Canonical distribution functions in polymer dynamics. (I). Dilute solutions of flexible polymers

Ilg, Patrick; Karlin, Iliya V.; Öttinger, Hans Christian

doi:10.1016/s0378-4371(02)01017-8

Cited by 35 publications

(87 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A very general discussion of the maximum entropy principle with applications to dissipative kinetics is given in the review [231]. Recently the quasiequilibrium approximation with some further correction was applied to description of rheology of polymer solutions [254,266] and of ferrofluids [267,268]. Quasiequilibrium approximations for quantum systems in the Wigner representation [36,37] was discussed very recently [232].…”

Section: Entropy and Quasiequilibriummentioning

confidence: 99%

See 1 more Smart Citation

Invariant Manifolds for Physical and Chemical Kinetics

Gorban¹,

Karlin²

2005

Lecture Notes in Physics

212

416

View full text Add to dashboard Cite

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...

show abstract

Section: Entropy and Quasiequilibriummentioning

confidence: 99%

“…In most of the works (of us and of other people on similar problems), analytic forms were required to represent manifolds (see, however, the method of Legendre integrators [254,266,369]). However, in order to construct manifolds of a relatively low dimension, grid-based representations of manifolds become a relevant option.…”

Section: Invariant Gridsmentioning

confidence: 99%

Invariant Manifolds for Physical and Chemical Kinetics

Gorban¹,

Karlin²

2005

Lecture Notes in Physics

212

416

View full text Add to dashboard Cite

show abstract

“…Numerical experiments [12,13] have proven the effectiveness of this idea. The main computational challenge in this method is to calculate integrals of the form…”

Section: Macroscopic Variables and Boundary Conditionsmentioning

confidence: 88%

“…The methods of the first order, based on this idea, were suggested and tested in Refs. [12][13][14]. The method of the thermodynamic projector lets us represent every ansatz-manifold as the solution to the variational problem (2) with specially chosen constraints.…”

Section: S(ψ)mentioning

confidence: 99%

Legendre integrators, post-processing and quasiequilibrium

Gorban

Karlin

2004

Journal of Non-Newtonian Fluid Mechanics

View full text Add to dashboard Cite

A toolbox for the development and reduction of the dynamical models of nonequilibrium systems is presented. The main components of this toolbox are: Legendre integrators, dynamical post-processing, and the thermodynamic projector. The thermodynamic projector is the tool to transform almost any anzatz to a thermodynamically consistent model. The post-processing is the cheapest way to improve the solution obtained by the Legendre integrators. Legendre integrators give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium ("maximum entropy", MaxEnt) approximations. The essentially new element of this toolbox, the method of thermodynamic projector, is demonstrated on application to the FENE-P model of polymer kinetic theory. The multi-peak model of polymer dynamics is developed.

show abstract

“…A very general discussion of the maximum entropy principle with applications to dissipative kinetics is given in the review [25]. Recently the quasiequilibrium approximation with some further correction was applied to description of rheology of polymer solutions [26,27] and offerrofluids [28,29]. Quasiequilibrium approximations for quantum systems in the Wigner representation [30,31] was discussed very recently [32].…”

Section: Article In Pressmentioning

confidence: 99%

Quasi-equilibrium closure hierarchies for the Boltzmann equation

Gorban

Karlin

2006

Physica A: Statistical Mechanics and its Applications

Self Cite

View full text Add to dashboard Cite

In this paper, explicit method of constructing approximations (the triangle entropy method) is developed for nonequilibrium problems. This method enables one to treat any complicated nonlinear functionals that fit best the physics of a problem (such as, for example, rates of processes) as new independent variables.The work of the method is demonstrated on the Boltzmann's-type kinetics. New macroscopic variables are introduced (moments of the Boltzmann collision integral, or scattering rates). They are treated as independent variables rather than as infinite moment series. This approach gives the complete account of rates of scattering processes. Transport equations for scattering rates are obtained (the second hydrodynamic chain), similar to the usual moment chain (the first hydrodynamic chain). Various examples of the closure of the first, of the second, and of the mixed hydrodynamic chains are considered for the hard sphere model. It is shown, in particular, that the complete account of scattering processes leads to a renormalization of transport coefficients.The method gives the explicit solution for the closure problem, provides thermodynamic properties of reduced models, and can be applied to any kinetic equation with a thermodynamic Lyapunov function. r

show abstract

Canonical distribution functions in polymer dynamics. (I). Dilute solutions of flexible polymers

Cited by 35 publications

References 26 publications

Invariant Manifolds for Physical and Chemical Kinetics

Invariant Manifolds for Physical and Chemical Kinetics

Legendre integrators, post-processing and quasiequilibrium

Quasi-equilibrium closure hierarchies for the Boltzmann equation

Contact Info

Product

Resources

About