differential enthalpy states that the following holds for all thermodynamic systems.
\[dH = TdS + \mathbf{V}\cdot d\mathbf{P} + \mathbf{\mu}\cdot d\mathbf{N}\]\[T = \left( \frac{\partial H}{\partial S} \right)_{\mathbf{P},\mathbf{N}},\quad V_i = \left(\frac{\partial H}{\partial P_i}\right)_{S,P_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial H}{\partial N_\alpha} \right)_{S,\mathbf{P},N_{\beta\neq \alpha}}\]
Definition 1.5.1 Isothermal processes are processes where the temperature is constant.
Definition 1.5.2 The Helmholtz free energy is a state function $F$ defined by
\[F = E-TS\]
Result 1.5.3 For isothermal systems with no external work, the Helmholtz free energy is minimized.
\[dF \leq 0\]
Result 1.5.4 Differential Helmholtz free energy states that the following holds for all thermodynamic systems.
\[dF = -SdT - \mathbf{P}\cdot d\mathbf{V} + \mathbf{\mu}\cdot d\mathbf{N}\]\[-S = \left( \frac{\partial F}{\partial T} \right)_{\mathbf{V},\mathbf{N}},\quad -P_i = \left(\frac{\partial F}{\partial V_i}\right)_{T,V_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial F}{\partial N_\alpha} \right)_{T,\mathbf{V},N_{\beta\neq \alpha}}\]
Definition 1.6.1 The Gibbs free energy is a state function $G$ defined by
\[G = E-TS+\mathbf{P}\cdot\mathbf{V}\]
Result 1.6.2 For isothermal systems with constant external forces, Gibbs free energy is minimized.
\[dG\leq 0\]
Result 1.6.3 Differential Gibbs free energy states that the following holds for all closed thermodynamic systems.
\[dG =-SdT + \mathbf{V}\cdot d\mathbf{P} + \mathbf{\mu}\cdot d\mathbf{N}\]\[-S = \left(\frac{\partial G}{\partial T}\right)_{\mathbf{P},\mathbf{N}},\quad V_i = \left(\frac{\partial G}{\partial P_i}\right)_{T,P_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial G}{\partial N_\alpha} \right)_{T,\mathbf{P},N_{\beta\neq \alpha}}\]
Definition 1.7.1 The grand potential is a state function $\mathcal{G}$ defined by
\[\mathcal{G} = E - TS - \mathbf{\mu}\cdot\mathbf{N}\]
Result 1.7.2 For isothermal systems in chemical equilibrium with no external work, the grand potential is minimized
\[d\mathcal{G}\leq 0\]
Result 1.7.3 Differential grand potential states that the following holds for all thermodynamic systems.
\[d\mathcal{G} = -SdT - \mathbf{P}\cdot d\mathbf{V} - \mathbf{N}\cdot d\mathbf{\mu}\]\[-S = \left(\frac{\partial \mathcal{G}}{\partial T}\right)_{\mathbf{V},\mathbf{\mu}},\quad -P_i = \left(\frac{\partial \mathcal{G}}{\partial V_i}\right)_{T,V_{j\neq i},\mathbf{\mu}},\quad -N_\alpha=\left( \frac{\partial \mathcal{G}}{\partial \mu_\alpha} \right)_{T,\mathbf{V},\mu_{\beta\neq \alpha}}\]
Law 1.8.1 The 3rd law of thermodynamics states that the limit of entropy as temperature approaches to zero is a universal constant.
\[\lim_{T\to 0}{S(T,\dots)} = C\]
Definition 2.1.1 The multiplicity function $\Omega$ of a system is the number of possible microstates for a given macrostate.
Law 2.1.2 The Boltzmann Hypothesis states that probability of all possible microstates are equal for a particular macrostate
\[\mathscr{p}_i = \frac{1}{\Omega}\]
Definition 2.1.3 The microcanonical ensemble is the ensemble of statistical mechanics where the macrostates are described by the energy $E$, volumes $\mathbf{V}$ and particle numbers $\mathbf{N}$.
\[\mathscr{p}_i = \frac{1}{\Omega(E,\mathbf{V},\mathbf{N})}\]
Definition 2.1.4 The microcanonical entropy $S$ of a system is the Boltzmann constant times the natural log of the multiplicity function.
\[S = k_B\log\Omega\]
Definition 2.1.5 The temperature $T$ and thermodynamic temperature $\beta$ of a system are defined in terms of the derivative of energy with respect to entropy.
\[T = \frac{\partial E}{\partial S} = \frac{\partial E}{\partial k_B \log\Omega} = \frac{1}{k_B\beta}\]\[\beta = \frac{\partial}{\partial E}\log\Omega = \frac{1}{k_B T}\]
Proposition 2.1.6 Stirling's Approximation states that for sufficiently large $N$, the natural log of $N!$ can be approximated.
\[\log(N!) \approx N\log(N) - N\]
Definition 2.1.7 The ensemble average denoted $\langle \mathscr{O}\rangle$ of a variable $\mathscr{O}$ is sum of the value for all microstates weighted by their probabilities.
\[\langle \mathscr{O}\rangle = \sum_{i}{\mathscr{O}_i\mathscr{p}_i}\]
Definition 2.2.1 The canonical ensemble is the ensemble of statistical mechanics where the macrostates are described by the temperature $T$, volumes $\mathbf{V}$ and particles numbers $\mathbf{N}$. The probability of a particular microstate $i$ is written in terms of the energy of the microstate $E_i$, the thermodynamic temperature $\beta$ and the partition function $z$.
\[\mathscr{p}_i = \frac{1}{\Omega(T,\mathbf{V},\mathbf{N})}=\frac{e^{-\beta E_i}}{\sum_{j}{e^{-\beta E_j}}} = \frac{e^{-\beta E_i}}{z}\]\[z = \sum_{j}{e^{-\beta E_j}} = \sum_{j}{e^{-E_j/(k_BT)}}\]
Definition 2.2.2 The canonical energy E of a system in the canonical ensemble is the ensemble average of energy.
