**Abstract: **
Consider a three dimensional 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 that in
virtually all physically interesting situations, the conductivity
tensor is given by a Landauer-Buttiker type formula. This is a work
obtained in collaboration with H. Cornean, G. Nenciu and R. Purice,
*arXiv:0708.0303*. If time enough, we shall review some possible
applications to the computation of the current in various devices of the
type studied by the Exner's group.