# Statistical Mechanics $\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}$ ## 1 Thermodynamics ### 1.1 1st Law of Thermodynamics **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. ### 1.2 2nd Law of Thermodynamics **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. \[\oint\frac{\dj Q(\mathbf{\lambda})}{T(\lambda)}\partial\mathbf{\lambda} > S_f - S_i > 0\quad\text{ (for irreversible processes)}\] ### 1.3 Open and Closed Systems **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 Enthalpy **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. \[dH = TdS + \mathbf{V}\cdot d\mathbf{P} + \mathbf{\mu}\cdot d\mathbf{N}\] \[T = \left( \frac{\partial H}{\partial S} \right)_{\mathbf{P},\mathbf{N}},\quad V_i = \left(\frac{\partial H}{\partial P_i}\right)_{S,P_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial H}{\partial N_\alpha} \right)_{S,\mathbf{P},N_{\beta\neq \alpha}}\] ### 1.5 Helmholtz Free Energy **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. \[dF = -SdT - \mathbf{P}\cdot d\mathbf{V} + \mathbf{\mu}\cdot d\mathbf{N}\] \[-S = \left( \frac{\partial F}{\partial T} \right)_{\mathbf{V},\mathbf{N}},\quad -P_i = \left(\frac{\partial F}{\partial V_i}\right)_{T,V_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial F}{\partial N_\alpha} \right)_{T,\mathbf{V},N_{\beta\neq \alpha}}\] ### 1.6 Gibbs Free Energy **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. \[dG =-SdT + \mathbf{V}\cdot d\mathbf{P} + \mathbf{\mu}\cdot d\mathbf{N}\] \[-S = \left(\frac{\partial G}{\partial T}\right)_{\mathbf{P},\mathbf{N}},\quad V_i = \left(\frac{\partial G}{\partial P_i}\right)_{T,P_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial G}{\partial N_\alpha} \right)_{T,\mathbf{P},N_{\beta\neq \alpha}}\] ### 1.7 Grand Potential **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. \[d\mathcal{G} = -SdT - \mathbf{P}\cdot d\mathbf{V} - \mathbf{N}\cdot d\mathbf{\mu}\] \[-S = \left(\frac{\partial \mathcal{G}}{\partial T}\right)_{\mathbf{V},\mathbf{\mu}},\quad -P_i = \left(\frac{\partial \mathcal{G}}{\partial V_i}\right)_{T,V_{j\neq i},\mathbf{\mu}},\quad -N_\alpha=\left( \frac{\partial \mathcal{G}}{\partial \mu_\alpha} \right)_{T,\mathbf{V},\mu_{\beta\neq \alpha}}\] ### 1.8 3rd Law of Thermodynamics **Law 1.8.1 **  The **3rd law of thermodynamics** states that the limit of entropy as temperature approaches to zero is a universal constant. \[\lim_{T\to 0}{S(T,\dots)} = C\] ## 2 Fundamental Statistical Mechanics ### 2.1 Microcanonical Ensemble **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. \[\langle \mathscr{O}\rangle = \sum_{i}{\mathscr{O}_i\mathscr{p}_i}\] ### 2.2 Canonical Ensemble **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 $1/(1-r)$. \[\sum_{k=0}^\infty{r^k} = \frac{1}{1-r}\] ### 2.3 Ideal Gas in the Canonical Ensemble **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 \[S = -\left(\frac{\partial F}{\partial T}\right)_{N,V} = k_BN\left[\frac{5}{2}-\log\frac{N\ell_Q^3}{V}\right]\] ### 2.4 Grand Canonical Ensemble **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. \[\mathcal{G} = -\frac{1}{\beta}\log \mathscr{z}\] ### 2.5 Classical Statistical Mechanics **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 $B_j$. ### 2.6 Quantum Statistical Mechanics **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 $\mathbf{\mu}$. \[w_{i,N} = \frac{e^{-\beta (E_{i,N}-\mu{N})}}{\text{Tr}_{FS}(e^{-\beta (\hat{H}-\mu\hat{N}) })} = \frac{e^{-\beta (E_{i,N}-\mu N)}}{\sum_N\sum_{j}e^{-\beta (E_j-\mu N)}} = \frac{e^{-\beta (E_{i,N}-\mu N)}}{\mathscr{z}}\] \[\mathscr{z} = \text{Tr}_{FS}(e^{-\beta (\hat{H}-\mu\hat{N}) }) = \sum_N\sum_{j}e^{-\beta (E_j-\mu N)}\] ## 3 Quantum Gases ### 3.1 Identical Particles **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. \[I = \sum_{k_1\leq k_2\leq \dots \leq k_N} \ket{k_1,k_2,\dots,k_N}_\eta\ \prescript{}{\eta}{\bra{k_1,k_2,\dots,k_N}} = \frac{1}{N! \prod_{k}n_k!