  The linear conductivity tensor for generic homogeneous, microscopic quantum
models was formulated as a noncommutative Kubo formula in Refs.
\cite{BELLISSARD:1994xj,Schulz-Baldes:1998vm,Schulz-Baldes:1998oq}. This
formula was derived directly in the thermodynamic limit, within the framework
of $C^*$-algebras and noncommutative calculi defined over infinite spaces. As
such, the numerical implementation of the formalism encountered fundamental
obstacles. The present work defines a $C^*$-algebra and an approximate
noncommutative calculus over a finite real-space torus, which naturally leads
to an approximate finite-volume noncommutative Kubo formula, amenable on a
computer. For finite temperatures and dissipation, it is shown that this
approximate formula converges exponentially fast to its thermodynamic limit,
which is the exact noncommutative Kubo formula. The approximate noncommutative
Kubo formula is then deconstructed to a form that is implementable on a
computer and simulations of the quantum transport in a 2-dimensional disordered
lattice gas in a magnetic field are presented.
