# Advanced Statistical Mechanics in a nutshell

In case anybody is interested in the (very entertaining) lecture `Advanced Statistical Mechanics' by Jens Eisert, here is a short summary of the content. In fact, this is part of my exam preparation ;) I omitted most of the introduction on basic quantum mechanics, as this knowledge is required for the course anyway.

# Introduction

• Classification: isolated thermodynamic system: no exchange of energy or matter. closed system: no exchange of matter. open system: the rest.
• extensive quantities scale with the system size, intensive quantities do not
• Law of Boyle-Mariotte: $pV=c$
• Maxwell's demon

# Thermodynamics

• differential form $\delta F=\sum_{j=1}^kF_jdG_j$ and exact differential $dG$
• syntactic sugar: $\left(\frac{\partial G}{\partial x_1}\right)_{x_2, \dots, x_n}$ (explicit version of partial derivative)
• First law: $\delta A=pdV$ (or $MdB$, $\mu dN$, depending on control parameter)
• adiabatic process: $dU+\delta A=0$
• with heat flow: $dU+\delta A=\delta Q$
• ideal gas: $U=cpV=\frac{f}{2}pV=\frac{f}{2}NkT$
• thermal equilibrium as definition of the temperature
• control parameter spaces are added for joint thermodynamic systems
• definition heat bath
• definition perpetuum mobile (first kind: efficiency > 1, second kind: work from inner energy)
• empirical entropy relevant for reversibility of processes: $dS=0$ means reversible
• Carnot process: isothermal and adiabatic steps starting with low entropy and low temperature
• Carnot engine either heat engine or heat pump
• Carnot efficiency: $\eta = 1-\frac{|Q_3|}{Q_1}=1-\frac{T_-}{T_+}$
• Carnot process as a basis for absolute temperature scale
• entropy only grows with $dS=\frac{d Q}{T}$
• Gibbs' fundamental equation: $dS = \frac{dU-\delta A}{T}$
• side note: Lagrange parameters
• free energy $F=U-TS$
• chemical potential $\mu = \frac{\partial U}{\partial N}$
• Gibbs-Duhem equation: $U=TS-pV+\sum_{i=1}^s\mu_iN_i$
• free enthalpy: $G=U-TS+pV=\sum_{i=1}^s\mu_iN_i$
• definition convex functions: $\forall \mathbf{x}, \mathbf{y} \in S: \forall 0\le \lambda \le 1: \lambda\mathbf{x}+(1-\lambda)\mathbf{y}\in S$
• f is concave iff -f is convex. Iff f is both concave and convex, it is an affine function
• definition convex hull
• Legendre transform $\tilde f(\xi) := \sup_\mathbf{x}\{\xi\mathbf{x}-f(\mathbf{x})\}$
• $\tilde{\tilde f} = f$
• $F=-\tilde U$
• van der Waals gas: $\left(p+\frac{a}{(V/N)^2}\right)(V/N-b)=cT$
• Guggenheim-Quadrat: "Suff hilft Fhysikern pei großen Taten."

