25 research outputs found
Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics
We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we 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 induce a non-linear relation between thermodynamic fluxes and free energy driving force
Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics
We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we 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 induce a non-linear relation between thermodynamic fluxes and free energy driving force
A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility
Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows
Deriving effective models for multiscale systems via evolutionary -convergence
We discuss possible extensions of the recently established theory of evolutionary Gamma convergence for gradient systems to nonlinear dynamical systems obtained by perturbation of a gradient systems. Thus, it is possible to derive effective equations for pattern forming systems with multiple scales. Our applications include homogenization of reaction-diffusion systems, the justification of amplitude equations for Turing instabilities, and the limit from pure diffusion to reaction-diffusion. This is achieved by generalizing the Gamma-limit approaches based on the energy-dissipation principle or the evolutionary variational estimate
Decay to equilibrium for energy-reaction-diffusion systems
We derive thermodynamically consistent models of reaction-diffusion equations
coupled to a heat equation. While the total energy is conserved, the total
entropy serves as a driving functional such that the full coupled system is a
gradient flow. The novelty of the approach is the Onsager structure, which is
the dual form of a gradient system, and the formulation in terms of the
densities and the internal energy. In these variables it is possible to assume
that the entropy density is strictly concave such that there is a unique
maximizer (thermodynamical equilibrium) given linear constraints on the total
energy and suitable density constraints.
We consider two particular systems of this type, namely, a diffusion-reaction
bipolar energy transport system, and a drift-diffusion-reaction energy
transport system with confining potential. We prove corresponding
entropy-entropy production inequalities with explicitely calculable constants
and establish the convergence to thermodynamical equilibrium, at first in
entropy and further in using Cziszar-Kullback-Pinsker type inequalities.Comment: 40 page
A large-deviations approach to gelation
A large-deviations principle (LDP) is derived for the state at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdős-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. The proofs rely on estimates and asymptotics for the probability that smaller Erdős-Rényi graphs are connected
Gradient and Generic systems in the space of fluxes, applied to reacting particle systems
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 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
Gradient and Generic systems in the space of fluxes, applied to reacting particle systems
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 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
Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck operator
We introduce a family of various finite volume discretization schemes for the
Fokker--Planck operator, which are characterized by different weight functions
on the edges. This family particularly includes the well-established
Scharfetter--Gummel discretization as well as the recently developed
square-root approximation (SQRA) scheme. We motivate this family of
discretizations both from the numerical and the modeling point of view and
provide a uniform consistency and error analysis. Our main results state that
the convergence order primarily depends on the quality of the mesh and in
second place on the quality of the weights. We show by numerical experiments
that for small gradients the choice of the optimal representative of the
discretization family is highly non-trivial while for large gradients the
Scharfetter--Gummel scheme stands out compared to the others