Nonlinear-closed equations for the first and the second moments of macroscopic variables

A quantum-mechanical theory of describing systems far from equilibrium is developed. A set of time evolution equations for every moment of macroscopic variables is derived with the aid of the new idempotent operator. From this set of equations nonlinear but closed equations for the first and the second moments are obtained directly. The theory is applied to the problem of a spin interacting with its surroundings. The Bloch equation with the Landau-Lifshitz friction term is derived quantum mechanically. The relation between this method and that of system size expansion is discussed.