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- ItemGradient and Generic systems in the space of fluxes, applied to reacting particle systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Renger, D.R. MichielIn a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager-Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well.
- ItemDynamical large deviations of countable reaction networks under a weak reversibility condition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Patterson, Robert I.A.; Renger, D.R. MichielA dynamic large deviations principle for a countable reaction network including coagulation-fragmentation models is proved. The rate function is represented as the infimal cost of the reaction fluxes and a minimiser for this variational problem is shown to exist. A weak reversibility condition is used to control the boundary behaviour and to guarantee a representation for the optimal fluxes via a Lagrange multiplier that can be used to construct the changes of measure used in standard tilting arguments. Reflecting the pure jump nature of the approximating processes, their paths are treated as elements of a BV function space.
- ItemNon-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mielke, Alexander; Patterson, Robert I.A.; Peletier, Mark A.; Renger, D.R. MichielWe study stochastic interacting particle systems that model chemical reaction networks on the microscopic scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we also study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations enables us to derive a non-linear relation between thermodynamic fluxes and free energy driving force.
- ItemThe space of bounded variation with infinite-dimensional codomain(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Heida, Martin; Patterson, Robert I.A.; Renger, D.R. MichielWe study functions of bounded variation with values in a Banach or in a metric space. We provide several equivalent notions of variations and provide the notion of a time derivative in this abstract setting. We study four distinct topologies on the space of bounded variations and provide some insight into the structure of these topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin-Lions theorem. We finally provide some useful applications to stochastic processes.
- ItemFrom large deviations to Wasserstein gradient flows in multiple dimensions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Erbar, Matthias; Maas, Jan; Renger, D.R. MichielWe study the large deviation rate functional for the empirical measure of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of -convergence) to the JordanKinderlehrerOtto functional arising in the Wasserstein gradient flow structure of the FokkerPlanck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions.
- ItemOn the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Peletier, Mark A.; Renger, D.R. MichielMotivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions L that induce a flow, given by L(pt, pt) = 0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when L is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.
- ItemLarge deviations of reaction fluxes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Patterson, Robert I.A.; Renger, D.R. MichielWe study a system of interacting particles that randomly react to form new particles. The reaction flux is the rescaled number of reactions that take place in a time interval. We prove a dynamic large-deviation principle for the reaction fluxes under general assumptions that include mass-action kinetics. This result immediately implies the dynamic large deviations for the empirical concentration.
- ItemOn microscopic origins of generalized gradient structures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Liero, Matthias; Mielke, Alexander; Peletier, Mark A.; Renger, D.R. MichielClassical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.
- ItemLarge deviations of specific empirical fluxes of independent Markov chains, with implications for Macroscopic Fluctuation Theory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Renger, D.R. MichielWe consider a system of independent particles on a finite state space, and prove a dynamic large-deviation principle for the empirical measure-empirical flux pair, taking the specific fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finite-space setting.