The chemical Langevin equation and the associated chemical Fokker-Planck equation are well-known continuous approximations of the discrete stochastic evolution of reaction networks. In this work, we show that these approximations suffer from a physical inconsistency, namely, the presence of nonphysical probability currents at the thermal equilibrium even for closed and fully detailed-balanced kinetic schemes. An illustration is given for a model case.

In this work, we deal with general reactive systems involving N species and M elementary reactions under applicability of the mass-action law. Starting from the dynamic variables introduced in two previous works [P. Nicolini and D. Frezzato, J. Chem. Phys. 138(23), 234101 (2013); 138(23), 234102 (2013)], we turn to a new representation in which the system state is specified in a (N × M)(2)-dimensional space by a point whose coordinates have physical dimension of inverse-of-time. By adopting hyper-spherical coordinates (a set of dimensionless "angular" variables and a single "radial" one with physical dimension of inverse-of-time) and by examining the properties of their evolution law both formally and numerically on model kinetic schemes, we show that the system evolves towards the equilibrium as being attracted by a sequence of fixed subspaces (one at a time) each associated with a compact domain of the concentration space. Thus, we point out that also for general non-linear kinetics there exist fixed "objects" on the global scale, although they are conceived in such an abstract and extended space. Moreover, we propose a link between the persistence of the belonging of a trajectory to such subspaces and the closeness to the slow manifold which would be perceived by looking at the bundling of the trajectories in the concentration space.

Dimensionality reduction for the modeling of reacting chemical systems can represent a fundamental achievement both for a clear understanding of the complex mechanisms under study and also for the practical calculation of quantities of interest. To tackle the problem, different approaches have been proposed in the literature. Among them, particular attention has been devoted to the exploitation of the so-called slow manifolds (SMs). These are lower\ud dimensional hypersurfaces where the slow part of the evolution takes place. In this study, we present a low-computational-cost algorithm (based on a previously developed theoretical framework) for the localization of candidate points in the proximity of the SM. A parallel implementation (called DRIMAK) of such an approach has been developed, and the source code is made freely available. We tested the performance of the code on two model schemes\ud for hydrogen combustion, being able to localize points that fall very close to the perceived SM with limited computational effort. The method can provide starting points for other more accurate but computationally demanding strategies; this can be a great help especially when no information about the SM is available a priori, and very many species are involved in the reaction mechanism

In this work, we focus on the possibility to recast the ordinary differential equations (ODEs) governing the evolution of deterministic autonomous dynamical systems (conservative or damped and generally non-linear) into a parameter-free universal format. We term such a representation “hyper-spherical” since the new variables are a “radial” norm having physical units of inverse-of-time and a normalized “state vector” with (possibly complex-valued) dimensionless components. Here we prove that while the system evolves in its physical space, the mirrored evolution in the hyper-spherical space is such that the state vector moves monotonically towards fixed “attracting subspaces” (one at a time). Correspondingly, the physical space can be split into “attractiveness regions.” We present the general concepts and provide an example of how such a transformation of ODEs can be achieved for a class of mechanical-like systems where the physical variables are a set of configurational degrees of freedom and the associated velocities in a phase-space representation. A one-dimensional case model (motion in a bi-stable potential) is adopted to illustrate the procedure

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