ALEX 2018 Workshop: Abstracts

Gradient and GENERIC structures in the space of fluxes

D.R. Michiel Renger,

Weierstraß-Institut Berlin (Germany)

The chemical reaction rate equation

${\dot{\rho }}_t=\mathrm{\Gamma }\overline{k}({\rho }_t),$ ${\overline{k}}_r=k_{r,\mathrm{fw}}-k_{r,\mathrm{bw}},r\in ℛ,$ (1)

models the evolution of chemical concentrations of different species $𝒴$ under a set $ℛ$ of chemical reactions. Here $k_{r,\mathrm{fw}},k_{r,\mathrm{bw}}$ are the forward and backward reaction rates and the matrix $\mathrm{\Gamma }\in {ℝ}^{𝒴\times ℛ}$ contains the stoichiometric coefficients of all reactions.

A classical underlying microscopic model describes the concentration of random reaction particles in a large volume $V$, which converges as $V\to \mathrm{\infty }$ to the solution of (1). The corresponding large-deviation cost for a path to deviate from the expected path can be written as ${\int }_0^T\widehat{ℒ}({\rho }_t,{\dot{\rho }}_t)𝑑t$ for some cost function $\widehat{ℒ}$. In [2, 3] we showed how this cost can be related to a generalised gradient system for the evolution (1) by making the Ansatz that the cost has the form of an energy-dissipation balance:

$$\widehat{ℒ}({\rho }_t,{\dot{\rho }}_t)=\widehat{\mathrm{\Psi }}({\rho }_t,{\dot{\rho }}_t)+{\widehat{\mathrm{\Psi }}}^{*}({\rho }_t,-D\widehat{ℱ}({\rho }_t))+⟨D\widehat{ℱ}({\rho }_t),{\dot{\rho }}_t⟩.$$ (2)

More information about microscopic fluctuations can be retrieved by studying particle/reaction net fluxes, i.e. by bookkeeping the amount $W_t^V$ of forward minus backward reactions that have taken place up to time $t$. The concentrations can be retrieved from the fluxes via the continuity equation ${\rho }_t^V={\rho }_0^V+\mathrm{\Gamma }{W}_t^V$. Now the large-particle limit evolution is ${\dot{w}}_t=\overline{k}({\rho }_t)$, with corresponding large-deviation cost ${\int }_0^Tℒ(w_t,{\dot{w}}_t)𝑑t$ [4].

It turns out that for a network of fast and slow reactions, the flux cost can induce a generalised GENERIC structure in the spirit of [1], similarly to (2):

$$ℒ(w_t,{\dot{w}}_t)=\mathrm{\Psi }(w_t,{\dot{w}}_t-L(w_t)Dℰ(w_t))+{\mathrm{\Psi }}^{*}(w_t,-Dℱ(w_t))+⟨Dℱ(w_t),{\dot{w}}_t⟩,$$ (3)

the Hamiltonian part $LDℰ$ corresponds to the fast reactions whereas the dissipative elements $\mathrm{\Psi },{\mathrm{\Psi }}^{*},ℱ$ corresponds to the slow reactions.

From the fact that the two cost functions are related by a contraction principle $\widehat{ℒ}(\rho ,s)={inf}_{\rho ={\rho }_0+\mathrm{\Gamma }w,s=\mathrm{\Gamma }j}ℒ(w,j)$, we can in fact derive a more general theory about the relation of gradient/GENERIC structures in the space of fluxes with gradient/GENERIC structures in the space of concentrations [5].

Acknowledgments: This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems”, Project C08 “Stochastic spatial coagulation particle processes”.