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Iliya V. Karlin

Publications and source records attributed to Iliya V. Karlin.

17 recordsLinked to original sources
arXiv2012

Hydrodynamics from Grad's equations: What can we learn from exact solutions?

A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad's moment equations. Grad's systems are considered as the minimal kinetic models where the Chapman-Enskog method can be studied exactly, thereby providing the basis to compare various approximations in extending the hydrodynamic description beyond the Navier-Stokes approximation. Various techniques, such as the method of partial summation, Pade approximants, and invariance principle are compared both in linear and nonlinear situations.

cond-mat.stat-mech↗
arXiv2005

Thermodynamic Theory of Incompressible Hydrodynamics

The grand potential for open systems describes thermodynamics of fluid flows at low Mach numbers. A new system of reduced equations for the grand potential and the fluid momentum is derived from the compressible Navier-Stokes equations. The incompressible Navier-Stokes equations are the quasi-stationary solution to the new system. It is argued that the grand canonical ensemble is the unifying concept for the derivation of models and numerical methods for incompressible fluids, illustrated here with a simulation of a minimal Boltzmann model in a microflow setup.

cond-mat.stat-mech↗
arXiv2004

Kinetic Theory of Turbulence Modeling: Smallness Parameter, Scaling and Microscopic Derivation of Smagorinsky Model

A mean-field approach (filtering out subgrid scales) is applied to the Boltzmann equation in order to derive a subgrid turbulence model based on kinetic theory. It is demonstrated that the only Smagorinsky type model which survives in the hydrodynamic limit on the viscosity time scale is the so-called tensor-diffusivity model. Scaling of the filter-width with Reynolds number and Knudsen number is established. This sets the first rigorous step in deriving turbulence models from kinetic theory.

cond-mat.stat-mech↗
arXiv2004

Combined micro--macro integration scheme from an invariance principle: application to ferrofluid dynamics

A method for the combination of microscopic and macroscopic simulations is developed which is based on the invariance of the macroscopic relative to the microscopic dynamics. The method recognizes the onset and breakdown of the macroscopic description during the integration. We apply this method to the case of ferrofluid dynamics, where it switches between direct Brownian dynamics simulations and integration of the constitutive equation.

cond-mat.soft↗
arXiv2003

Generalized Additive Entropies in Fully Developed Turbulence

We explore a possible application of the additive generalization of the Boltzmann-Gibbs-Shannon entropy proposed in [A.N. Gorban, I.V. Karlin, Phys. Rev. E, 67:016104 (2003)] to fully developed turbulence. The predicted probability distribution functions are compared with those obtained from Tsallis' entropy and with experimental data. Consequences of the existence of hidden subsystems are illustrated.

physics.flu-dyn↗
arXiv2003

Irreversibility in the short memory approximation

A recently introduced systematic approach to derivations of the macroscopic dynamics from the underlying microscopic equations of motions in the short-memory approximation [Gorban et al, Phys. Rev. E, 63, 066124 (2001)] is presented in detail. The essence of this method is a consistent implementation of Ehrenfest's idea of coarse-graining, realized via a matched expansion of both the microscopic and the macroscopic motions. Applications of this method to a derivation of the nonlinear Vlasov-Fokker-Planck equation, diffusion equation and hydrodynamic equations of the fluid with a long-range mean field interaction are presented in full detail. The advantage of the method is illustrated by the computation of the post-Navier-Stokes approximation of the hydrodynamics which is shown to be stable unlike the Burnett hydrodynamics.

cond-mat.stat-mech↗
arXiv2002

Galilean-Invariant Lattice-Boltzmann Models with H-Theorem

We demonstrate that the requirement of galilean invariance determines the choice of H function for a wide class of entropic lattice Boltzmann models for the incompressible Navier-Stokes equations. The required H function has the form of the Burg entropy for D=2, and of a Tsallis entropy with q=1-2/D for D>2, where D is the number of spatial dimensions. We use this observation to construct a fully explicit, unconditionally stable, galilean invariant, lattice-Boltzmann model for the incompressible Navier-Stokes equations, for which attainable Reynolds number is limited only by grid resolution.

cond-mat.stat-mech↗
arXiv2002

Minimal entropic kinetic models for simulating hydrodynamics

We derive minimal discrete models of the Boltzmann equation consistent with equilibrium thermodynamics, and which recover correct hydrodynamics in arbitrary dimensions. A simple analytical procedure of constructing the equilibrium for the nonisothermal hydrodynamics is established. A new discrete velocity model is proposed for the simulation of the Navier-Stokes-Fourier equation and is tested in the set up of Taylor vortex flow. For the lattice Boltzmann method of isothermal hydrodynamics, the explicit analytical form of the equilibrium distribution is presented.

cond-mat↗
arXiv2002

The additive generalization of the Boltzmann entropy

There exists only one generalization of the classical Boltzmann-Gibbs-Shannon entropy functional to a one-parametric family of additive entropy functionals. We find analytical solution to the corresponding extension of the classical ensembles, and discuss in some detail the example of the deformation of the uncorrelated state.