\[\langle E\rangle = \sum_{i}{E_i \mathscr{p}_i} = \frac{1}{z}\sum_{i}{\frac{-\partial}{\partial \beta}e^{-\beta E_i}} = -\frac{1}{z}\frac{\partial z}{\partial \beta} = -\frac{\partial}{\partial \beta}\log z\]
Definition 2.2.3 The heat capacity of a system $C_V$ is the derivative of canonical energy in terms of temperature.
\[C_V = \left(\frac{\partial E}{\partial T}\right)_{\mathbf{V},\mathbf{N}} = -k_B \beta^2 \frac{\partial^2}{\partial\beta^2}\log z = -k_B \beta^2 \frac{\partial^2}{\partial\beta^2}(\beta F) = k_B\beta^2(\langle E^2\rangle - \langle E\rangle^2)\]
Result 2.2.4 The Helmholtz free energy $F$ can be written in terms of the temperature and the partition function.
\[F = -\frac{1}{\beta}\log z\]
Result 2.2.5 The pressure $P$ of a system can be written as the ensemble average of pressure for each microstate.
\[P = -\left(\frac{\partial F}{\partial V}\right)_{T,\mathbf{N}} = \frac{\partial}{\partial V}\left(\frac{1}{\beta}\log z\right) = \frac{1}{z}\sum_i{\left(\frac{-\partial E}{\partial V}\right)e^{-\beta E_i}} = \langle P_i\rangle\]
Result 2.2.6 The entropy $S$ of a system can be written in terms of the ensemble average of log of the probability of each microstate.
\[S = -\left(\frac{\partial F}{\partial T}\right)_{\mathbf{V},\mathbf{N}} = k_B\beta^2\left(\frac{\partial F}{\partial \beta}\right)_{\mathbf{V},\mathbf{N}} = -k_B\beta^2\left(\frac{\partial}{\partial \beta}\frac{1}{\beta}\log z\right)_{\mathbf{V},\mathbf{N}} = \frac{k_B}{z}\sum_i{e^{-\beta E_i}(\log z +\beta E_i)}\]
Proposition 2.2.7 Geometric series convergence states that for $|r|<1$ the following infinite series converges to $1/(1-r)$.
\[\sum_{k=0}^\infty{r^k} = \frac{1}{1-r}\]
Definition 2.3.1 A free particle in the canonical ensemble is a system of a single quantum mechanical particle is a cubic box of volume $V=L^3$.
\[\varepsilon_{\vec{n}} = \frac{\hbar^2\pi^2}{2mL}\left(n_x^2 + n_y^2+ n_z^2\right)\]\[z = \sum_{\vec{n}}e^{-\beta\varepsilon_\vec{n}} = \frac{V}{\ell_Q^3}\]
Definition 2.3.2 The Debroglie thermal wavelength denoted $\ell_Q$ is the average wavelength of particles in a free particles system or an ideal gas.
\[\ell_Q = \sqrt{\frac{2\pi\hbar^2}{mk_BT}}\]
Result 2.3.3 The total partition function of non-interacting systems is the product of their individual partition functions.
\[z_{\text{tot}} = z_1z_2\dots\]
Definition 2.3.4 An ideal gas in the canonical ensemble is a system of $N$ indistinguishable non-interacting free particles with the Gibbs factor $\frac{1}{N!}$ to account for the indistinguishably of the quantum particles.
\[z = \frac{1}{N!}z_1^N = \frac{1}{N!}\left(\frac{V}{\ell_Q^3}\right)^N\]
Result 2.3.5 The Helmholtz free energy of an ideal gas with the Gibbs factor in the canonical ensemble is
\[F = k_BT\log\frac{\ell_Q^3}{V} + k_BTN\log N - k_BTN\]
Result 2.3.6 The ideal gas law states that for an ideal gas in the canonical ensemble,
\[PV=Nk_BT\]
Result 2.3.7 The average energy of an ideal gas in the canonical ensemble is
\[\langle E\rangle = \frac{3}{2}Nk_BT\]
Result 2.3.8 The entropy of an ideal gas with the Gibbs factor in the canonical ensemble is
\[S = -\left(\frac{\partial F}{\partial T}\right)_{N,V} = k_BN\left[\frac{5}{2}-\log\frac{N\ell_Q^3}{V}\right]\]
Definition 2.4.1 The grand-canonical ensemble is the ensemble of statistical mechanics where the macrostates are described by the temperature $T$, volumes $\mathbf{V}$, and chemical potentials $\mathbf{\mu}$. The probability of a particular microstate $i$ is written in terms of the energy of the microstate $E_i$, the particle numbers of the microstate $\mathbf{N}_i$, the thermodynamic temperature $\beta$, the chemical potentials $\mathbf{\mu}$ and the grand partition function $z$.
\[\mathscr{p}_i = \frac{1}{\Omega(\mathbf{T},\mathbf{V},\mathbf{\mu})}=\frac{e^{-\beta(E_i-\mathbf{\mu}\cdot\mathbf{N}_i)}}{\sum_\mathbf{N}{\sum_j{e^{-\beta(E_j-\mathbf{\mu}\cdot\mathbf{N})}}}}=\frac{e^{-\beta(E_i-\mathbf{\mu}\cdot\mathbf{N}_i)}}{\mathscr{z}}\]\[\mathscr{z} = \sum_\mathbf{N}{\sum_j{e^{-\beta(E_j-\mathbf{\mu}\cdot\mathbf{N})}}}\]
Result 2.4.2 The grand potential $\mathcal{G}$ can be written in terms of the temperature and the grand partition function.
\[\mathcal{G} = -\frac{1}{\beta}\log \mathscr{z}\]
Definition 2.5.1 The Hamiltonian denoted $\mathcal{H}$ of a classical system is a function that represents the total energy of the system.
Law 2.5.2 Hamilton's Equations state that classical systems with Hamiltonian $\mathcal{H}$ evolve according to the following differential equations, where $q$ is the position and $p$ is the momentum.
\[\frac{\partial q_i}{\partial t}=\frac{\partial \mathcal{H}}{\partial p_i},\quad \frac{\partial p_i}{\partial t} = -\frac{\partial \mathcal{H}}{\partial q_i}\]
Definition 2.5.3 The probability density function denoted $\rho(q,p)$ is the function whose integral represents the probability of finding a classical system in a given region of phase space.
