Mathematical strategies for the coarse -graining of interacting particle systems

In this thesis we derive deterministic and stochastic models that describe physical processes and that can be efficiently used in large scale molecular simulations. The motivation comes from applications in materials science. We derive deter ministic mesoscopic theories for model continuous spin lattice systems both at equilibrium and non-equilibrium in the presence of thermal fluctuations. The full magnetic Hamiltonian that includes singular integral (dipolar) interactions is also considered at equilibrium. The non-equilibrium microscopic models we consider are relaxation-type dynamics arising in kinetic Monte Carlo or Langevin-type simulations of lattice systems. In this context we also employ the derived mesoscopic models to study the relaxation of such algorithms to equilibrium. We also introduce higher order numerical methods for the coarse-graining of stochastic lattice systems. The coarse-graining is equivalent to the usual renormalization group map of the block averaging transformation. Using cluster expansion techniques we present a systematic way of calculating explicit approximations of the renormalized Hamiltonian. We also propose higher order methods for the case of adsorption-desorption processes by constructing coarse-grained rates which are consistent with the equilibrium theory