Reduced description in the reaction kinetics

Abstract

Models of complex reactions in thermodynamically isolated systems often demonstrate evolution towards low-dimensional manifolds in the phase space. For this class of models, we suggest a direct method to construct such manifolds, and thereby to reduce the e ective dimension of the problem. The approach realizes the invariance principle of the reduced description, it is based on iterations rather than on a small parameter expansion, it leads to tractable linear problems, and is consistent with thermodynamic requirements. The approach is tested with a model of catalytic reaction.

Corrections and enhancements of quasi-equilibrium states

“…Construction of the self-adjoined operator makes use of the detail balance principle and Onsager relations as applied to rapid motions, and the explicit realization is given in [15].…”

“…It should be stressed that the method of invariant manifolds seeks true solutions of the original dynamics (or at least sufficient approximations thereof). In spite of the fact that the method is rather elaborative, it has been successful in a set of problems, in particular, the derivation of higher-order hydrodynamics from the Boltzmann and related equations [13,16,17], the construction of a reduced description for non-linear reaction kinetic equations [15,18], and the establishment of the universal limit in the dilute polymer dynamics [19]. It is essential that the method of invariant manifold assumes dissipativity of the microscopic dynamics, and cannot be used to introduce dissipativity if it is absent from the input microscopic dynamics.…”

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

“…The family of distributions (14) can be improved to include the proper equilibrium (this is important condition: the equilibrium should belong to the ansatz manifold). There may be different further refinements, some of them are discussed below.…”

“…The Gaussian parallelepiped (14) seems to be a "rigid" structure: the possibilities to extend this ansatz, to make it more exact, but with preservation of more or less transparent structure, are not obvious. The simple transformation can improve this situation.…”

“…There are different physically motivated ways to select the scalar product and create the symmetrization [14,15,16]. But symmetrization does not provide termodinamicity and the entropy for the projected equations can decrease.…”