Theorem 2.5.4 The Liouville Theorem states that for classical systems with Hamiltonian $\mathcal{H}$ and probability density $\rho$,
\[\frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + \{\rho,\mathcal{H}\},\]
where $\{\ ,\ \}$ is a Poisson bracket.
Definition 2.5.5 The classical microcanonical ensemble is the ensemble of statistical mechanics for classical systems where the macrostates are described by the energy $E$, volumes $\mathbf{V}$ and particle numbers $\mathbf{N}$.
\[\rho(p,q) =\frac{\delta(E-\mathcal{H})}{(2\pi\hbar)^{3N}}\]\[\Omega(E) = \int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}\delta(E-\mathcal{H})\]
Result 2.5.6 For large $N$, the following multiplicity functions are equivalent.
\[\Omega(E) = \int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}\delta(E-\mathcal{H})\]\[\Omega(E) = \int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}(\Theta(E-\mathcal{H})-\Theta(E-\Delta-\mathcal{H}))\]\[\Omega(E) = \int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}\Theta(E-\mathcal{H})\]
Definition 2.5.7 A classical ideal gas in the microcanonical ensemble is the system with hamiltonian $\mathcal{H}$ defined by
\[\mathcal{H} = \sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}\]\[\Omega \approx \frac{2\pi^{3N/2}}{(3N/2 - 1)!} \frac{2m^{3N/2}}{N!(2\pi\hbar)^{3N}3N} \frac{3N\Delta}{2E}V^NE^{3N/2}\]
Definition 2.5.8 The classical canonical ensemble is the ensemble of statistical mechanics for classical systems where the macrostates are described by temperature $T$, volumes $\mathbf{V}$, and particles numbers $\mathbf{N}$.
\[\rho(\mathbf{p},\mathbf{q}) = \frac{e^{-\beta \mathcal{H}}}{z}\]\[z = \frac{1}{N!}\int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}e^{-\beta\mathcal{H}}\]\[z = \frac{1}{N!}(z_1)^N,\quad z_1 = \int\frac{d^{3}qd^{3}p}{(2\pi\hbar)^{3}}e^{-\beta\mathcal{H}}\]
Definition 2.5.9 The classical grand-canonical ensemble is the ensemble of statistical mechanics for classical systems where the macrostates are described by temperature $T$, volumes $\mathbf{V}$, and chemical potentials $\mathbf{\mu}$.
\[\rho(\mathbf{p},\mathbf{q},\mathbf{N}) = \frac{e^{-\beta (\mathcal{H}_\mathbf{N}-\mathbf{\mu}N)}}{\mathscr{z}}\]\[\mathscr{z} = \sum_N\frac{1}{N!}\int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}e^{-\beta(\mathcal{H}_\mathbf{N}-\mathbf{\mu}\mathbf{N})}\]
Theorem 2.5.10 The equipartition theorem states that the average energy is proportional to temperature times the number of nonzero coefficients for a Hamiltonian of the following form.
\[\mathcal{H} = \sum_jA_jp_j^2 + \sum_jB_jq_j^2\]\[\langle\mathcal{H}\rangle = \frac{1}{2}fk_BT\]
where $f$ is the number of non-vanishing coefficients $A_j$ and $B_j$.
Definition 2.6.1 The density operator denoted $\hat{\rho}$ is the quantum mechanical operator that represents the probability of a state.
\[\hat{\rho} = \sum_\alpha\mathscr{p}_\alpha\ket{\Psi_\alpha(t)}\bra{\Psi_\alpha(t)}\]
where $\mathscr{p}_\alpha$ is the probability of the state $\ket{\Psi_\alpha(t)}$.
Result 2.6.2 The trace of the density operator is one, $\text{Tr}(\hat{\rho}) = \sum_\alpha\mathscr{p}_\alpha = 1$.
Result 2.6.3 The density operator is Hermitian with eigenstates $\ket{\rho_i}$ and corresponding real eigenvalues $w_i$, such that
\[\hat{\rho} = \sum_iw_i\ket{\rho_i}\bra{\rho_i}.\]
Result 2.6.4 The square of the density operator is itself, if and only if the system is a pure quantum state $\ket{\Psi}$. \[\hat{\rho}^2=\hat{\rho} \quad \Leftrightarrow \quad \hat{\rho} = \ket{\Psi}\bra{\Psi}\]
Result 2.6.5 The time dependence of the density operator is given by the commutation relation of the Hamiltonian with the density operator.
\[i\hbar\frac{\partial }{\partial t}\hat{\rho}(t) = [\hat{H},\hat{\rho}(t)]\]
Result 2.6.6 For systems in equilibrium, there exist simultaneous eigenvalues $\ket{E_i}$ of $\hat{H}$ and $\hat\rho$ such that
\[\hat{\rho}\ket{E_i} = w_i\ket{E_i},\quad \hat{H}\ket{E_i} = E_i\ket{E_i}.\]\[\hat{\rho} = \sum_iw_i\ket{E_i}\bra{E_i}.\]
Definition 2.6.7 The quantum ensemble average denoted $\langle \mathscr{O}\rangle$ of a variable $\mathscr{O}$ is sum of the quantum expectation value for all microstates weighted by their probabilities.
\[\langle \mathscr{O}\rangle = \sum_{\alpha}{\mathscr{p}_\alpha \bra{\Psi_\alpha(t)} \hat{\mathscr{O}} \ket{\Psi_\alpha(t)} } = \text{Tr}(\hat{\rho}\hat{\mathscr{O}}) = \sum_n{\bra{E_n}\hat{\rho}\hat{\mathscr{O}}\ket{E_n}} = \sum_n{w_n\bra{E_n}\hat{\mathscr{O}}\ket{E_n}}\]
Definition 2.6.8 The quantum microcanonical ensemble is the ensemble of statistical mechanics for quantum systems where the macrostates are described by the energy $E$, volumes $\mathbf{V}$ and particle numbers $\mathbf{N}$.
