Stochastic Time-Dependent Current-Density Functional Theory

Static and dynamical density functional methods have been applied with a certain degree of success to a variety of closed quantum mechanical systems, i.e., systems that can be described via a Hamiltonian dynamics. However, the relevance of open quantum systems - those coupled to external environments, e.g., baths or reservoirs - cannot be overestimated. To investigate open quantum systems with DFT methods we have introduced a new theory, we have named Stochastic Time-Dependent Current Density Functional theory (S-TDCDFT) $[1]$: starting from a suitable description of the system dynamics via a {\it stochastic} Schr\"odinger equation $[2]$, we have proven that given an initial quantum state and the coupling between the system and the environment, there is a one-to-one correspondence between the ensemble-averaged current density and the external vector potential applied to the system.\\ In this talk, I will introduce the stochastic formalism needed for the description of open quantum systems, discuss in details the theorem of Stochastic TD-CDFT, and provide few examples of its applicability like the dissipative dynamics of excited systems, quantum-measurement theory and other applications relevant to charge and energy transport in nanoscale systems.\\ $[1]$ M. Di Ventra and R. D'Agosta, Physical Review Letters {\bf 98}, 226403 (2007)\\ $[2]$ N.G. van Kampen, {\it Stochastic processes in Physics and Chemistry}, (North Holland, 2001), 2nd ed.