Dynamical Decomposition of a Large System

The equations for the statistical matrix ${\rho }_{\theta }$ and the correlation matrix ${\rho }_{\theta }(t_1,t_2)$ of a large system subject to time-dependent external forces are cast in a new form based on a continuation of Schrödinger's equation to complex times. It is shown that these equations also apply when the time dependence of the Hamiltonian is due to a coupling with another system (heat bath) which is at equilibrium at a temperature ${\theta }^{\prime }$, but that such time dependence must be expressed in terms of random functions of the complex argument $z^{\prime }=\frac{t-i}{{2\theta }^{\prime }}$. A generalization to the case that the external time-dependent fields and the coupling with a heat bath exist simultaneously is straightforward and leads to a Schrödinger-type equation with non-Hermitian Hamiltonian which describes the dynamical and statistical aspects of the motion.