\[w_i = \frac{\delta_{E,E_i}}{\Omega(E)}\]
Definition 2.6.9 The quantum canonical ensemble is the ensemble of statistical mechanics for quantum systems where the macrostates are described by temperature $T$, volumes $\mathbf{V}$, and particles numbers $\mathbf{N}$.
\[w_i = \frac{e^{-\beta E_i}}{\text{Tr}(e^{-\beta \hat{H}})} = \frac{e^{-\beta E_i}}{\sum_{j}\bra{E_j}e^{-\beta \hat{H}}\ket{E_j}} = \frac{e^{-\beta E_i}}{\sum_{j}{e^{-\beta E_j}}} = \frac{e^{-\beta E_i}}{z}\]\[z = \text{Tr}(e^{-\beta \hat{H}}) = \sum_{j}\bra{E_j}e^{-\beta \hat{H}}\ket{E_j} = \sum_{j}{e^{-\beta E_j}}\]
Definition 2.6.10 The quantum grand-canonical ensemble is the ensemble of statistical mechanics for quantum systems where the macrostates are described by temperature $T$, volumes $\mathbf{V}$, and chemical potentials $\mathbf{\mu}$.
\[w_{i,N} = \frac{e^{-\beta (E_{i,N}-\mu{N})}}{\text{Tr}_{FS}(e^{-\beta (\hat{H}-\mu\hat{N}) })} = \frac{e^{-\beta (E_{i,N}-\mu N)}}{\sum_N\sum_{j}e^{-\beta (E_j-\mu N)}} = \frac{e^{-\beta (E_{i,N}-\mu N)}}{\mathscr{z}}\]\[\mathscr{z} = \text{Tr}_{FS}(e^{-\beta (\hat{H}-\mu\hat{N}) }) = \sum_N\sum_{j}e^{-\beta (E_j-\mu N)}\]
Definition 3.1.1 The exchange operator is the operator $P_{a,b}$ that exchanges the quantum states of particles $a,b$.
\[P_{a,b}\Psi(a,b) = \Psi(b,a)\]
Result 3.1.2 For identical particles, the exchange operator can introduce a phase factor of $\phi = 0$ or $\phi = \pi$.
\[P_{a,b}\Psi(a,b) = \Psi(b,a) = e^{i\phi}\Psi(a,b) = \eta\Psi(a,b)\quad\text{ where }\eta = \pm 1\]
Definition 3.1.3 The permutation operator is the exchange operator generalized to $N$ particle systems that performs a permutation $P$ on the quantum states of particles in the system.
\[\hat{P}\Psi(1,2,\dots, N) = \Psi(P(1),P(2),\dots,P(N))\]
Definition 3.1.4 The parity of a permutation denoted $\sigma(P)$ is the minimum number of pairwise swaps of the permutation $P$.
Result 3.1.5 For identical particles, the permutation operator can introduce a phase factor of $\phi=0$ or $\phi = \pi$ for each pairwise swap.
\[\hat{P}\Psi(1,2,\dots, N) = \Psi(P(1),P(2),\dots,P(N)) = \eta^{\sigma(P)} \Psi(1,2,\dots,N)\quad\text{ where }\eta = \pm 1\]
Definition 3.1.6 A fermion is a particles where a sign flip is introduced by the exchange operator, that is $\eta = -1$.
Definition 3.1.7 A boson is a particle where no sign flip is introduced by the exchange operator, that is $\eta = +1$.
Definition 3.1.8 The antisymmetrizer operator denoted $\mathcal{A}$ is the operator that creates an antisymmetric quantum state.
\[\ket{k_1,k_2,\dots,k_N}_{-} = \mathcal{A}\ket{k_1,k_2,\dots,k_N} = \frac{1}{N!}\sum_{P}{(-1)^{\sigma(P)}\hat{P}\ket{k_1,k_2,\dots,k_N}}\]
Result 3.1.9 The Pauli Principle states that it is impossible to construct an antisymmetric quantum states where two particles are in the same state.
Definition 3.1.10 The symmetrizer operator denoted $\mathcal{S}$ is the operator that create a symmetric quantum state.
\[\ket{k_1,k_2,\dots,k_N}_{+} = \mathcal{S}\ket{k_1,k_2,\dots,k_N} = \frac{1}{\sqrt{N!\prod_{k}{n_k!}}}\sum_{p}{\hat{P}\ket{k_1,k_2,\dots,k_N}}\]
where $n_k$ is the number of particles in state $k$.
Result 3.1.11 Generalized Quantum States for Identical Particles states that the antisymmetrized and symmetrized quantum states for identical fermions or bosons can be written with generalized notation
\[\ket{k_1,k_2,\dots,k_N}_{\eta} = \frac{1}{\sqrt{N! \prod_{k}n_k!}}\sum_{p}\eta^{\sigma(P)}\hat{P}\ket{k_1,k_2,\dots,k_N}\]\[\eta = -1 \text{ for Fermions},\quad \eta = +1 \text{ for Bosons}\]\[n_k \in \{0,1\}\text{ for Fermions},\quad n_k \in \{0,1,2,3,\dots\} \text{ for Bosons}\]\[\sum_k{n_k} = N\]
Result 3.1.12 The completeness relation for identical particles states that for identical particles the completeness relation can be written as a sum of quantum states.
\[I = \sum_{k_1\leq k_2\leq \dots \leq k_N} \ket{k_1,k_2,\dots,k_N}_\eta\ \prescript{}{\eta}{\bra{k_1,k_2,\dots,k_N}} = \frac{1}{N! \prod_{k}n_k!}\sum_{k_1,k_2,\dots,k_N} \ket{k_1,k_2,\dots,k_N}_\eta\ \prescript{}{\eta}{\bra{k_1,k_2,\dots,k_N}}\]
Definition 3.2.1 The identical free particles in a box is the quantum system of $N$ particles in a large box with the following approximate eigenstates.
\[\Psi_{k_1,\dots,k_N}(x_1,\dots,x_N) = \prod_{a=1}^N\frac{e^{ik_a\cdot x_a}}{\sqrt{V}}\]\[k_a = \frac{\pi}{L}(n_{a,x},n_{a,y},n_{a,z}),\quad n_{a,x},n_{a,y},n_{a,z} \in \{1,2,\dots\}\]
Result 3.2.2 For identical free particles, the matrix element of the density operator multiplied by the partition function $\tilde{\rho}_{B,F}$ can be written in terms of the density operator for distinguishable free particles multiplied by the partition function $\tilde{\rho}_D$.
