Lagrange Instability of Geodesics in Curved Double Twisted Liquid Crystals

Garcia de Andrade

Home/Archive/arXiv

arXiv · 0708.3378

Lagrange Instability of Geodesics in Curved Double Twisted Liquid Crystals

Garcia de Andrade

View original source ↗Read source document ↗Export metadata ↓

Abstract

It is shown that curved and flat helical double twisted liquid crystal (DTLC) in blue phase, can be unstable (stable) depending of the sign, negative (positive) of sectional curvature, depending on the pitch of the helix of the nematic crystal. In both cases Cartan torsion is presented. It is also shown that the instability or stability depends on the value of the pitch of the helix in nematic crystals. Frank energy stability A similar result using the method of Frank energy stability in the twist of cholesteric liquid crystal was given by Kiselev and Sluckin [PRE 71(2005)], where the free twist number determines the equilibrium value of the cholesteric liquid crystals (CLC) pitch of the helix. As a final example we solve the geodesic equations in twisted nematics with variable pitch helix and non-constant torsion. This non-Riemannian geometrical approach, also seems to unify two recent analysis of cylindrical columns given by Santangelo et al [PRL 99,(2007)] and the curved crystal endowed with torsion, given by Vitelli et al [Proc Nat Acad Sci (2006)]. Stability of toroidal curved surfaces were also previously considered by Bowick et al [PRE 69,(2004)], as an example of curvature-induced defect. Investigation of the Lagrangean instability may be useful in the investigation of HIV viruses and proteins.

Explore related subjects

cond-mat.soft cond-mat.mtrl-sci

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Garcia de Andrade. 2007-08-24. Lagrange Instability of Geodesics in Curved Double Twisted Liquid Crystals. https://arxiv.org/abs/0708.3378

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Dirac Representation for Lattice Spin Operators: Spin-$1/2$ and Spin-$1$ cases

We introduce a quantum representation of lattice spin operators (LSOs) by mapping quantum spins to their classical analogues for spin values $S=1/2$ and $S=1$. The resulting Dirac (bra--ket) forms are derived from the structure of binary and ternary digit distributions in non-negative integers. \color{blue} The construction requires only an enumeration of the lattice sites and is therefore independent on the spatial dimension and lattice geometry. \color{black} Analytical expressions for extracting the $j$th bit or trit of an integer without performing an explicit binary or ternary decomposition are obtained. \color{blue}As an application of the representation, \color{black} we analyze the one-dimensional spin-$1/2$ {XXZ} Heisenberg model with open boundaries in a magnetic field in both absence (uniform magnetic field) and presence of disorder (random magnetic field). In the absence of disorder (clean system), we demonstrate that the corresponding eigenvalue problem can be reduced to a tight-binding problem on a graph. For the isotropic case, this is solved analytically without resorting to any spinless transformation or the Bethe Ansatz. In the presence of disorder, a convergent perturbation theory is elaborated. Our analytical results are benchmarked against exact diagonalization for systems up to $K\leq 18$ spins. These computations exploit the global $U(1)$ symmetry to block-diagonalize the Hamiltonian and the spin-inversion symmetry to further decompose the zero-magnetization sector. Quantitative agreement between analytical and numerical results is observed.

cond-mat.soft↗

arXiv2026

Density and shape govern the dynamical self-organization of active matter on a closed surface

Morphogenesis emerges from dynamic feedback among geometry, mechanics, and chemistry; however, disentangling these contributions in living systems remains challenging. Here, we focus on the interplay between geometry and mechanics by developing a minimal in vitro model in which purified microtubules and kinesin motor clusters self-organize into a two-dimensional active nematic cortex at the surface of spherical water-in-oil droplets. The spherical geometry enforces a total topological charge of +2, here realized by four +1/2 defects whose trajectories reveal robust, self-sustained oscillations. Using full-surface reconstructions, we show that the collective dynamics of the defects lead to a periodic switching between planar and tetrahedral arrangements through alternating coiling and hemisphere-crossing phases. By tuning microtubule density, the system spans a continuum from a classic defect-dominated active nematic to a regime resembling an extensile filament confined to a curved surface, where low density is associated with increased trajectory variability and direction reversals. Geometric perturbations introduced through controlled squeezing redistribute curvature and induce the nucleation of additional defects, thereby reorganizing the entire topological landscape while preserving total charge. Together, these results show that periodic morphogenetic-like cycles, defect topology, and material organization can arise solely from the interplay of activity, density, and curvature. This reconstituted system provides a versatile platform for elucidating the coupling between mechanics and geometry underlying shape formation in active biological matter.

cond-mat.soft↗

arXiv2026

Exact dynamics and dimensional effects in torque-driven trapped active Brownian particles

The interplay of chirality, self-propulsion, and confinement generates pronounced non-equilibrium fluctuations, with higher-order moments encoding the shape of the position distribution. We present an exact analytical framework, based on the Laplace transform of the Fokker-Planck equation, for the transient dynamics of a chiral active Brownian particle in a harmonic trap in two and three dimensions. Closed-form expressions for time-dependent moments up to fourth order allow a full characterization of the excess kurtosis across transient and steady-state regimes. In two dimensions, the excess kurtosis exhibits damped oscillations with multiple re-entrant crossovers, evolving from negative values, indicative of off-centered ring-like distributions, to small positive values characteristic of near-Gaussian distributions with a weak tail enhancement; increasing trap stiffness progressively suppresses this oscillatory response. In three dimensions, the excess kurtosis of a torque-driven active Brownian particle stays negative, reflecting robust non-Gaussian fluctuations with half-ring- to band-like position distributions in the plane defined by the torque axis. These results reveal how chirality, propulsion, confinement, and dimensionality jointly shape transient dynamics and provide experimentally accessible signatures of confined chiral active motion.

cond-mat.soft↗