Potts Models on Feynman Diagrams

We investigate numerically and analytically Potts models on ``thin'' random graphs -- generic Feynman diagrams, using the idea that such models may be expressed as the N --> 1 limit of a matrix model. The thin random graphs in this limit are locally tree-like, in distinction to the ``fat'' random graphs that appear in the planar Feynman diagram limit, more familiar from discretized models of two dimensional gravity. The interest of the thin graphs is that they give mean field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree-like structures such as the Bethe lattice. q-state Potts models display a first order transition in the mean field for q>2, so the thin graph Potts models provide a useful test case for exploring discontinuous transitions in mean field theories in which many quantities can be calculated explicitly in the saddle point approximation.Comment: 10 pages, latex, + 6 postscript figure

Coarse-graining schemes for stochastic lattice systems with short and long-range interactions

We develop coarse-graining schemes for stochastic many-particle microscopic models with competing short- and long-range interactions on a d-dimensional lattice. We focus on the coarse-graining of equilibrium Gibbs states and using cluster expansions we analyze the corresponding renormalization group map. We quantify the approximation properties of the coarse-grained terms arising from different types of interactions and present a hierarchy of correction terms. We derive semi-analytical numerical schemes that are accompanied with a posteriori error estimates for coarse-grained lattice systems with short and long-range interactions.Comment: 31 pages, 2 figure

Statistical equilibrium measures in micromagnetics

We derive an equilibrium statistical theory for the macroscopic description of a ferromagnetic material at positive finite temperatures. Our formulation describes the most-probable equilibrium macrostates that yield a coherent deterministic large-scale picture varying at the size of the domain, as well as it captures the effect of random spin fluctuations caused by the thermal noise. We discuss connections of the proposed formulation to the Landau-Lifschitz theory and to the studies of domain formation based on Monte Carlo lattice simulations

ERROR ANALYSIS OF COARSE-GRAINED KINETIC MONTE CARLO METHOD

In this paper we investigate the approximation properties of the coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice stochastic dynamics. We provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long-time simulations. We demonstrate that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse-graining ratio and that the natural small parameter is the coarse-graining ratio over the range of particle/particle interactions. The error estimate is shown to hold in the weak convergence sense. We employ the derived analytical results to guide CGMC algorithms and we demonstrate a CPU speed-up in demanding computational regimes that involve nucleation, phase transitions and metastability

Error analysis of coarse-graining for stochastic lattice dynamics

The coarse‐grained Monte Carlo (CGMC) algorithm was originally proposed in the series of works [M. A. Katsoulakis, A. J. Majda, and D. G. Vlachos, J. Comput. Phys., 186 (2003), pp. 250–278; M. A. Katsoulakis, A. J. Majda, and D. G. Vlachos, Proc. Natl. Acad. Sci. USA, 100 (2003), pp. 782–787; M. A. Katsoulakis and D. G. Vlachos, J. Chem. Phys., 119 (2003), pp. 9412–9427]. In this paper we further investigate the approximation properties of the coarse‐graining procedure and provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long‐time simulations. We demonstrate that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse‐graining ratio and that the natural small parameter is the coarse‐graining ratio over the range of particle/particle interactions. The error estimate is shown to hold in the weak convergence sense. We employ the derived analytical results to guide CGMC algorithms and demonstrate a CPU speed‐up in demanding computational regimes that involve nucleation, phase transitions, and metastability

Mesoscopic modeling for continuous spin lattice systems: Model problems and micromagnetics applications

In this paper we derive deterministic 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 equilibriu

Numerical and statistical methods for the coarse-graining of many-particle stochastic systems

In this article we discuss recent work on coarse-graining methods for microscopic stochastic lattice systems. We emphasize the numerical analysis of the schemes, focusing on error quantification as well as on the construction of improved algorithms capable of operating in wider parameter regimes. We also discuss adaptive coarse-graining schemes which have the capacity of automatically adjusting during the simulation if substantial deviations are detected in a suitable error indicator. The methods employed in the development and the analysis of the algorithms rely on a combination of statistical mechanics methods (renormalization and cluster expansions), statistical tools (reconstruction and importance sampling) and PDE-inspired analysis (a posteriori estimates). We also discuss the connections and extensions of our work on lattice systems to the coarse-graining of polymers