# Classical statistics

• phase space $\Gamma =(\mathbb{R}^3\times\Omega)^N$
• Liouville measure: $d\gamma = dp_1dq_1\dots dq_{3N}$
• definition energy surface: $\{\gamma: H(\gamma)=E\}$ or $\{\gamma: E-\varepsilon \le H(\gamma)\le E\}$
• expectation value: $\langle f \rangle_\rho = \int d\gamma\rho(\gamma)f(\gamma)$
• variance: $\langle (f-\langle f\rangle_\rho)^2\rangle_\rho = \langle f^2 \rangle_\rho-\langle f \rangle_\rho^2$
• definition statistically independent: $\rho(\gamma_1, \gamma_2) = \rho_1(\gamma_1)\rho_2(\gamma_2)$ (only for non-interacting Hamiltonians)
• micro canonical, canonical and grand-canonical ensemble
• ensemble necessary in order to find equilibrium states due to "Umkehreinwand" and "Wiederkehreinwand"
• definition ergodic: systems without other constants of motion without $H=E$
• expectation value micro canonical ensemble: $\langle f\rangle =\int d\gamma \delta(H(\gamma)-E)f(\gamma)\cdot\left[\int d\gamma \delta(H(\gamma)-E)\right]^{-1}$
• expectation value canonical ensemble: $\langle f\rangle =\int d\gamma \omega_B(E-H(\gamma))f(\gamma)\cdot\left[Z\right]^{-1}$
• canonical ensemble inverse temperature $\beta=1/(kT)$: $\langle f\rangle_\beta = \int d\gamma \exp(-\beta H(\gamma))f(\gamma)\cdot\left[Z\right]^{-1}$
• ensembles different, but mathematically equal for $N, V\rightarrow \infty$
• the canonical ensemble minimizes the free energy
• equipartition theorem: $\langle p_i\frac{\delta H}{\delta p_j}\rangle_\beta = \frac{1}{\beta}\delta _{ij}$
• equipartition theorem gives fkT/2 for kinetic energy

# Quantum statistical physics

• bounded operator: $\|A|\phi\rangle\|\le c\||\phi\rangle\|$
• Pauli matrices
• trace is cyclic
• expectation value: $\langle A\rangle_\rho=\text{tr}(\rho A)$
• density operator for pure states: $\rho = |\psi\rangle\langle\psi|$, $\text{tr}(\rho^2)=1$
• density operator for mixed states: $\rho = \sum_{i=1}^np_i|\psi\rangle\langle\psi|$, $\text{tr}(\rho^2)<1$
• von Neumann equation: $i\hbar \frac{d}{dt}\rho(t)=[H, \rho(t)]$
• $\rho(t)=U_t\rho(0)U_t^\dagger$, $U_t=\exp(-iHt)$
• product states: $\rho=\rho_1\otimes\rho_2$
• microcanonical ensemble: $\rho(E)=Z(E)^{-1}\sum_{E-\varepsilon\le E_j\le E}|E_j\rangle\langle E_j|$
• canonical ensemble: $\rho_\beta=Z^{-1}\exp(-\beta H)$, $Z=\text{tr}(\exp(-\beta H))$
• free energy: $F=-\frac{1}{kT}\log\text{tr}(\exp(-\beta H))$
• von Neumann entropy: $\tilde S(\rho)=\text{tr}(\rho\log \rho)$
• for pure states, the entropy vanishes, entropy monotonous function of the temperature
• $\tilde S(\rho_1\otimes\rho_2)=\tilde S(\rho_1)+\tilde S(\rho_2)$
• for $T\rightarrow 0$, the entropy converges to 0
• definition symplectic transformations: $S\sigma S^T=\sigma$
• Hamiltonian of the harmonic chain: $H=\sum_i\left(\frac{P_i^2}{2m}+(D/2)(Q_i-Q_{i+1})^2\right)$
• continuum limit: $\sum_\mathbf{p} \rightarrow \frac{V}{h^3}\int d^3p$
• canonical partition function for both Bose and Fermi gas $Z=\sum\exp(-\beta\sum_p E_pn_p)$
• grand canonical ensemble: $\rho = Z^{-1}\exp(-\beta(H-\mu N))$, $pV=kT\log Z$
• $\langle n_p\rangle=\exp(-\beta(E_p-\mu))/\left[1\mp\exp(-\beta(E_p-\mu))\right]$ (minus sign for bosons)
• $U=-\frac{\partial }{\partial \beta}\log G=(3/2)pV$
• Fermi pressure due to quantum effects
• BEC

# Lattice models and phase transitions

• local Hamiltonians
• low temperatures: lattice tends to be ordered
• definition contour, length
• 1D Ising model: no phase transition due to vanishing order paramter
• mean field approximation: good results for high dimensions, allows for phase transitions