arXiv subjects

Fabian Parsch

Publications and source records attributed to Fabian Parsch.

8 recordsLinked to original sources

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↗

arXiv2025

Towards Constructing Geodesic Nets with Four Boundary Vertices and an Increasing Number of Balanced Vertices

We construct a geodesic net in the plane with four boundary (unbalanced) vertices that has 25 balanced vertices and that is irreducible, i.e. it does not contain nontrivial subnets. This net is novel and remarkable for several reasons: (1) It increases the previously known maximum for balanced vertices of nets of this kind from 16 to 25. (2) It is, to our knowledge, the first such net that includes balanced vertices whose incident edges are not exhibiting symmetries of any kind. (3) The approach taken in the construction is quite promising as it might have the potential for generalization. This would allow to construct a series of irreducible geodesic nets with four boundary vertices and an arbitrary number of balanced vertices, answering a conjecture that the number of balanced vertices is in fact unbounded for nets with four boundary vertices. This would stand in stark contrast to the previously proven theorem that for three boundary vertices, there can be at most one single balanced vertex.

math.MG↗

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

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↗

arXiv2019

Geodesic Nets: Some Examples and Open Problems

Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round $2$-sphere. In the first half of this paper we survey some results and open questions (old and new) about geodesic nets on Riemannian manifolds. Many of these open questions are about geodesic nets on the Euclidean plane. The second half contains a partial answer for one of these questions, namely, a description of a new infinite family of geodesic nets on the Euclidean plane with 14 boundary (or unbalanced) vertices and arbitrarily many inner (or balanced) vertices of degree $\geq 3$.

math.MG↗

arXiv2019

Geodesic nets with three boundary vertices

We prove that a geodesic net with three boundary (= unbalanced) vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degrees of unbalanced vertices. The result seems to be new even in the Euclidean case. We demonstrate by examples that the result is not true for metrics of positive curvature on the plane, and that there are no immediate generalizations of this result for geodesic nets with four unbalanced vertices.

math.MG↗

arXiv2019

An example for a nontrivial irreducible geodesic net in the plane

We construct a geodesic net in the plane with four unbalanced (boundary) vertices that has 16 balanced vertices and does not contain proper geodesic subnets. This is the first example of an irreducible geodesic net in the Euclidean plane with 4 boundary vertices that is not a tree.

math.MG↗