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Nonlinear Langevin equations and the time dependent density functional method.

To study the time dependent density functional method (TDDFM), two streaming velocity (reversible) terms are reformulated in the nonlinear Langevin equation. Mori's [Prog. Theor. Phys. 33, 423 (1965)] projection operator method shows a variety of nonlinear Langevin equations. This is because the equations depend on the choice of phase space functions employed in the projection. If phase space functions include particular functions, however, the streaming velocity term has an invariable form. The form is independent of the choice of other phase space functions. Since the invariable streaming velocity term does not introduce the TDDFM, the second viewpoint is presented. In this, the linearization of the streaming velocity term agrees with the frequency term in the linear Langevin equation. Since only the second streaming velocity term introduces the TDDFM, one needs to be cautious in the derivation of the TDDFM.