\[\text{Let } \tilde{\rho}(x_1,\dots,x_N|x_1',\dots,x_N') = z\rho(x_1,\dots,x_N|x_1',\dots,x_N') = \bra{x_1,\dots,x_N}e^{-\beta\hat{H}}\ket{x_1',\dots,x_N' }\]\[\tilde{\rho}_{B,F}(x_1,\dots,x_N|x_1',\dots,x_N') = \frac{1}{N!}\sum_P{\eta^{\sigma(P)}\tilde{\rho}_D}(x_1,\dots,x_N|x_1',\dots,x_N')\]
Result 3.2.3 The partition function for identical free particles can be written as a sum of integrals over all permutations of the $N$ particles.
\[z_{B,F} = \frac{1}{N!}\frac{1}{\ell_Q^{3N}}\sum_P{\eta^{\sigma(P)}\int{e^{\frac{-\pi}{\ell_Q^2}\sum_{a=1}^N{(x_a-x_{P(a)})^2}} dx_1,\dots,dx_N}}\]
Result 3.2.4 The partition function for identical free particles can be written as Gibbs term and the quantum exchange correction term.
\[z_{B,F} = \frac{1}{N}\left[\frac{V^N}{\ell_Q^{3N}} + \int\prod_{a=1}^N{d^3x_a}\sum_{P\neq\text{ identity}}\eta^{\sigma(P)}e^{\frac{-\pi}{\ell_Q^2}\sum_a{(x_a-x_{P(a)})^2}}\right]\]
Theorem 3.2.5 The ideal gas approximation theorem states that the ideal gas is a valid approximation when density is much larger than the square of the Debroglie thermal wavelength.
\[\ell_Q^2 >> \left(\frac{V}{N}\right)^{2/3}\]
Result 3.3.1 The grand partition function for identical particles can be written as a product of the grand partition functions for each single partition state, where the possible values of $n_k$ depends on whether the particles are fermions or bosons.
\[\mathscr{z} = \prod_{k}\sum_{n_k}e^{e^{-\beta(E_k-\mu)n_k}} = \prod_k{(1-\eta e^{-\beta(E_k - \mu)})^{-\eta}}\]\[\eta = \begin{cases}
-1 & \text{for fermions} \\
1 & \text{for bosons}
\end{cases}\]
where for bosons we find that $e^{-\beta(\epsilon_k - \mu)} < 1$.
Result 3.3.2 The following thermodynamic quantities can be computed for identical particles in the grand canonical ensemble.
\[\mathcal{G} = \frac{\eta}{\beta}\sum_k{\log(1-\eta e^{-\beta(E_k-\mu)})}\]\[\langle n_k \rangle = \frac{1}{e^{\beta (E_k - \mu)}-\eta}\]\[\langle E \rangle = \sum_k E_k\langle n_k \rangle = \sum_k{\frac{E_k}{e^{\beta(E_k - \mu)}-\eta}}\]\[\langle N\rangle = \sum_k \langle n_k \rangle = \sum_{k}\frac{1}{e^{\beta(E_k-\mu)}-\eta}\]
Definition 3.4.1 The number of accessible states denoted $\Sigma(E)$ is a function of energy that represents the number of energy states with energy less than $\epsilon$.
\[\Sigma(E) = \sum_{\alpha}\theta(E-E_\alpha)\]
Definition 3.4.2 The density of states denoted $g(E)$ is a function of energy that represents the density of states at energy $E$.
\[g(E) = \frac{\partial \Sigma(E)}{\partial E}\]
Result 3.4.3 Any sum over discrete quantum states of a function that depends on energy can be written as an energy integral of that function weighted by the density of states.
\[\sum_{\alpha}f(E_\alpha) \to \int g(E) f(E) dE\]
For identical particles the following thermodynamic quantities can be written in terms of integrals over density of states:
\[\mathcal{G} = \frac{\eta}{\beta} \int_0^\infty g(\epsilon) \log(1-\eta e^{-\beta(\epsilon-\mu)}) d\epsilon\]\[\langle E \rangle = \int_0^\infty g(\epsilon) \frac{\epsilon}{e^{\beta(\epsilon - \mu)}-\eta} d\epsilon\]\[\langle N\rangle = \int_0^\infty g(\epsilon) \frac{1}{e^{\beta(\epsilon-\mu)}-\eta} d\epsilon\]
Result 3.4.4 The density of states $g(\epsilon)$ for a spin-$S$ gas with spin degeneracy $g_S$ in a $D$-dimensional box with energy relation $\epsilon(\mathbf{p})$ is
\[g(\epsilon) = g_s\left(\frac{L}{2\pi\hbar}\right)^D \int{d^Dp\delta(\epsilon - \epsilon(p))} = g_s\left(\frac{L}{2\pi\hbar}\right)^D \int d\Omega_D \frac{p(\epsilon)^{D-1}}{\left|\frac{\partial \epsilon}{\partial p}(p(\epsilon))\right|}\]
Definition 3.4.5 The fugacity is defined for a partitcular temperature and chemical potential as $\mathbb{z} = e^{\beta\mu}$.
