Integrodifferential Equation for Response Theory

The problem of response theory in statistical mechanics involves the determination of the density matrix $\rho$ from the Liouville equation and the subsequent computation of the response $r$ from this $\rho$. Projection techniques are applied to avoid the entire complicated problem of the full dynamics of $\rho$ and to select only that part of $\rho$ which is relevant to the response $r$. The procedure replaces an inhomogeneous equation by a linear homogeneous integrodifferential equation for response theory. This is a very general equation which can be analyzed in different ways to yield a variety of results. It is shown that the Kubo theory of linear response emerges as the lowest-order approximation. The general equation is solved without approximations for a step-function stimulus, and it is discussed in the context of the steady state.