Invariant Manifolds for Physical and Chemical Kinetics

Abstract

Of course, it is physics. Model reduction in kinetics requires physical concepts and structures; it is impossible to make an expedient reduction of a kinetic model without thermodynamics, for example. The entropy, the Legendre transformation generated by the entropy, and the Riemann structure defined by the second differential of the entropy provide the elementary geometrical basis for the first approximation. The physical sense of the models gives many hints for their further processing. So, it is not mathematics; we care about the physical sense more than about rigorous proofs. We should deal with equations even in the absence of theorems about existence and uniqueness of solutions. Mathematics assimilates the physical notions with a considerable delay in time, but any such an assimilation leads to further insights. 1But, without doubt, it is mathematics. The story about invariant manifolds for differential equations began inside mathematics. The first significant steps were taken by two great mathematicians, A.M. Lyapunov and H. Poincaré, at the end of the XIXth century. Then N.M. Krylov and N.N. Bogolyubov, A.N. Kolmogorov, V.I. Arnold and J. Moser, J.E. Marsden, M.I. Vishik, R. Temam, and many other mathematicians developed this field of science, and many elegant theorems and useful methods were created. This is not only pure mathematics, the wide field of applications was developed too, from hydrodynamics to process engineering and control theory and methods. This is pure and applied dynamics. The language of model reduction, the basic notions that we use, the theorems and methods, all this either came from 1 The closest example: after mathematicians discovered how the entropy functional may be important for the theory of the Boltzmann equation, then they proved the existence theorem (P.L. Lions and R. DiPerna, this work was awarded the Fields medal in 1994). VIIIPreface pure and applied dynamics directly, or bears the visible imprint of its ideas and methods. Maybe the book presents a specific chapter on this subject? But, of course, the problems came from physics, from engineering. Maybe it is both physics and mathematics? Or perhaps it is something different, but what can it be? It is not so easy to answer the question, what is the subject of our book, even for the authors. But we can say what we want it to be. We want it to be a special "meeting point" of pure and applied dynamics, of physics, and of engineering sciences. This meeting point has a sufficient number of specific problems, methods and results to deserve a special name. We propose the name Model Engineering. As long as it is engineering, it is synthetic subject: if it is possible to prove something exactly, this is great, and we should follow this possibility, but if the physical sense gives us a seminal hint, well, we should use it even if the rigorous foundations are far from complete. The result is the model that works. In this enormous field of intellectual activity our book tends to be in the theoretical corner; we focus our study on c...

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

“…Further numerical experiments are presented in [64]. The statistics of FENE-P solutions with random strains was studied recently by J.-L. Thiffeault [65] In accordance with [59] the ansatz for Ψ can be suggested in the following form:…”

“…It is not the whole truth, even for the FENE-P equation, as it was shown in ref. [20,59]. The Fokker-Planck equation describes the shape of a probability cloud in the space of conformations.…”

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“…[3,4,5,6,7]. In the sequel, we focus on the aspects related to MIG application to non-isothermal cases.…”

Renormalization of the lattice Boltzmann hierarchy

“…(ii) Although the simple one-step renormalization is quite reliable, a rigorous approach to the non-perturbative renormalization can be based on the invariance equation [23], ∆(q) = ∂ t q − ∂q ∂ρ ∂ t ρ + ∂q ∂j ∂ t j + ∂q ∂P ∂ t P = 0. (14) A stable fixed point of (14) is a fully renormalized q. Owing to a specific feature of the LB hierarchy (linearity of propagation), a way to solve Eq.…”

“…Substituting q lin into (14), we compute the defect of invariance ∆ lin = ∆(q lin ). With this, a refinement can be written, q ≈ q lin + a∆ lin , where a can be estimated via a relaxation method [23].…”