Computing minimal entropy production trajectories: An approach to model reduction in chemical kinetics

Abstract

Advanced experimental techniques in chemistry and physics provide increasing access to detailed deterministic mass action models for chemical reaction kinetics. Especially in complex technical or biochemical systems the huge amount of species and reaction pathways involved in a detailed modeling approach call for efficient methods of model reduction. These should be automatic and based on a firm mathematical analysis of the ordinary differential equations underlying the chemical kinetics in deterministic models. A main purpose of model reduction is to enable accurate numerical simulations of even high dimensional and spatially extended reaction systems. The latter include physical transport mechanisms and are modeled by partial differential equations. Their numerical solution for hundreds or thousands of species within a reasonable time will exceed computer capacities available now and in a foreseeable future. The central idea of model reduction is to replace the high dimensional dynamics by a low dimensional approximation with an appropriate degree of accuracy. Here I present a global approach to model reduction based on the concept of minimal entropy production and its numerical implementation. For given values of a single species concentration in a chemical system all other species concentrations are computed under the assumption that the system is as close as possible to its attractor, the thermodynamic equilibrium, in the sense that all modes of thermodynamic forces are maximally relaxed except the one, which drives the remaining system dynamics. This relaxation is expressed in terms of minimal entropy production for single reaction steps along phase space trajectories.

Entropy-Related Extremum Principles for Model Reduction of Dissipative Dynamical Systems

“…In the original work [21] the entropy production rate (1) has been chosen as a criterion Φ in (2a) and following the fundamental idea to search for an entropy-related extremum principle that characterizes trajectories on or near slow attracting manifolds we discuss geometric criteria in [31]. In Section 4. we show that the latter are superior to the previously used entropy production rate and yield accurate approximations of slow invariant manifolds for an example model of chemical kinetics.…”

“…For isolated systems with constant internal energy and volume, the entropy is a Lyapunov function of the dynamical system following the Second Law of Thermodynamics. In [21,22], numerical approximations of slow attracting manifolds are obtained by computation of trajectories along which the total (time integral over entropy production rate) entropy production (1) summing over all elementary reaction steps is minimal while chemical equilibrium is approached as time progresses. The approach yields approximations of slow manifolds, however, they lack invariance.…”

Approximation of Slow Attracting Manifolds in Chemical Kinetics by Trajectory-Based Optimization Approaches

“…Under isolated conditions the attractor of a chemical system is the thermodynamic equilibrium. In Lebiedz's model reduction approach, a special trajectory (called Minimal Entropy Production Trajectory (MEPT)) converging towards equilibrium is calculated such that the sum of affinities of the entropy production rates of single reaction steps is minimized 28,35,36 . The entropy production rate is closely related to the concept of chemical affinity which was first introduced by de Donder 37 as the driving force of chemical reactions.…”

Investigation of the Hierarchical Structure of Kinetic Models in Ignition Problems

“…There are other recent approaches based on different minimization strategies such as Rate-Controlled Constrained Equilibrium (RCCE) [22] and Minimal Entropy Production Trajectories (MEPT) [23], and the Lumping Method (LM) [24], but they use knowledge about the hierarchical.decomposed structure that has to be provided by other methods and, consequently, require a time (human resources) consuming analysis of the hierarchy. The Method of Invariant.Integral Manifolds (MIM) [25][26][27][28][29][30][31] gives a mathematical basis for the most approaches listed above as well as for different optimization strategies (see e.g.…”