Fachbereich Mathematik

Effective Description of 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 dimensions. 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 that show the validity of such effective equations for many different systems. Examples are

  • The derivation of the Vlasov equation for interacting, Newtonian particles (long range)
  • The derivation of the Boltzmann equation for interacting, Newtonian particles (short range)
  • The derivation of the Hartree equation from interacting quantum systems
  • The derivation of Maxwell's equations from QED
  • Depending on the interests of the participants other systems from physics or biology.

Place and Time

The seminar takes place on Mondays from 4:15 p.m. to 5:45 pm in C4H33 (aquarium). In our first meeting on October 16th we will discuss details of the different topics and distribute the talks among the participants. Students who are interested in the seminar are asked to come to the first meeting, pre-registration is not required.