# Mathematical Models for Transport in Macroscopic and Mesoscopic Systems - Abstract

**Duclos, Pierre**

*On the current in continuous systems with an adiabatically switched-on electrical bias*

Consider a system which looks like a cross connected pipe system, i.e. a small sample coupled to a finite number of leads. We investigate the current running through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of a finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We finally prove the conductivity tensor is given by a Landauer-Büttiker type formula. This is a work obtained in collaboration with H. Cornean, G. Nenciu and R. Purice, arXiv:0708.0303.