Local approximations for effective scalar field equations of motion

Fluctuation and dissipation dynamics is examined at all temperature ranges for the general case of a background time evolving scalar field coupled to heavy intermediate quantum fields which in turn are coupled to light quantum fields. The evolution of the background field induces particle production from the light fields through the action of the intermediate catalyzing heavy fields. Such field configurations are generically present in most particle physics models, including grand unified and supersymmetry theories, with application of this mechanism possible in inflation, heavy ion collision, and phase transition dynamics. The effective evolution equation for the background field is obtained and a fluctuation-dissipation theorem is derived for this system. The effective evolution, in general, is nonlocal in time. Appropriate conditions are found for when these time nonlocal effects can be approximated by local terms. Here careful distinction is made between a local expansion and the special case of a derivative expansion to all orders, which requires analytic behavior of the evolution equation in Fourier space.