# Gradient and GENERIC Systems in the Space of Fluxes, Applied to Reacting Particle Systems

@article{Renger2018GradientAG,
title={Gradient and GENERIC Systems in the Space of Fluxes, Applied to Reacting Particle Systems},
author={Michiel Renger},
journal={Entropy},
year={2018},
volume={20}
}
• M. Renger
• Published 27 June 2018
• Computer Science
• Entropy
In 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…

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## References

SHOWING 1-10 OF 42 REFERENCES
From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage
• Computer Science
• 2010
A new connection is established between a system of many independent Brownian particles and the deterministic diffusion equation by proving that Jh and Kh are equal up to second order in h as h → 0.
Flux Large Deviations of Independent and Reacting Particle Systems, with Implications for Macroscopic Fluctuation Theory
We consider a system of independent particles and a system of reacting particles on a discrete state space. For the independent case, we rigorously prove a dynamic large-deviation principle for the
On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion
• Mathematics
• 2013
Motivated by the occurrence 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(ρt,ρ̇t)=0\$\mathcal{L} (\rho
GENERIC formalism of a Vlasov-Fokker-Planck equation and connection to large-deviation principles
• Mathematics
• 2013
In this paper we discuss the connections between a Vlasov–Fokker–Planck equation and an underlying microscopic particle system, and we interpret those connections in the context of the GENERIC
Formulation of thermoelastic dissipative material behavior using GENERIC
We show that the coupled balance equations for a large class of dissipative materials can be cast in the form of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). In
Large Deviation Principle For Finite-State Mean Field Interacting Particle Systems
• Mathematics
• 2016
We establish a large deviation principle for the empirical measure process associated with a general class of finite-state mean field interacting particle systems with Lipschitz continuous transition
WASSERSTEIN GRADIENT FLOWS FROM LARGE DEVIATIONS OF MANY-PARTICLE LIMITS
• Mathematics
• 2013
We study the Fokker-Planck equation as the many-particle limit of a stochastic particle system on one hand and as a Wasserstein gradient flow on the other. We write the path-space rate functional,
Non-equilibrium Thermodynamical Principles for Chemical Reactions with Mass-Action Kinetics
• Mathematics
SIAM J. Appl. Math.
• 2017
For both systems, the corresponding large deviations are calculated and it is shown 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.
Continuous Time Markov Chain Models for Chemical Reaction Networks
• Mathematics
• 2011
This chapter develops much of the mathematical machinery needed to describe the stochastic models of reaction networks and shows how to derive the deterministic law of mass action from the Markov chain model.
Canonical Structure and Orthogonality of Forces and Currents in Irreversible Markov Chains
• Mathematics
Journal of statistical physics
• 2018
We discuss a canonical structure that provides a unifying description of dynamical large deviations for irreversible finite state Markov chains (continuous time), Onsager theory, and Macroscopic