Orthogonality of fluxes in general nonlinear reaction networks

Abstract

We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.

On decompositions of non-reversible processes

“…Here we start from a general decomposition of the Hamiltonian into two Hamiltonians and obtain a general decomposition for the Lagrangian valuated at 0, see Proposition 3.3. The decomposition may come from different ways that have been studied in the literature, such as from a decomposition of fluxes and forces [5,6,7,8], hydrodynamic limits of many-particle systems [9] (which often impose some specific structures on the functionals) or from a symmetric-antisymmetric decomposition of the generator [10].…”

“…The formulations (2a), (2b) and (2c) give rise to different ways of decomposing a nonreversible dynamics into symmetric and anti-symmetric parts studied in recent years: (i) A decomposition of the generator [15,10]: L = L S + L A (and thus of the dual operator L ′ ), (ii) A decomposition of the fluxes [5]: j = j S + j A , (iii) A decomposition of the forces [5,6,7]:…”

GENERIC framework for reactive fluid flows

“…We point out that evolution equations () and () are rate equations, where the Onsager operator $${\skew{1.5}\bar{{\mathbb{K}}}}$$, or more generally, the non‐linear operator $$\mathrm{D}_{{\bar{\bm{\xi }}}}\bar{\Psi }^*({\bar{\bm{q}}};\cdot )$$ maps the thermodynamic driving force $$\mathrm{D}\bar{{\mathcal {S}}}({\bar{\bm{q}}})\in {\mathcal {Q}}^{*}$$ to a rate $$\partial _t{\bar{\bm{q}}}\in \bar{{\mathcal {Q}}}$$. Other examples for gradient systems can be found for example in [12] for the porous medium equation, in [9, 10] for complex fluids, in [39, 58–60] for (slow & fast) reaction‐diffusion systems with detailed balance, in [61] for the Fokker–Planck equation, and in [62, 63] in the context of large deviations. Mathematical properties of gradient flows are discussed in [11].…”

Dynamical Phase Transitions for Flows on Finite Graphs

“…Note that due to monotonicity of κ, the function V m is convex and has unique minimiser ρ. By macroscopic fluctuation techniques [31,Prop. 5.3] it can be proved that the functional V m decreases along solutions ρ(t) of (4.3), which can also be seen from the direct computation:…”

“…Entropic cost functions are in many aspects more challenging than quadratic ones. For example, they can only induce generalised gradient flows [26], they can only be decomposed using a generalised notion of orthogonality [18,31], and it is not clear whether they relate to some manifold or more general geometry.…”