Microscopic derivation of time-dependent density functional methods

Time-dependent density functional methods (TDDFM) are studied from the microscopic viewpoint using projection operator methods in classical liquids. A density field is defined without averaging, so that a time evolution equation of the density field is derived with a random force. The derived equation includes a free energy functional, which is different from that defined in the TDDFM. The projection operator method provides the exact expression of the free energy functional. Another definition of the density field by an average leads to the equation of the TDDFM. In addition, an equation describing fluctuations is also derived.