Foundation of classical dynamical density functional theory: uniqueness of time-dependent density-potential mappings

Abstract

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.

Figures (2)

• Figure 1: Schematics of non-unique potentials for bounded domains (illustrated as within hard walls): We start from the equilibrium solution $\rho(x,t)$ for non-interacting particles (a) without an external potential, i.e., $V(x,t)=0$; or (b) with a diverging potential $V(x,t)=\tan^2(x)$. Since the systems are in equilibrium, $j(x,t)=0$. When a time-dependent potential $d_V(x,t)$ is added to the system (with $c(t)>0$), the density profile remains unchanged, $\rho'(x,t)\equiv\rho(x,t)$, but we obtain a spatially constant current $j'(x,t)\not\equiv0$, as indicated by the arrows at the bottom). Such a current requires a source and a sink at the boundary, in violation of our conditions in Theorem \ref{['thm:no-normal-flux']}.
• Figure 2: Schematics of non-unique potentials for unbounded domains, analogous to Fig. \ref{['fig_bounded']} but now with potentials (a) $V(x,t)=x^2$ and (b) $V(x,t)=\log(1+x^2)$.

Theorems & Definitions (23)

• Theorem 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Definition 4.1
• Definition 4.2
• Proposition 4.3
• proof
• Lemma 4.4