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: $$G=U-TS+pV=\sum_{i=1}^s\mu_iN_i$$
  • inner energy of joint systems is sub-additive, entropy is super-additive
  • 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

Leave a comment

Your email address will not be published.