arXiv subjects

N. Destainville

Publications and source records attributed to N. Destainville.

12 recordsLinked to original sources

arXiv2017

Random tilings of high symmetry: II. Boundary conditions and numerical studies

We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence with algorithms for sorting lists in computer science. We obtain statistics on path counting and vertex coordination which compare well with predictions of mean-field theory and allow estimation of the configurational entropy, which tends to the value 0.568 per vertex in the limit of continuous symmetry. Tilings with phason strain appear to share the same entropy as unstrained tilings, as predicted by mean-field theory. We consider the thermodynamic limit and argue that the limiting fixed boundary entropy equals the limiting free boundary entropy, although these differ for finite rotational symmetry.

cond-mat.stat-mech↗

arXiv2007

Thermal Denaturation of Fluctuating DNA Driven by Bending Entropy

A statistical model of homopolymer DNA, coupling internal base pair states (unbroken or broken) and external thermal chain fluctuations, is exactly solved using transfer kernel techniques. The dependence on temperature and DNA length of the fraction of denaturation bubbles and their correlation length is deduced. The thermal denaturation transition emerges naturally when the chain fluctuations are integrated out and is driven by the difference in bending (entropy dominated) free energy between broken and unbroken segments. Conformational properties of DNA, such as persistence length and mean-square-radius, are also explicitly calculated, leading, e.g., to a coherent explanation for the experimentally observed thermal viscosity transition.

cond-mat.soft↗

arXiv2006

Detection of confinement and jumps in single molecule membrane trajectories

We propose a novel variant of the algorithm by Simson et al. [R. Simson, E.D. Sheets, K. Jacobson, Biophys. J. 69, 989 (1995)]. Their algorithm was developed to detect transient confinement zones in experimental single particle tracking trajectories of diffusing membrane proteins or lipids. We show that our algorithm is able to detect confinement in a wider class of confining potential shapes than Simson et al.'s one. Furthermore it enables to detect not only temporary confinement but also jumps between confinement zones. Jumps are predicted by membrane skeleton fence and picket models. In the case of experimental trajectories of $μ$-opioid receptors, which belong to the family of G-protein-coupled receptors involved in a signal transduction pathway, this algorithm confirms that confinement cannot be explained solely by rigid fences.

q-bio.QM↗

arXiv2005

Random tilings of high symmetry: I. Mean-field theory

We study random tiling models in the limit of high rotational symmetry. In this limit a mean-field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions. We base our mean-field theory on an iterative tiling construction inspired by the work of de Bruijn. In addition to the entropy, we consider correlation functions, phason elasticity and the thermodynamic limit. Tilings of dimension other than two are considered briefly.

cond-mat.stat-mech↗

arXiv2004

A formula for the number of tilings of an octagon by rhombi

We propose the first algebraic determinantal formula to enumerate tilings of a centro-symmetric octagon of any size by rhombi. This result uses the Gessel-Viennot technique and generalizes to any octagon a formula given by Elnitsky in a special case.

math.CO↗

arXiv2004

Slow dynamics due to entropic barriers in the one-dimensional `descent model'

We propose a novel one-dimensional simple model without disorder exhibiting slow dynamics and aging at the zero temperature limit. This slow dynamics is due to entropic barriers. We exactly solve the statics of the model. We derive an evolution equation for the slow modes of the dynamics which are responsible for the aging. This equation is equivalent to a random walker on the energetic landscape. This latter elementary model can be solved analytically up to some basic approximations and is eventually shown to present aging by itself, as well as a slow logarithmic relaxation of the energy: e(t) ~ 1/ln(t) at large t.

cond-mat.stat-mech↗

arXiv2003

Quantum dynamics in high codimension tilings: from quasiperiodicity to disorder

We analyze the spreading of wavepackets in two-dimensional quasiperiodic and random tilings as a function of their codimension, i.e. of their topological complexity. In the quasiperiodic case, we show that the diffusion exponent that characterizes the propagation decreases when the codimension increases and goes to 1/2 in the high codimension limit. By constrast, the exponent for the random tilings is independent of their codimension and also equals 1/2. This shows that, in high codimension, the quasiperiodicity is irrelevant and that the topological disorder leads in every case, to a diffusive regime, at least in the time scale investigated here.

cond-mat.dis-nn↗

arXiv2003

Arctic octahedron in three-dimensional rhombus tilings and related integer solid partitions

Three-dimensional integer partitions provide a convenient representation of codimension-one three-dimensional random rhombus tilings. Calculating the entropy for such a model is a notoriously difficult problem. We apply transition matrix Monte Carlo simulations to evaluate their entropy with high precision. We consider both free- and fixed-boundary tilings. Our results suggest that the ratio of free- and fixed-boundary entropies is $σ_{free}/σ_{fixed}=3/2$, and can be interpreted as the ratio of the volumes of two simple, nested, polyhedra. This finding supports a conjecture by Linde, Moore and Nordahl concerning the ``arctic octahedron phenomenon'' in three-dimensional random tilings.

cond-mat.stat-mech↗

arXiv2002

Bounding spectral gaps of Markov chains: a novel exact multi-decomposition technique

We propose an exact technique to calculate lower bounds of spectral gaps of discrete time reversible Markov chains on finite state sets. Spectral gaps are a common tool for evaluating convergence rates of Markov chains. As an illustration, we successfully use this technique to evaluate the ``absorption time'' of the ``Backgammon model'', a paradigmatic model for glassy dynamics. We also discuss the application of this technique to the ``Contingency table problem'', a notoriously difficult problem from probability theory. The interest of this technique is that it connects spectral gaps, which are quantities related to dynamics, with static quantities, calculated at equilibrium.

cond-mat.stat-mech↗

arXiv2000

Fixed-boundary octagonal random tilings: a combinatorial approach

Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus generalizing previous results on the more restricted class of codimension-one tilings. In particular this method gives access to counting formulas, which are directly related to questions of entropy in these statistical systems. Methods and tools from the field of enumerative combinatorics are used.

cond-mat.stat-mech↗

arXiv1999

Two-dimensional random tilings of large codimension: new progress

Two-dimensional random tilings of rhombi can be seen as projections of two-dimensional membranes embedded in hypercubic lattices of higher dimensional spaces. Here, we consider tilings projected from a $D$-dimensional space. We study the limiting case, when the quantity $D$, and therefore the number of different species of tiles, become large. We had previously demonstrated [ICQ6] that, in this limit, the thermodynamic properties of the tiling become independent of the boundary conditions. The exact value of the limiting entropy and finite $D$ corrections remain open questions. Here, we develop a mean-field theory, which uses an iterative description of the tilings based on an analogy with avoiding oriented walks on a random tiling. We compare the quantities so-obtained with numerical calculations. We also discuss the role of spatial correlations.

cond-mat.stat-mech↗

arXiv1998

Entropy and Boundary Conditions in Random Lozenge Tilings

The tilings of lozenges in 2 dimensions and of rhomboedra in 3 dimensions are studied when they are constrained by fixed boundary conditions. We establish a link between those conditions and free or periodic boundary ones: the entropy is written as a functional integral which is treated via a saddle-point method. Then we can exhibit the dominant states of the statistical ensemble of tilings and show that they can display a strong structural inhomogeneity caused by the boundary. This inhomogeneity is responsible for a difference of entropy between the studied fixed boundary tilings and free boundary ones. This method uses a representation of tilings by membranes embedded in a higher-dimensional hypercubic lattice. It is illustrated in the case of 60 degree lozenge tilings.

cond-mat.stat-mech↗