  A time-dependent current-density-functional theory for many-particle systems
in interaction with arbitrary external baths is developed. We prove that, given
the initial quantum state $|\Psi_0>$ and the particle-bath interaction
operator, two external vector potentials ${\bf A}({\bf r},t)$ and ${\bf
A}'({\bf r},t)$ that produce the same ensemble-averaged current density,
$\bar{{\bf j}({\bf r},t)}$, must necessarily coincide up to a gauge
transformation. This result greatly expands the applicability of time-dependent
density-functional theory to open quantum systems, and allows for
first-principles calculations of many-particle time evolution beyond
Hamiltonian dynamics.
