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- ItemModeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures(New York, NY [u.a.] : Springer Science + Business Media B.V., 2020) Maas, Jan; Mielke, AlexanderWe consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary Γ-convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.
- ItemTraveling Fronts in a Reaction–Diffusion Equation with a Memory Term(New York, NY [u.a.] : Springer Science + Business Media B.V., 2022) Mielke, Alexander; Reichelt, SinaBased on a recent work on traveling waves in spatially nonlocal reaction–diffusion equations, we investigate the existence of traveling fronts in reaction–diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction–diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory. The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising traveling front, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations.
- ItemFast Reaction Limits via Γ-Convergence of the Flux Rate Functional(New York, NY [u.a.] : Springer Science + Business Media B.V., 2023) Peletier, Mark A.; Renger, D. R. MichielWe study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant, and ‘fast’ rates are scaled as 1/ϵ, and we prove the convergence in the fast-reaction limit ϵ→0. We establish a Γ-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterises both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Γ-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.
- ItemLarge Deviations of Continuous Regular Conditional Probabilities(New York, NY [u.a.] : Springer Science + Business Media B.V., 2016) van Zuijlen, W.We study product regular conditional probabilities under measures of two coordinates with respect to the second coordinate that are weakly continuous on the support of the marginal of the second coordinate. Assuming that there exists a sequence of probability measures on the product space that satisfies a large deviation principle, we present necessary and sufficient conditions for the conditional probabilities under these measures to satisfy a large deviation principle. The arguments of these conditional probabilities are assumed to converge. A way to view regular conditional probabilities as a special case of product regular conditional probabilities is presented. This is used to derive conditions for large deviations of regular conditional probabilities. In addition, we derive a Sanov-type theorem for large deviations of the empirical distribution of the first coordinate conditioned on fixing the empirical distribution of the second coordinate.
- ItemRelating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials(New York, NY [u.a.] : Springer Science + Business Media B.V., 2021) Mielke, AlexanderWe consider a non-negative and one-homogeneous energy functional J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional E(t,u)=tJ(u) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.
- ItemExponential Moments for Planar Tessellations(New York, NY [u.a.] : Springer Science + Business Media B.V., 2020) Jahnel, Benedikt; Tóbiás, AndrásIn this paper we show existence of all exponential moments for the total edge length in a unit disk for a family of planar tessellations based on stationary point processes. Apart from classical tessellations such as the Poisson–Voronoi, Poisson–Delaunay and Poisson line tessellation, we also treat the Johnson–Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.
- ItemThe Taste of Waste: The Edge of Eggshell Over Calcium Carbonate in Acrylonitrile Butadiene Rubber(New York, NY [u.a.] : Springer Science + Business Media B.V., 2019) Bhagavatheswaran, Eshwaran Subramani; Das, Amit; Rastin, Hadi; Saeidi, Hoda; Jafari, Seyed Hassan; Vahabi, Henri; Najafi, Farhood; Khonakdar, Hossein Ali; Formela, Krzysztof; Jouyandeh, Maryam; Zarrintaj, Payam; Saeb, Mohammad RezaRubber technology experiences a new age by the use of biowaste or natural fillers. In this regard, taking properties of reinforcing agents from biowaste fillers remains as the challenging matter. Chicken eggshell (ES) biowaste has recently been introduced to substitute calcium carbonate (CaCO3) duo to its superior properties and low price. In this work, composites based on acrylonitrile butadiene rubber (NBR) reinforced with ES and CaCO3 microfillers at various loading levels were prepared and characterized. To improve the interactions between fillers and the NBR matrix, ES and CaCO3 were surface-functionalized using a terpolymer, namely poly(vinyl 2-pyrrolidone-co-maleic acid-co-acrylic acid). Fourier-transform infrared spectroscopy (FTIR) and thermogravimetric analysis (TGA) were used to characterize the modified fillers. The incorporation of the functionalized fillers resulted in a significant rise in the maximum torque according to the rheometric measurements. The Young’s modulus of the ES-based and CaCO3-based compounds showed a mild improvement over a wide range of filler contents. The elongation at break of the NBR composites, however, was dependent on the filler content. This work provides exciting opportunities for the design of novel and innovative coupling agents to be used in rubber applications. © 2019, The Author(s).
- ItemReconstruction of Quasi-Local Numerical Effective Models from Low-Resolution Measurements(New York, NY [u.a.] : Springer Science + Business Media B.V., 2020) Caiazzo, A.; Maier, R.; Peterseim, D.We consider the inverse problem of reconstructing an effective model for a prototypical diffusion process in strongly heterogeneous media based on coarse measurements. The approach is motivated by quasi-local numerical effective forward models that are provably reliable beyond periodicity assumptions and scale separation. The goal of this work is to show that an identification of the matrix representation related to these effective models is possible. On the one hand, this provides a reasonable surrogate in cases where a direct reconstruction is unfeasible due to a mismatch between the coarse data scale and the microscopic quantities to be reconstructed. On the other hand, the approach allows us to investigate the requirement for a certain non-locality in the context of numerical homogenization. Algorithmic aspects of the inversion procedure and its performance are illustrated in a series of numerical experiments. © 2020, The Author(s).
- ItemDynamical Phase Transitions for Flows on Finite Graphs(New York, NY [u.a.] : Springer Science + Business Media B.V., 2020) Gabrielli, Davide; Renger, D.R. MichielWe study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.