arXiv subjects

Ruslan L. Davidchack

Publications and source records attributed to Ruslan L. Davidchack.

26 records · Page 2Linked to original sources

arXiv2009

On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain

The continuous and discrete symmetries of the Kuramoto-Sivashinsky system restricted to a spatially periodic domain play a prominent role in shaping the invariant sets of its chaotic dynamics. The continuous spatial translation symmetry leads to relative equilibrium (traveling wave) and relative periodic orbit (modulated traveling wave) solutions. The discrete symmetries lead to existence of equilibrium and periodic orbit solutions, induce decomposition of state space into invariant subspaces, and enforce certain structurally stable heteroclinic connections between equilibria. We show, on the example of a particular small-cell Kuramoto-Sivashinsky system, how the geometry of its dynamical state space is organized by a rigid `cage' built by heteroclinic connections between equilibria, and demonstrate the preponderance of unstable relative periodic orbits and their likely role as the skeleton underpinning spatiotemporal turbulence in systems with continuous symmetries. We also offer novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space flow through projections onto low-dimensional, PDE representation independent, dynamically invariant intrinsic coordinate frames, as well as in terms of the physical, symmetry invariant energy transfer rates.

nlin.CD↗

arXiv2009

Langevin Thermostat for Rigid Body Dynamics

We present a new method for isothermal rigid body simulations using the quaternion representation and Langevin dynamics. It can be combined with the traditional Langevin or gradient (Brownian) dynamics for the translational degrees of freedom to correctly sample the NVT distribution in a simulation of rigid molecules. We propose simple, quasi-symplectic second-order numerical integrators and test their performance on the TIP4P model of water. We also investigate the optimal choice of thermostat parameters.

physics.comp-ph↗

arXiv2008

On the use of stabilising transformations for detecting unstable periodic orbits in the Kuramoto-Sivashinsky equation

In this paper we develop further a method for detecting unstable periodic orbits (UPOs) by stabilising transformations, where the strategy is to transform the system of interest in such a way that the orbits become stable. The main difficulty of using this method is that the number of transformations, which were used in the past, becomes overwhelming as we move to higher dimensions (Davidchack and Lai 1999; Schmelcher et al. 1997, 1998). We have recently proposed a set of stabilising transformations which is constructed from a small set of already found UPOs (Crofts and Davidchack 2006). The main benefit of using the proposed set is that its cardinality depends on the dimension of the unstable manifold at the UPO rather than the dimension of the system. In a typical situation the dimension of the unstable manifold is much smaller than the dimension of the system so the number of transformations is much smaller. Here we extend this approach to high-dimensional systems of ODEs and apply it to the model example of a chaotic spatially extended system -- the Kuramoto-Sivashinsky equation. A comparison is made between the performance of this new method against the competing methods of Newton-Armijo (NA) and Levernberg-Marquardt (LM). In the latter case, we take advantage of the fact that the LM algorithm is able to solve under-determined systems of equations, thus eliminating the need for any additional constraints.

nlin.CD↗

arXiv2003

Direct calculation of the crystal-melt interfacial free energies for continuous potentials: Application to the Lennard-Jones system

Extending to continuous potentials a cleaving wall molecular-dynamics simulation method recently developed for the hard-sphere system [Phys.Rev.Lett 85, 4751 (2000)], we calculate the crystal-melt interfacial free energies, $γ$, for a Lennard-Jones system as functions of both crystal orientation and temperature. At the triple point, T* = 0.617, the results are consistent with an earlier cleaving potential calculation by Broughton and Gilmer [J. Chem. Phys. {\bf 84}, 5759 (1986)], however, the greater precision of the current calculation allows us to accurately determine the anisotropy of $γ$. From our data we find that, at all temperatures studied, $γ_{111} < γ_{110} < γ_{100}$. Comparison is made to the results from our previous hard-sphere calculation and to recent results for Ni by Asta, Hoyt and Karma [Phys. Rev. B, 66 100101(R) (2002)].

cond-mat.mtrl-sci↗

arXiv2000

Direct calculation of the hard-sphere crystal/melt interfacial free energy

We present a direct calculation by molecular-dynamics computer simulation of the crystal/melt interfacial free energy, $γ$, for a system of hard spheres of diameter $σ$. The calculation is performed by thermodynamic integration along a reversible path defined by cleaving, using specially constructed movable hard-sphere walls, separate bulk crystal and fluid systems, which are then merged to form an interface. We find the interfacial free energy to be slightly anisotropic with $γ$ = 0.62$\pm 0.01$, 0.64$\pm 0.01$ and 0.58$\pm 0.01 k_BT/σ^2$ for the (100), (110) and (111) fcc crystal/fluid interfaces, respectively. These values are consistent with earlier density functional calculations and recent experiments measuring the crystal nucleation rates from colloidal fluids of polystyrene spheres that have been interpreted [Marr and Gast, Langmuir {\bf 10}, 1348 (1994)] to give an estimate of $γ$ for the hard-sphere system of $0.55 \pm 0.02 k_BT/σ^2$, slightly lower than the directly determined value reported here.

cond-mat.mtrl-sci↗

arXiv1999

Efficient algorithm for detecting unstable periodic orbits in chaotic systems

We present an efficient method for fast, complete, and accurate detection of unstable periodic orbits in chaotic systems. Our method consists of a new iterative scheme and an effective technique for selecting initial points. The iterative scheme is based on the semi-implicit Euler method, which has both fast and global convergence, and only a small number of initial points is sufficient to detect all unstable periodic orbits of a given period. The power of our method is illustrated by numerical examples of both two- and four-dimensional maps.

chao-dyn↗

arXiv1999

Molecular dynamics simulation of binary hard-sphere crystal/melt interfaces

We examine, using molecular dynamics simulation, the structure and thermodynamics of the (100) and (111) disordered face-centered cubic (FCC) crystal/melt interfaces for a binary hard-sphere system. This study is an extension of our previous work, [Phys. Rev. E 54, R5905 (1996)], in which preliminary data for the (100) interface were reported. Density and diffusion profiles on both fine- and course-grained scales are calculated and analyzed leading to the conclusion that equilibrium interfacial segregation is minimal in this system.

cond-mat.stat-mech↗

arXiv1999

Weighted-density approximation for general nonuniform fluid mixtures

In order to construct a general density-functional theory for nonuniform fluid mixtures, we propose an extension to multicomponent systems of the weighted-density approximation (WDA) of Curtin and Ashcroft [Phys. Rev. A 32, 2909 (1985)]. This extension corrects a deficiency in a similar extension proposed earlier by Denton and Ashcroft [Phys. Rev. A 42, 7312 (1990)], in that that functional cannot be applied to the multi-component nonuniform fluid systems with spatially varying composition, such as solid-fluid interfaces. As a test of the accuracy of our new functional, we apply it to the calculation of the freezing phase diagram of a binary hard-sphere fluid, and compare the results to simulation and the Denton-Ashcroft extension.

cond-mat.stat-mech↗