The present volume analyzes mathematical models of time-dependent physical phenomena on three levels: microscopic, mesoscopic, and macroscopic. We provide a
rigorous derivation of each level from the preceding level and the resulting mesoscopic equations are analyzed in detail. Following Haken (1983, Sect. 1.11.6) we
deal, “at the microscopic level, with individual atoms or molecules, described by
their positions, velocities, and mutual interactions. At the mesoscopic level, we
describe the liquid by means of ensembles of many atoms or molecules. The extension of such an ensemble is assumed large compared to interatomic distances
but small compared to the evolving macroscopic pattern... . At the macroscopic
level we wish to study the corresponding spatial patterns.” Typically, at the macroscopic level, the systems under consideration are treated as spatially continuous
systems such as fluids or a continuous distribution of some chemical reactants, etc.
In contrast, on the microscopic level, Newtonian mechanics governs the equations of
motion of the individual atoms or molecules.1 These equations are cast in the form
of systems of deterministic coupled nonlinear oscillators. The mesoscopic level2
is probabilistic in nature and many models may be faithfully described by stochastic
ordinary and stochastic partial differential equations (SODEs and SPDEs),3 where
the latter are defined on a continuum. The macroscopic level is described by timedependent partial differential equations (PDE’s) and its generalization and simplifications