Definition 3.5.1 A non-relativistic gas is a quantum gas where the energy eigenstates are related to momentum by the following relation.
\[\epsilon(p) = \frac{p^2}{2m}\]
Definition 3.5.2 The wave vector $k$ is position of the energy eigenstates in reciprocal space and is related to the momentum of the energy eigenstates.
\[k = \frac{p}{\hbar}\]\[k = \frac{\pi}{L}(n_1,n_2,\dots,n_D),\quad \text{ for }n_i\in\mathbb{N}\]\[p = \frac{\hbar\pi}{L}(n_1,n_2,\dots,n_D),\quad \text{ for }n_i\in\mathbb{N}\]
Result 3.5.3 For free particles in a box, we can convert sums of many particles into integrals of momentum or wave vectors.
\[\sum_{n_i} \to \frac{L}{2\pi\hbar}\int_{-\infty}^{\infty}{dp_i},\quad \sum_{n_i} \to \frac{L}{2\pi}\int_{-\infty}^{\infty}{dk_i}\]
Result 3.5.4 The density of states of a non-relativistic 3d Fermi and Bose gas can be derived by applying this result to $\epsilon(p) = \frac{p^2}{2m}$, $p(\epsilon) = \sqrt{2m\epsilon}$.
\[g(\epsilon) = \frac{g_sV}{\sqrt{2}\pi^2\hbar^3}m^{3/2}\sqrt{\epsilon}\]
Definition 3.5.5 The Riemann Zeta Functions for Non-relativistic Quantum Gases is the class of functions $f_m^\eta(\mathbb{z})$ of the following form.
\[f_m^\eta(\mathbb{z}) = \frac{1}{\Gamma(m)} \int_0^\infty{\frac{dx\ x^{m-1}}{\mathbb{z}^{-1}e^x -\eta}}\]
Result 3.5.6 The pressure, energy density, and density of a non-relativistic 3d Fermi and Bose gas are given by
\[\beta P = \beta \frac{\eta}{V\beta}\int_0^\infty{d\epsilon\ g(\epsilon)\log(1-\eta e^{-\beta(\epsilon-\mu)})} = \frac{g_s}{\ell_Q^3}\frac{4}{3\sqrt{\pi}}\int_0^\infty{\frac{dx\ x^{3/2}}{\mathbb{z}^{-1}e^x - \eta}} = \frac{g_s}{\ell_Q^3}f_{5/2}^\eta(\mathbb{z})\]\[\beta\varepsilon = \beta \frac{E}{V} = \beta \int_0^\infty{\frac{d\epsilon\ g(\epsilon)\epsilon}{e^{\beta(\epsilon - \mu)-\eta}}}= \frac{g_s}{\ell_Q^3}\frac{2}{\sqrt{\pi}}\int_0^\infty{\frac{dx\ x^{3/2}}{\mathbb{z}^{-1}e^x - \eta}} = \frac{3}{2}\frac{g_s}{\ell_Q^3}f_{5/2}^\eta(\mathbb{z})\]\[n = \frac{N}{V} = \frac{1}{V}\int_0^\infty{\frac{d\epsilon\ g(\epsilon)}{e^{\beta(\epsilon-\mu)-\eta}}} = \frac{g_s}{\ell_Q^3}\frac{2}{\sqrt{\pi}}\int_0^\infty{\frac{dx\ x^{1/2}}{\mathbb{z}^{-1}e^x - \eta}} = \frac{g_s}{\ell_Q^3}f_{3/2}^\eta(\mathbb{z})\]
Result 3.5.7 The $f_m^\eta(\mathbb{z})$ can be expanded as a geometric series for $\mathbb{z} << 1$
\[f_m^\eta(\mathbb{z}) \approx \sum_{\alpha = 1}^\infty{\eta^{\alpha + 1}\frac{\mathbb{z}^\alpha}{\alpha^m}} = \mathbb{z} + \eta\frac{\mathbb{z}^2}{2^m} + \frac{\mathbb{z}^3}{3^m} + \cdots\]
Definition 3.6.1 A degenerate Fermi gas is a Fermi gas at the low temperature limit $T\to 0$. In this limit the occupation function approaches a step function centered at $\epsilon_\alpha = \mu$.
\[\langle n_\alpha \rangle = f(\epsilon_\alpha) = \frac{1}{e^{\beta(\epsilon - \mu)} + 1} \to \Theta(\mu(T=0) - \epsilon_\alpha)\]
Result 3.6.2 The Fermi level $\epsilon_F$ is the chemical potential of a Fermi gas at temperature goes to zero.
Proposition 3.6.3 The Fermi level can be calculated from the density of states and N by solving one of the following expressions:
\[N = g_s\sum_p\Theta(p_f-|\vec{p}|) \to g_s\left(\frac{L}{2\pi\hbar}\right)^D\int d\Omega_D\int_0^{p_F}{p^{d-1}dp}\]\[N = \int_0^{\epsilon_F}{g(\epsilon) d\epsilon}\]
The energy and degeneracy pressure can also be found with similar expressions.
\[E = g_s\sum_p\Theta(p_f-|\vec{p}|) \to g_s\left(\frac{L}{2\pi\hbar}\right)^D\int d\Omega_D\int_0^{p_F}{p^{d-1}\epsilon(p)dp}\]\[N = \int_0^{\epsilon_F}{g(\epsilon)\epsilon d\epsilon}\]
Result 3.6.4 For a degenerate non-relativistic Fermi gas the Fermi level, average energy, and degeneracy pressure are given by the following expressions.
\[\epsilon_F = \frac{\hbar^2}{2m}\left( \frac{6\pi^2 n}{g_s} \right)^{2/3}\]\[E = \frac{3}{5}N\epsilon_F\]\[P = \frac{2}{5}\frac{N}{V}\epsilon_F = \frac{2}{3}\frac{E}{V}\]
Definition 3.6.5 The Sommerfeld expansion is the expansion of the occupancy function for $T<<0$ of the following form.
\[\langle n_\alpha \rangle = f(\epsilon_\alpha) = \Theta(\mu - \epsilon) + \delta f(\epsilon)\]\[\delta f(\epsilon) = \begin{cases}f(\epsilon)=\frac{1}{e^{\beta(\epsilon - \mu)} + 1} & \epsilon > \mu \\ -f(\epsilon) = \frac{1}{e^{\beta(\epsilon - \mu)} + 1} & \epsilon < \mu\end{cases}\]
Result 3.6.6 The sommerfeld expansion can be applied to the chemical potential to derive the following expression.
\[\mu(T) \approx \epsilon_F(1 - \frac{\pi^2}{12}\frac{k_B^2T^2}{\epsilon_F^2})\]
Definition 3.7.1 A Bose Einstein condensate is a boson gas where a macroscopic number of particles are in the ground state for temperatures much greater than the ground state energy $k_B\tau >> \epsilon_0$.
