  We derive solutions to the Schwinger-Dyson equations on the Closed-Time-Path
for a scalar field in the limit where backreaction is neglected. In Wigner
space, the two-point Wightman functions have the curious property that the
equilibrium component has a finite width, while the out-of equilibrium
component has zero width. This feature is confirmed in a numerical simulation
for scalar field theory with quartic interactions. When substituting these
solutions into the collision term, we observe that an expansion including terms
of all orders in gradients leads to an effective finite-width. Besides, we
observe no breakdown of perturbation theory, that is sometimes associated with
pinch singularities. The effective width is identical with the width of the
equilibrium component. Therefore, reconciliation between the zero-width
behaviour and the usual notion in kinetic theory, that the out-of-equilibrium
contributions have a finite width as well, is achieved. This result may also be
viewed as a generalisation of the fluctuation-dissipation relation to
out-of-equilibrium systems with negligible backreaction.
