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## Transport Theory Lecture Notes

### Part I: Statistical Mechanics in a Nutshell[ps-file]

#### Contents:

1. Some probability theory
• Constrained distributions

• random experiments, relative frequencies, constraints
• Concentration theorem

• a priori distribution, (meta-)probability for occurrence of relative frequencies, maximal point
• Frequency estimation

• variational equation, Lagrange multipliers, "principle of insufficient reason", iterated estimation
• Hypothesis testing

• theoretical model vs. experimental data, fit parameters, statistical fluctuations, acceptance bound, chi^2-test, meaning of a rejection
• Jaynes' analysis of Wolf's die data

• loaded die, likely imperfections, iteratively improved hypotheses, paradigm for the experimental method
• Conclusion

• crucial concept: "entropy"
2. Macroscopic Systems in Equilibrium
• Macrostate

• phase space distribution, incoherent mixture, expectation values, types of macroscopic data (data given with certainty, prescribed expectation values, control parameters), partition function, thermodynamic variables, conjugates, equilibrium, constants of the motion, internal energy, microcanonical, canonical, grand canonical distribution
• First law of thermodynamics

• work, heat, Boltzmann constant, temperature, entropy, volume, pressure, particle number, chemical potential, magnetic induction, magnetization, electric field, electric polarization, momentum, velocity, angular momentum, angular velocity
• Example: Ideal quantum gas

• bosons, fermions, Fock space, partition function, average occupation numbers, entropy
• Thermodynamic potentials

• grand potential, free energy, internal energy, enthalpy, free enthalpy, Legendre transformation, Born diagram, homogeneous systems, Gibbs-Duhem relation
• Correlations

• canonical correlation function, correlation matrix, correlation of occupation numbers
3. Linear Response
• Liouvillian and evolution

• equation of motion, Hamilton function / Hamilton operator, Poisson bracket / commutator, constants of the motion, stationary states, causal evolver, integral equation, time-dependent perturbation theory
• Kubo formula

• weak external fields, first-order perturbation theory, dynamical susceptibility
• Example: Electrical conductivity

• current density, external electric field

### Part II: Projected Dynamics

[PR ....] = Section .... of: J. Rau and B. Müller, From Reversible Quantum Microdynamics to Irreversible Quantum Transport, Physics Reports 272, 1 (1996) [ps-file]

#### Contents:

1. Beyond Equilibrium - Warmup
• Prologue [html-file / PR 1]

• Why study transport theory?
• Decay of a single resonance [ps-file]

• occupation probability, non-Markovian equation of motion; memory time, Markovian and quasistationary limits; narrow resonance approximation
• Level transitions [ps-file]

• perturbing external potential; Fermi's golden rule, rate equation
2. Projection technique
• Transport equation [PR 3.1 / 3.2.1 / 3.2.2 / 3.2.3]

• selected observables, level of description, (time-dependent) projectors; first (mean field) term, memory term, residual force; problem of initial state: transport equation closed only if it vanishes
• Time scales [PR 3.2.4]

• relevant time, memory time; quasistationary limit (t_mem<<t), Markovian limit (t_mem<<t_rel)
• Approximations [PR 3.2.4]

• gain (t_mem/t_rel) as additional expansion parameter; expansion around the Markovian limit (memory corrections); perturbation theory: decomposition of the Liouvillian, mean field (decoupling) approximation, random phase approximation, second order perturbation theory
• Special projectors [PR 3.2.2 / 3.4.1]

• Mori projector, Langevin-Mori equation, frequency matrix, memory matrix, stochastic force, dynamical correlations; Robertson projector, time-dependent macrostate, relevant part of the statistical operator, Robertson equation; equivalence close to equilibrium
• Recipe for applying the projection technique [PR 3.3 / 3.4.2 / 4.1]

• selecting observables, choosing the projector, time scale analysis; iterative procedure
3. Example: Quantum Boltzmann equation
• Preliminaries [PR 4.1]

• level of description, representation of the projector, Hartree form, Wick theorem
• Collision term [PR 4.5]

• interacting many-particle systems, Hamiltonian, perturbation theory, non-Markovian collision term, time scale analysis, Markovian limit