Result 3.7.2 The number of particles in excited states $N_e$ for a boson gas can be written in terms of the density of states.
\[N_e(\tau) = \int_0^\infty{\frac{d\epsilon\ g(\epsilon)}{e^{\epsilon/\tau} - 1}}\]
Theorem 3.7.3 A Bose Einstein condensate is possible when the integral approximation of $\langle N\rangle$ as $\mu \to 0$ is less than $N$.
\[N \leq \int_0^\infty{\frac{d\epsilon\ g(\epsilon)}{e^{\epsilon/\tau} - 1}} \quad \Rightarrow \quad \text{No BEC}\]\[N \geq \int_0^\infty{\frac{d\epsilon\ g(\epsilon)}{e^{\epsilon/\tau} - 1}}\quad \Rightarrow \quad \text{BEC}\]
Result 3.7.4 For a non-relativistic Bose Gas the Bose Einstein condensate occurs at a critical temperature $T_C$ or critical density $n_C$.
\[N \geq \int_0^\infty{\frac{d\epsilon\ g(\epsilon)}{e^{\epsilon/\tau} - 1}} = 2.612\frac{V}{\ell_Q^3}\]\[T_C = \frac{2\pi\hbar}{k_B m}\left(\frac{N}{2.612 V}\right)^{2/3}\]\[n_C = \frac{2.612}{\ell_Q^3}\]
Result 3.7.5 For $T\leq T_C$, the pressure can be approximated is terms of this expansion.
\[P \approx f_{5/2}^{+1}(1) \frac{k_B T}{\ell_Q^3}\]
Definition 3.7.6 The Riemann zeta function denoted $\zeta(s)$ is a function $\zeta:\mathbb{C}\to\mathbb{C}$ defined by
\[\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{\Gamma(s)}\int_0^\infty{\frac{x^{s-1}}{e^x-1}dx}\]
where $\Gamma(s) = \int_0^\infty{x^{s-1}e^{-x}dx}$ is the gamma function.
Definition 3.8.1 A photon gas is a system consisting of photons in a box with the following energy levels and two polarization modes $\lambda = \pm 1$. This system can be considered at as a Bose gas with $g_s=2$ and $\epsilon(p) = pc$ and $\mu = 0$.
\[k = \frac{2\pi}{L}\sqrt{n_x^2 + n_y^2 + n_z^2}\]\[\epsilon_k = \hbar c k = \frac{2\pi\hbar c}{L}\sqrt{n_x^2 + n_y^2 + n_z^2}\]
Result 3.8.2 For a photon gas in canonical ensemble, the partition function, Helmholtz free energy, pressure, energy and entropy are given by the following equations.
\[z_{k,\lambda} = \sum_{n_{k,\lambda}=0}^\infty{e^{-\beta\epsilon_k}} = \frac{1}{1-e^{-\beta\hbar\omega_k}}\]\[z = \prod_{k,\lambda}z_{k,\lambda} =\prod_{k,\lambda}\frac{1}{1-e^{-\beta\hbar ck}}\]\[F = \frac{\pi^2 Vk_B^4 T^4}{45(\hbar c)^3},\quad P = \frac{\pi^2 k_B^4 T^4}{45(\hbar c)^3}\]\[E = \frac{3\pi^2 Vk_B^4T^4}{45(\hbar c)^3} = 3 PV,\quad S = \frac{4\pi^2Vk_B^4 T^3}{45(\hbar c)^3}\]
Law 3.8.3 Plank's law states that the spectral energy density $\mu(\omega)$ can be written in terms of temperature and frequency $\omega$.
\[\frac{E}{V} = \int d\omega \mu(\omega) = \frac{\hbar V}{\pi^2 c^3}\int{\frac{d\omega\ \omega^3}{e^{\beta\hbar\omega} - 1}}\]\[\mu(\omega) = \frac{\hbar}{\pi^2 c^3}\frac{\omega^3}{e^{\beta\hbar\omega} - 1}\]
Definition 3.8.4 A black body is a material that perfectly absorbs electromagnetic radiation of all frequencies.
Definition 3.8.5 The absorptivity $\alpha$ of a material is the fraction of photons absorbed by the object at temperature $T$.
Definition 3.8.6 The emmisivity $e$ of a material is the fraction of black body radiation emitted by an object at temperature $T$.
Law 3.8.7 Kirchoff's law states that the absorptivity $a$ and emmisivity $e$ of a material are equal $e=a$.
Definition 3.9.1 Phonons are a system of $3N$ harmonic oscillators with frequencies $\omega_i$.
\[E = \sum_{i=1}^{3N}{\epsilon_i} = \sum_{i=1}^{3N}{n_i\hbar\omega_i}\]
Definition 3.9.2 The Einstein model of a phonon gas simplifies the harmonic oscillators to all have the same frequency $\omega$.
Definition 3.9.3 The Debye frequency denoted $\omega_D$ is the maximum frequency in a material due to the lattice spacing.
Definition 3.9.4 The Debye model models phonons as an elastic wave with speed of sound $c_s$ and a maximum frequency $\omega_D$.
\[\omega_{n_x,n_y,n_z} = c_sk = \frac{2\pi c_s}{L}\sqrt{n_x^2 + n_y^2 + n_z^2}\]
Result 3.9.5 The Debye frequency for phonons $\omega_D$ in a solid with speed of sound $c_s$ in $m$-dimensional space can be derived from the number of particles $N$.
\[mN = m\sum_{n}\Theta(\omega_D-\omega_n) = \frac{mL^m}{(2\pi c_s)^m}\int d\Omega_m\int_0^{\omega_D}\omega^{m-1}d\omega\]\[\omega_D^{m} = 2m\pi^{m-1}\frac{N}{L^m}c_s^m\]
Result 3.9.6 The total energy $E$ for phonons in a solid with speed of sound $c_s$ in $m$-dimensional space can be derived in terms of the Debye frequency $\omega_D$.
