arXiv subjects

Vakhtang Putkaradze

Publications and source records attributed to Vakhtang Putkaradze.

44 records · Page 3Linked to original sources

arXiv2007

Kinetic models of heterogeneous dissipation

We suggest kinetic models of dissipation for an ensemble of interacting oriented particles, for example, moving magnetized particles. This is achieved by introducing a double bracket dissipation in kinetic equations using an oriented Poisson bracket, and employing the moment method to derive continuum equations for magnetization and density evolution. We show how our continuum equations generalize the Debye-Hueckel equations for attracting round particles, and Landau-Lifshitz-Gilbert equations for spin waves in magnetized media. We also show formation of singular solutions that are clumps of aligned particles (orientons) starting from random initial conditions. Finally, we extend our theory to the dissipative motion of self-interacting curves.

nlin.AO↗

arXiv2007

Geometric dissipation in kinetic equations

A new symplectic variational approach is developed for modeling dissipation in kinetic equations. This approach yields a double bracket structure in phase space which generates kinetic equations representing coadjoint motion under canonical transformations. The Vlasov example admits measure-valued single-particle solutions. Such solutions are reversible; and the total entropy is a Casimir, and thus is preserved.

physics.plasm-ph↗

arXiv2006

Formation and Evolution of Singularities in Anisotropic Geometric Continua

Evolutionary PDEs for geometric order parameters that admit propagating singular solutions are introduced and discussed. These singular solutions arise as a result of the competition between nonlinear and nonlocal processes in various familiar vector spaces. Several examples are given. The motivating example is the directed self assembly of a large number of particles for technological purposes such as nano-science processes, in which the particle interactions are anisotropic. This application leads to the derivation and analysis of gradient flow equations on Lie algebras. The Riemann structure of these gradient flow equations is also discussed.

nlin.AO↗

arXiv2006

Geometric order parameter equations

Aggregation of particles whose interaction potential depends on their mutual orientation is considered. The aggregation dynamics is derived using a version of Darcy's law and a variational principle depending on the geometric nature of the physical quantities. The evolution equations that result separate into two classes: either characteristic equations, or gradient flow equations. We derive analytical solutions of both types of equations which are collapsed (clumped) states and show their dynamical emergence from smooth initial conditions in numerical simuations.

nlin.AO↗

arXiv2005

Formation of clumps and patches in self-aggregation of finite size particles

New model equations are derived for dynamics of self-aggregation of finite-size particles. Differences from standard Debye-Huckel and Keller-Segel models are: a) the mobility $μ$ of particles depends on the locally-averaged particle density and b) linear diffusion acts on that locally-averaged particle density. The cases both with and without diffusion are considered here. Surprisingly, these simple modifications of standard models allow progress in the analytical description of evolution as well as the complete analysis of stationary states. When $μ$ remains positive, the evolution of collapsed states in our model reduces exactly to finite-dimensional dynamics of interacting particle clumps. Simulations show these collapsed (clumped) states emerging from smooth initial conditions, even in one spatial dimension. If $μ$ vanishes for some averaged density, the evolution leads to spontaneous formation of \emph{jammed patches} (weak solution with density having compact support). Simulations confirm that a combination of these patches forms the final state for the system.

nlin.PS↗

arXiv2005

Aggregation of finite size particles with variable mobility

New model equations are derived for dynamics of self-aggregation of finite-size particles. Differences from standard Debye-Huckel and Keller-Segel models are: the mobility of particles depends on the configuration of their neighbors and linear diffusion acts on locally-averaged particle density. The evolution of collapsed states in these models reduces exactly to finite-dimensional dynamics of interacting particle clumps. Simulations show these collapsed (clumped) states emerge from smooth initial conditions, even in one spatial dimension.

nlin.PS↗

arXiv2003

Rotating Concentric Circular Peakons

We study invariant manifolds of measure-valued solutions of the partial differential equation for geodesic flow of a pressureless fluid. These solutions describe interaction dynamics on lower-dimensional support sets; for example, curves, or filaments, of momentum in the plane. The 2+1 solutions we study are planar generalizations of the 1+1 peakon solutions of Camassa & Holm [1993] for shallow water solitons. As an example, we study the canonical Hamiltonian interaction dynamics of $N$ rotating concentric circles of peakons, whose solution manifold is $2N-$dimensional. Thus, the problem is reduced from infinite dimensions to a finite-dimensional, canonical, invariant manifold. The existence of this reduced solution manifold and many of its properties may be understood, by noticing that it is also the momentum map for the action of diffeomorphisms on the space of curves in the plane. We show both analytical and numerical results.

nlin.SI↗

arXiv1996

Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns

Spatiotemporally chaotic dynamics of a Kuramoto-Sivashinsky system is described by means of an infinite hierarchy of its unstable spatiotemporally periodic solutions. An intrinsic parametrization of the corresponding invariant set serves as accurate guide to the high-dimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics.

chao-dyn↗