# 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