cond-mat.stat-mech↗
arXiv2002

Method of invariant manifold for chemical kinetics

In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A review of existing alternative methods is extended by a thermodynamically consistent version of the method of intrinsic low-dimensional manifolds. A grid-based version of MIM is developed, and model extensions of low-dimensional dynamics are described. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.

cond-mat.stat-mech↗
arXiv2002

Kinetic boundary conditions in the lattice Boltzmann method

Derivation of the lattice Boltzmann method from the continuous kinetic theory [X. He and L. S. Luo, {\it Phys. Rev. E} {\bf 55}, R6333 (1997); X. Shan and X. He, {\it Phys. Rev. Lett.} {\bf 80}, 65 (1998)] is extended in order to obtain boundary conditions for the method. For the model of a diffusively reflecting moving solid wall, the boundary condition for the discrete set of velocities is derived, and the error of the discretization is estimated. Numerical results are presented which demonstrate convergence to the hydrodynamic limit. In particular, the Knudsen layer in the Kramers' problem is reproduced correctly for small Knudsen numbers.

nlin.CG↗
arXiv2002

Role of the H theorem in lattice Boltzmann hydrodynamic simulations

In the last decade, minimal kinetic models, and primarily the Lattice Boltmann equation, have met with significant success in the simulation of complex hydrodynamic phenomena, ranging from slow flows in grossly irregular geometries to fully developed turbulence, to flows with dynamic phase transitions. Besides their practical value as efficient computational tools for the dynamics complex systems, these minimal models may also represent a new conceptual paradigm in modern computational statistical mechanics: instead of proceeding bottom-up from the underlying microdynamic systems, these minimal kinetic models are built top-down starting from the macroscopic target equations. This procedure can provide dramatic advantages, provided the essential physics is not lost along the way. For dissipative systems, one such essential requirement is the compliance with the Second Law of thermodynamics. In this Colloquium, we present a chronological survey of the main ideas behind the Lattice Boltzmann method, with special focus on the role played by the $H$ theorem in enforcing compliance of the method with macroscopic evolutionary constraints (Second Law) as well as in serving as a numerically stable computational tool for fluid flows and other dissipative systems out of equilibrium.

cond-mat.stat-mech↗
arXiv2002

Canonical Distribution Functions in Polymer Dynamics: I. Dilute Solutions of Flexible Polymers

The quasi--equilibrium or maximum entropy approximation is applied in order to derive constitutive equations from kinetic models of polymer dynamics. It is shown in general and illustrated for an example how canonical distribution functions are obtained from the maximum entropy principle, how macroscopic and constitutive equations are derived therefrom and how these constitutive equations can be implemented numerically. In addition, a measure for the accuracy of the quasi--equilibrium approximation is proposed that can be evaluated while integrating the constitutive equations. In the example considered, it is confirmed that the accuracy of the approximation is increased by including more macroscopic variables. In steady elongational flow, it is found that more macroscopic variables need to be included above the coil--stretch transition to achieve the same accuracy as below.

cond-mat.stat-mech↗
arXiv2002

Canonical Distribution Functions in Polymer Dynamics: II. Liquid--Crystalline Polymers

The quasi--equilibrium approximation is employed as a systematic tool for solving the problem of deriving constitutive equations from kinetic models of liquid--crystalline polymers. It is demonstrated how kinetic models of liquid--crystalline polymers can be approximated in a systematic way, how canonical distribution functions can be derived from the maximum entropy principle and how constitutive equations are derived therefrom. The numerical implementation of the constitutive equations based on the intrinsic dual structure of the quasi--equilibrium manifold thus derived is developed and illustrated for particular examples. Finally, a measure of the accuracy of the quasi--equilibrium approximation is proposed that can be implemented into the numerical integration of the constitutive equation.

cond-mat.stat-mech↗
arXiv2002

Boundary layer variational principles: A case study

Considering the model heat conduction problem in the setting of Grad's moment equations, we demonstrate a crossover in the structure of minima of the entropy production within the boundary layer. Based on this observation, we formulate and compare variation principles for solving the problem of boundary conditions in nonequilibrium thermodynamics.

cond-mat.stat-mech↗
arXiv2002

Family of additive entropy functions out of thermodynamic limit

Starting with the additivity condition for Lyapunov functions of master equation, we derive a one-parametric family of entropy functions which may be appropriate for a description of certain effects of finiteness of statistical systems, in particular, distribution functions with long tails. This one-parametric family is different from Tsallis entropies, and is essentially a convex combination of the Boltzmann-Gibbs-Shannon entropy and the entropy function introduced by Burg. An example of how longer tails are described within the present approach is worked out for the canonical ensemble. In addition, we discuss a possible origin of a hidden statistical dependence, and give explicit recipes how to construct corresponding generalizations of master equation.

cond-mat.stat-mech↗