Foundation of classical dynamical density functional theory: uniqueness of time-dependent density-potential mappings
When can we map a classical density profile to an external potential? In equilibrium, without time dependence, the one-body density is known to specify the external potential that is applied to the many-body system. This mapping from a density to the potential is the cornerstone of classical density functional theory (DFT). Here, we consider non-equilibrium, time-dependent many-body systems that evolve from a given initial condition. We derive explicit conditions, for example, no flux at the boundary, that ensure that the mapping from the density to a time-dependent external potential is unique. We thus prove the underlying assertion of dynamical density functional theory (DDFT), without resorting to the so-called adiabatic approximation often used in applications. By ascertaining uniqueness for all $n$-body densities, we ensure that the proof and the physical conclusions drawn from it hold for general superadiabatic dynamics of interacting systems even in the presence of (known) non-conservative forces.