  We compare four different types of equations of motion for reduced density
matrix of a system of molecular excitons interacting with thermodynamic bath.
All four equations are of second order in the linear system-bath interaction
Hamiltonian, with different approximations applied in their derivation. In
particular we compare time-nonlocal equations obtained from so-called
Nakajima-Zwanzig identity and the time-local equations resulting from the
partial ordering prescription of the cummulant expansion. In each of these
equations we alternatively apply secular approximation to decouple population
and coherence dynamics from each other. We focus on the dynamics of intraband
electronic coherences of the excitonic system which can be traced by coherent
two-dimensional spectroscopy. We discuss the applicability of the four
relaxation theories to simulations of population and coherence dynamics, and
identify features of the two-dimensional coherent spectrum that allow us to
distinguish time-nonlocal effects.
