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 $|��_0>$ 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.

4 pages, no figure, RevTex4, final version