Fachbereich Mathematik

Seminar Effective Equations of Interacting Many-Body Systems


Interacting many body systems are the starting point for discussing many interesting effects in various sciences. Solving equations for many interacting particle-systems numerically or analytically is, however, often practically impossible. For quantum systems this is in particular difficult and with current techniques numerics is at its limits for systems of roughly ten particles in three dimension. To solve this problem, effective equations have been established in various situations that draw the main features of the system and are given by evolution equations on a much smaller space. Depending on the question one asks, in particular on the need of accuracy, finer descriptions are sought for which give a better approximation on the true dynamics with the price of increased -- but still manageable -- cost of solving the equation.

Contents

In the seminar we will discuss rigorous proofs of effective equations for different systems. Examples are

  • The derivation of the Vlasov equation from interacting Newtonian particles (long range)
  • The derivation of the Boltzmann equation from interacting Newtonian particles (short range)
  • The derivation of the Hartree equation for interacting quantum systems
  • Corrections to Hartree dynamics
  • Derivation of Maxwell's equations form QED
  • Many more depending on the interests of the participants.

Practically all these topics can be used as a basis for a scientific project that might end in a master thesis.

Prerequisites

Next to the requirements for entering the Master program in Mathematics or Mathematical Physics, basic knowledge in quantum mechanics and probability are helpful for the class.

Place and Time

The seminar takes place on Mondays 4:15 pm in C4H33. In the first meeting on October 16th details on the seminar/topics/literature will be discussed and the talks distributed among the participants. Students interested in the seminar are asked to come to the first meeting, registration is not required before the first meeting has taken place.