Narrow Results

We consider the $d=1$ Ising model with Kac potentials at inverse temperature $\beta >1$ where the mean field predicts a phase transition with two possible equilibrium magnetizations $\pm m_{\beta }$, $m_{\beta }>0$. We show that when the Kac scaling parameter $\gamma$ is sufficiently small, typical spin configurations are described (via a coarse graining) by an infinite sequence of successive plus and minus intervals where the empirical magnetization is “close” to $m_{\beta }$, and respectively, $-m_{\beta }$. We prove that the corresponding marginal of the unique DLR measure is a renewal process.

Coarse-grained analysis enhances our understanding of complex processes such as physical procedures, economic complexity. We collect the data sets from 31 regions of China in terms of the gross regional domestic product (GRDP) and the expense invested in Education and R&D between 1998 and 2013, then employ the coarse-grained method to analyze the causal direction according to the cross-section data with time-series information. Specifically, the empirical results suggest that the share of the GRDP invested in Education and R&D in large time scale reveals a dynamical process due to economic complexity, but limits around to base lines.

Dissipative Particle Dynamics (DPD) simulations were performed to investigate resolution or "coarse graining" effects on the simulation results. Fluid flow through a periodic array of spheres has been studied as a model for fluid filtration into a porous medium. In our model system, it appears that quantitatively correct results for the dimensionless drag can be obtained for relatively small system sizes. For higher solid volume fractions, it is necessary to increase the system size to avoid finite size and resolution effects. Simulations of colloidal spheres suspended in a DPD fluid show effective attraction between the large colloid particles, causing depletion aggregation. This effect may be expected as a consequence of the coarse-grained nature of the DPD fluid. By imposing a steady shear rate the aggregation can be suppressed. The results show that for dilute suspensions, the Brownian noise in the particle interactions causes an effective colloid polydispersity, which suppresses aggregation effects.

A connection between dynamical systems and network theory is outlined based on a mapping of the dynamics into a discrete probabilistic process, whereby the phase space is partitioned into finite size cells. It is found that the connectivity patterns of networks generated by deterministic systems can be related to the indicators of the dynamics such as local Lyapunov exponents. The procedure is extended to networks generated by stochastic processes.

Coarse graining techniques and their associated symbolic dynamics are reviewed with a focus on probabilistic aspects of complex dynamical systems. The probabilistic approach initiated by Nicolis and coworkers has been elaborated. One of the major issues when dealing with the dynamics of complex nonlinear systems, the fact that the inherent time-scales of the unfolding phenomena are not well separated, is brought into focus. Recent results related to this interdependence, which is one of the most characteristic aspects of complexity and a major challenge in prediction, error estimates and monitoring of nonlinear complex systems, are discussed.

We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.

The main mechanisms that control the organization of multicellular tissues are still largely open. A commonly used tool to study basic control mechanisms are in vitro experiments in which the growth conditions can be widely varied. However, even in vitro experiments are not free from unknown or uncontrolled influences. One reason why mathematical models become more and more a popular complementary tool to experiments is that they permit the study of hypotheses free from unknown or uncontrolled influences that occur in experiments. Many model types have been considered so far to model multicellular organization ranging from detailed individual-cell based models with explicit representations of the cell shape to cellular automata models with no representation of cell shape, and continuum models, which consider a local density averaged over many individual cells. However, how the different model description may be linked, and, how a description on a coarser level may be constructed based on the knowledge of the finer, microscopic level, is still largely unknown. Here, we consider the example of monolayer growth in vitro to illustrate how, in a multi-step process starting from a single-cell based off-lattice-model that subsumes the information on the sub-cellular scale by characteristic cell-biophysical and cell-kinetic properties, a cellular automaton may be constructed whose rules have been chosen based on the findings in the off-lattice model. Finally, we use the cellular automaton model as a starting point to construct a multivariate master equation from a compartment approach from which a continuum model can be derived by a systematic coarse-graining procedure. We find that the resulting continuum equation largely captures the growth behavior of the CA model. The development of our models is guided by experimental observations on growing monolayers.

Levels of a complex system are characterized by the fact that they admit a closed functional description in terms of concepts and quantities intrinsic to that level. Several ideas have come up so far in order to make the notion of a closed description precise. In this paper, we present four of these approaches and investigate their mutual relationships. Our study is restricted to the case of discrete dynamical systems, where the different levels are linked by a coarse-graining of variables and states of the system.