arXiv subjects

John Çamkıran

Publications and source records attributed to John Çamkıran.

7 recordsLinked to original sources

arXiv2026

The maximum distance covariance of bounded random vectors

A sharp upper bound is obtained for the distance covariance of bounded random vectors in terms of the diameters of their supports alone. The bound is attained for all pairs of support diameter constraints and hence provides the maximum distance covariance subject to constraints of this kind. A normalized measure of dependence is derived by dividing distance covariance by the sharp bound when it is positive. Several properties of the resulting quotient are proved, including domination by distance correlation, invariance under similarities, and subadditivity under sums of independent pairs. The quotient is shown to be nonincreasing under the addition of independent noise, a property not shared by distance correlation. The usual sample distance covariance inherits the sharp bound from the population quantity and the associated normalized statistic is strongly consistent.

math.PR↗

arXiv2026

A statistical theory of structure in many-particle systems with local interactions

A theory of structure is formulated for systems of many structureless classical particles with stable local interactions in Euclidean space. Such systems are shown to have their structure in thermodynamic equilibrium determined exactly by a random field of fine local descriptions and approximately by coarsenings thereof. The degree of order in the local cluster consisting of a particle and its neighbors is identified as a universal source of coarse local descriptions and characterized by expressing the behavior of configurational entropy in local microscopic terms. A local measure of the angular redundancy in neighboring particle positions is found to satisfy this characterization and thereby established as a valid local order quantifier. A precise relationship between order and symmetry is obtained by bounding this quantifier sharply from below by a simple function of the local point group and the largest stabilizer under its action on the set of bond pairs. The marginal distribution of the quantifier is given in closed form for highly coordinated particles with broadly distributed bond angles. Applications are made to the ideal gas, perfect crystal, and simple liquid.

cond-mat.stat-mech↗

arXiv2024

Towards an information-based theory of structure

We call for a theory of the particle-scale structure of materials that is based on the general notion of information rather than its special case of symmetry. An inherent limitation to the symmetry-based understanding of structure is described. The rapid decay in interaction strength with interparticle distance is used to argue for the representability of a system locally through the neighbourhoods of its constituent particles. The extracopularity coefficient E is presented as a local quantifier of information and compared to point group order |G|, a local quantifier of symmetry. The former is found to have nearly double the resolution of the latter for a set of commonly encountered coordination geometries. A proof is given for the generality of extracopularity over point symmetry. Some practical challenges and future perspectives are discussed.

cond-mat.mtrl-sci↗

arXiv2022

Local-order fluctuations in Kob-Andersen-type glass formers

Notwithstanding decades of work, we still lack a satisfactory understanding of the structural relaxation that takes place as a liquid is rapidly cooled to form a glass. The present paper discusses a novel statistical characterization of this phenomenon in Kob-Andersen-type mixtures -- a simple yet powerful class of model glass formers. We use the variance of an order parameter called the extracopularity coefficient to measure the intensity of instantaneous fluctuations in local orientational order. This intensity is found to be nearly composition independent at the onset temperature of glassy dynamics for the standard Kob-Andersen mixture. We decompose these fluctuations into a density and symmetry contribution. Through the behavior of these contributions, we argue that the near composition independence prevails when the structure of the system equally resembles a liquid and a glass. We moreover report that the intensity of local-order fluctuations behaves like a heuristic measure of glass-forming ability, which could prove useful when exact methods are intractable.

physics.chem-ph↗

arXiv2022

On the topology of the space of coordination geometries

Coordination geometries describe how the neighbours of a central particle are arranged around it. Such geometries can be thought to lie in an abstract topological space; a model of this space could provide a mathematical basis for understanding physical transformations in crystals, liquids, and glasses. With this motivation, the present work proposes a metric model of the space of three-dimensional coordination geometries. This model is conceived through the generalisation of a local orientational order parameter and seems to be consistent with geometric intuition. It appears to suggest a taxonomy of coordination geometries with five main classes, each with a distinct character. A quantitative notion of orientational typicality is introduced and its interplay with orientational order is found to evidence a statistical regularity with respect to point symmetry. By the assertion of axioms on the topology of the space herein modelled, the range of structures that are possible to resolve with the order parameter in molecular dynamics simulations is greatly increased.

math-ph↗

arXiv2022

An entropy functional bounded from above by one

Shannon entropy is widely used to quantify the uncertainty of discrete random variables. But when normalized to the unit interval, as is often done in practice, it no longer conveys the alphabet sizes of the random variables being studied. This work introduces an entropy functional based on Jensen-Shannon divergence that is naturally bounded from above by one. Unlike normalized Shannon entropy, this new functional is strictly increasing in alphabet size under uniformity and is thus well suited to the characterization of discrete random variables.

cs.IT↗

arXiv2022

A local orientational order parameter for systems of interacting particles

Many physical systems are well modeled as collections of interacting particles. Nevertheless, a general approach to quantifying the absolute degree of order immediately surrounding a particle has yet to be described. Motivated thus, we introduce a quantity $E$ that captures the amount of pairwise informational redundancy among the bonds formed by a particle. Particles with larger $E$ have less diversity in bond angles and thus simpler neighborhoods. We show that $E$ possesses a number of intuitive mathematical properties, such as increasing monotonicity in the coordination number of Platonic polyhedral geometries. We demonstrate analytically that $E$ is, in principle, able to distinguish a wide range of structures and conjecture that it is maximized by the icosahedral geometry under the constraint of equal sphere packing. An algorithm for computing $E$ is described and is applied to the structural characterization of crystals and glasses. The findings of this study are generally consistent with existing knowledge on the structure of such systems. We compare $E$ to the Steinhardt order parameter $Q_6$ and polyhedral template matching (PTM). We observe that $E$ has resolution comparable to $Q_6$ and robustness similar to PTM despite being much simpler than the former and far more informative than the latter.

math-ph↗