}\sum_{k_1,k_2,\dots,k_N} \ket{k_1,k_2,\dots,k_N}_\eta\ \prescript{}{\eta}{\bra{k_1,k_2,\dots,k_N}}\] ### 3.2 Quantum Gases in the Canonical Ensemble **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](#99eh). \[\ell_Q^2 >> \left(\frac{V}{N}\right)^{2/3}\] ### 3.3 Quantum Gases in the Grand Canonical Ensemble **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. \[\mathcal{G} = \frac{\eta}{\beta}\sum_k{\log(1-\eta e^{-\beta(E_k-\mu)})}\] \[\langle n_k \rangle = \frac{1}{e^{\beta (E_k - \mu)}-\eta}\] \[\langle E \rangle = \sum_k E_k\langle n_k \rangle = \sum_k{\frac{E_k}{e^{\beta(E_k - \mu)}-\eta}}\] \[\langle N\rangle = \sum_k \langle n_k \rangle = \sum_{k}\frac{1}{e^{\beta(E_k-\mu)}-\eta}\] ### 3.4 Single Particle Density of States **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 $\mathbb{z} = e^{\beta\mu}$. ### 3.5 Non-relativistic Fermi and Bose Gases **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](https://www.kaedon.net/l/0j0H) 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 $\mathbb{z} << 1$ \[f_m^\eta(\mathbb{z}) \approx \sum_{\alpha = 1}^\infty{\eta^{\alpha + 1}\frac{\mathbb{z}^\alpha}{\alpha^m}} = \mathbb{z} + \eta\frac{\mathbb{z}^2}{2^m} + \frac{\mathbb{z}^3}{3^m} + \cdots\] ### 3.6 Degenerate Fermi Gases **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. \[\mu(T) \approx \epsilon_F(1 - \frac{\pi^2}{12}\frac{k_B^2T^2}{\epsilon_F^2})\] ### 3.7 Bose Einstein Condensate **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](https://www.kaedon.net/l/raa2). \[P \approx f_{5/2}^{+1}(1) \frac{k_B T}{\ell_Q^3}\] **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. ### 3.8 Photon Gas **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 $e=a$. ### 3.9 Phonon Gas **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 $\omega_D$. \[E = m\sum_{n}\epsilon_n \to \frac{mL^m}{(2\pi c_s)^m}\int d\Omega_m\int_0^{\omega_D}\frac{\hbar\omega^m}{e^{\hbar\omega/\tau}-1}d\omega\] ## 4 Interacting Systems and Phase Transitions ### 4.1 Virial Expansion **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> b$, we have the van der Waal equation of state \[\left(P + \frac{aN^2}{V^2}\right)\left(\frac{V}{N} - b\right) = \tau\] ### 4.3 Phase Transistions **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 $m(r)$. \[\xi_T = \beta V\int d^3r \langle m(r)m(0)\rangle\] ### 4.4 Landau Ginzburg Theory **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 $H_{eff}$. 1. *Locality* states that we should be able to write the effective Hamiltonian $H_{eff}$ in terms of a local energy density $\Phi$. \[\beta H_{eff} = \int d^dr \Phi[m(r), \nabla m(r), \dots]\] 2. *Analytic/Polynomial Expansion in m* states that the $\Phi$ can be expanded as a polynomial in terms of $m(r)$, \[\Phi = \text{const} + q_1m + q_2 m^2 +\dots + \kappa(\nabla m)^2 + \dots\] 3. $H_{eff}$ *should respect underlying symmetries*. For symmetric systems $H_{eff}[m] = H_{eff}[-m]$ so \[q_1 = q_3 = \dots = 0\] 3. *Stability* the coefficient of highest power in $m$ must be positive to avoid the infinite limit. 5. *Coefficient of gradient term* states that the gradient term $\kappa\nabla m$ should be positive because it is energetically favorable for nearby states to be aligned. **Result 4.4.3 **  **Landau Ginzburg Theory for Magnetization** states that \[\beta H_{eff}\approx \beta F_0 + \int d^dr \left[ q_2 m^2 + q_4 m^4 + \frac{\kappa}{2}\left(\nabla m\right)^2 \right], \quad q_4 > 0, \kappa > 0\] \[\beta H_{eff} \approx \beta F_0 V (q_2 m^2 + q_4m^4 - Hm)\] \[q_2 > 0 \text{ for } T>T_C, \quad q_2 < 0 \text{ for } T < T_c \text{ and }\bar{m}\neq 0\] \[q_2(t) \approx \frac{a}{2}t,\quad q_4 \approx b > 0\] \[m(t) \sim |t|^\beta,\quad \beta = \frac{1}{2}\]