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Transport Theory
Lecture Notes
Contents:

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^2test, 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"

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,
GibbsDuhem relation

Correlations
canonical correlation function, correlation matrix,
correlation of occupation numbers

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, timedependent perturbation theory

Kubo formula
weak external fields, firstorder 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) [psfile]
Contents:

Beyond Equilibrium  Warmup

Prologue [htmlfile / PR 1]
Why study transport theory?

Decay of a single resonance [psfile]
occupation probability, nonMarkovian equation of motion; memory
time, Markovian and quasistationary limits; narrow resonance approximation

Level transitions [psfile]
perturbing external potential; Fermi's golden rule, rate equation

Projection technique

Transport equation [PR 3.1 / 3.2.1 / 3.2.2 / 3.2.3]
selected observables, level of description, (timedependent) 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, LangevinMori equation, frequency matrix, memory
matrix, stochastic force, dynamical correlations; Robertson projector,
timedependent 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

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 manyparticle systems, Hamiltonian, perturbation theory,
nonMarkovian collision term, time scale analysis, Markovian limit