The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks

Abstract

In this paper, the applicability of the entropy method for the trend towards equilibrium for reaction-diffusion systems arising from first order chemical reaction networks is studied. In particular, we present a suitable entropy structure for weakly reversible reaction networks without detail balance condition.We show by deriving an entropy-entropy dissipation estimate that for any weakly reversible network each solution trajectory converges exponentially fast to the unique positive equilibrium with computable rates. This convergence is shown to be true even in cases when the diffusion coefficients of all but one species are zero.For non-weakly reversible networks consisting of source, transmission and target components, it is shown that species belonging to a source or transmission component decay to zero exponentially fast while species belonging to a target component converge to the corresponding positive equilibria, which are determined by the dynamics of the target component and the mass injected from other components. The results of this work, in some sense, complete the picture of trend to equilibrium for first order chemical reaction networks.2010 Mathematics Subject Classification. 35B35, 35B40, 35F35, 35K37, 35Q92.

Quasi-steady-state approximation and numerical simulation for a volume-surface reaction-diffusion system

“…In our understanding, these sharp yet smooth profiles of l and p are the combined effect of the volume diffusion of L and P and the reversible reactions between L and l, which transfer a diffusive effect from the volume Ω onto the boundary Γ. In the context of reaction-diffusion systems with partially degenerate diffusion, such an indirect diffusion effect has been analytically shown first in [16] and for general linear weakly-reversible systems recently in [21]. Figure 4 plots the volume concentrations L and P corresponding to Figure 3.…”

“…Moreover, the dynamics of Fig. 1 forms a so-called weakly-reversible or complex balance reaction network, for which the convergence towards a steady state and well as the structure of the steady states are significantly more subtle than for detailed balance models, see the discussion of the numerical examples in Section 4 or also [21,26] and the references therein. In the following, we propose a continuum model of partial differential equations, which describe the reactions and the diffusion processes of these species both on the domain Ω and on its surface Γ.…”

Coexistence in molecular communications

“…In order to answer this question, it will be necessary to make comparisons with a full reaction-diffusion framework. This can be achieved by simulation studies (e.g., [30]) or a study of qualitative features of reaction-diffusion systems by exploiting recent work in [31], although a full analytical treatment remains challenging as the results in [31] only hold for first order reactions. Motivated by this observation, we wait for a sufficiently long period of time t 1 until the concentration of information molecules lies in a δ 1 -neighborhood of zero, which it will never leave since no new information molecules are produced.…”

Hypocoercivity and Fast Reaction Limit for Linear Reaction Networks with Kinetic Transport

“…This work can be seen as an extension of the corresponding result for linear reaction diffusion models [8], which has recently been extended to general mass action kinetics [9], bringing the theory for reaction diffusion models close to the best results on the global attractor conjecture [13] for ODE models without transport [3].…”

“…Two alternative proofs are presented. In the first one, relaxation in velocity space is separated from relaxation to chemical equilibrium and known results for the latter [8] are used. The second proof extends the proof in [8] by introducing reaction paths in species-velocity space.…”