\[E = m\sum_{n}\epsilon_n \to \frac{mL^m}{(2\pi c_s)^m}\int d\Omega_m\int_0^{\omega_D}\frac{\hbar\omega^m}{e^{\hbar\omega/\tau}-1}d\omega\]
Definition 4.1.1 The virial expansion is a perturbative approach to finding an approximate canonical partition function for a system with Hamiltonian $H$ that consists of a Hamiltonian $H_0$ with a known partition function $z_0$ and a small perturbation Hamiltonian $V$ the following holds.
\[H=H_0+V\]\[z = z_0\left(1 + \sum_{n=1}^\infty\frac{(-\beta)^n}{n!}\langle V^n\rangle_0\right)\]\[\langle V^n \rangle_0 = \frac{\text{Tr}(e^{-\beta H_0}V^n)}{\text{Tr}(e^{-\beta H_0})}\]
Definition 4.1.2 An interacting ideal gas is a perturbative system for an ideal gas $(n\ell_Q^3 << 1)$ with some small interaction potential $u(r)$ between particles.
\[H = H_0 + V = \sum_{i=1}^N{\frac{p_i^2}{2m}} + \sum_{i<j}u(|r_i-r_j|)\]
Result 4.1.3 The virial expansion for an interacting ideal gas approximates the equation of state for an interacting ideal gas in the grand canonical ensemble.
\[\frac{PV}{N\tau} = 1 + \sum_{m=1}^{\infty}{a_m(\tau)\left(n\ell_Q^3\right)^m}\]\[\mathscr{z} = \sum_{N=0}^\infty{\frac{\mathfrak{z}^N}{N!\ell_Q^{3N}}Q_N}\]\[Q_N = \int d^{3N}r e^{-\beta \sum_{i<j}u(|r_i-r_j|)} \]\[Q_1 = \int d^3 N = V, \quad Q_2 = \int d^3 r_1 d^3 r_2 e^{-\beta u(|r_i-r_j|)} = \int d^3R \int d\Omega_3 \int_0^\infty r^2 e^{-\beta u(r)} dr\]\[a_1 = \frac{Q_1}{V} = 1,\quad a_2 = \frac{-1}{2\ell_Q^3 V}(Q_2-Q_1^2) = \frac{-2\pi}{\ell_Q^3}\int_0^\infty r^2 (e^{-\beta u(r)}-1) dr\]\[\frac{PV}{N\tau} \approx 1 - \frac{2\pi N}{V}\int_0^\infty r^2 (e^{-\beta u(r)}-1) dr\]
Definition 4.2.1 A Van der Waals gas is an interacting gas system with interacting potential $u(r)$ defined by
\[u(r) = \begin{cases}\infty & r<r_0\\ -u_0\left(\frac{r_0}{r}\right)^6 & r\geq r_0\end{cases}\]
Result 4.2.2 The virial expansion for a Van der Waals gas approximates the equation of states in the grand canonical ensemble.
\[a_2 = \frac{2\pi r_0^3}{3 \ell_Q^3} \left(1-\frac{u_0}{\tau}\right)\]\[\frac{PV}{N\tau} \approx a_1 + a_2 \frac{N\ell_Q^3}{V} = 1 + \frac{2\pi r_0^3N}{3 V} \left(1-\frac{u_0}{\tau}\right)\]\[b = \frac{2\pi r_0^3N}{3V},\quad a=bu_0 = \frac{2\pi r_0^3 u_0}{3}\]\[\frac{PV}{N\tau} \approx 1 + \frac{bN}{V}\left(1-\frac{a}{b\tau}\right)\]
For a dilute gas where $\frac{V}{N}>> b$, we have the van der Waal equation of state
\[\left(P + \frac{aN^2}{V^2}\right)\left(\frac{V}{N} - b\right) = \tau\]
Definition 4.3.1 A phase transition is any non analytic behavior of thermodynamic quantities, such as discontinuities or divergences.
Definition 4.3.2 An order parameter is a thermodynamic quantity that distinguishes the phases of a phase transition.
Definition 4.3.3 A discontinuous phase transition or a first order phase transition is a phase transition characterized by a discontinuity of the order parameter.
Definition 4.3.4 A continuous phase transition or a second order phase transition is a discontinuity in at least one of the derivative of the order parameter.
Definition 4.3.5 A critical exponent is the exponent $\beta$ that describes the limiting behavior $|t|^\beta$ of an order parameter $x(t)$ in terms of $t = \frac{T-T_C}{T_C}$.
\[x(t)\sim |t|^\beta\]
Theorem 4.3.6 The fluctuation dissipation theorem states that the magnetic susceptibility order parameter $\xi_T$ for the ferromagnet to paramagnet transition is related to spacial fluctuation in the magnetization $m(r)$.
\[\xi_T = \beta V\int d^3r \langle m(r)m(0)\rangle\]
Definition 4.4.1 Landau Ginzburg theory rewrites the partition function in terms of a larger scale effective hamiltonian $H_{eff}$ that depends on a slowly varying function $m(r)$.
\[\mathscr{z} = \int \prod_{I = 1}^{N_{cells}} dm(r_I) e^{-\beta H_{eff}[m(r)]}\]\[e^{-\beta H_{eff}[m(r)]} = \sum_{\{S_i\}} e^{-\beta H} \prod_I \delta\left(m(r_I)-\frac{1}{V_I}\sum_{i\in I}S_I\right)\]
however computation of exact $H_{eff}$ is not practically possible for large systems.
Result 4.4.2 The Landau Ginzburg rules describe how to approximate with mean field theory the form of $H_{eff}$.
Result 4.4.3 Landau Ginzburg Theory for Magnetization states that
\[\beta H_{eff}\approx \beta F_0 + \int d^dr \left[ q_2 m^2 + q_4 m^4 + \frac{\kappa}{2}\left(\nabla m\right)^2 \right], \quad q_4 > 0, \kappa > 0\]\[\beta H_{eff} \approx \beta F_0 V (q_2 m^2 + q_4m^4 - Hm)\]\[q_2 > 0 \text{ for } T>T_C, \quad q_2 < 0 \text{ for } T < T_c \text{ and }\bar{m}\neq 0\]\[q_2(t) \approx \frac{a}{2}t,\quad q_4 \approx b > 0\]\[m(t) \sim |t|^\beta,\quad \beta = \frac{1}{2}\]