Long-Range Interacting Particles on a Helix: A Statistical and Correlation Analysis of Equilibrium Configurations

Abstract

We provide a statistical and correlational analysis of the spatial and energetic properties of equilibrium configurations of a few-body system of two to eight equally charged classical particles that are confined on a one-dimensional helical manifold. The two-body system has been demonstrated to yield an oscillatory effective potential, thus providing stable equilibrium configurations despite the repulsive Coulomb interactions. As the system size grows, the number of equilibria increases, approximately following a power-law. This can be attributed to the increasing complexity in the highly non-linear oscillatory behavior of the potential energy surface. This property is reflected in a crossover from a spatially regular distribution of equilibria for the two-body system to a heightened degree of disorder upon the addition of particles. However, in accordance with the repulsion within a helical winding, the observed interparticle distances in equilibrium configurations cluster around values of odd multiples of half a helical winding, thus maintaining an underlying regularity. Furthermore, an energetic hierarchy exists based on the spatial location of the local equilibria, which is subject to increasing fluctuations as the system size grows.

Figures (12)

• Figure 1: Sketch of three equally charged particles on a helix with radius $R$ and pitch $h$ for $h/R = 0.8$. Particles are arranged in an EC, which means that neither particle can move without increasing the potential energy. The red dotted lines indicate the repulsive Coulomb interactions.
• Figure 2: (a) Potential energy $V$ for two charged particles confined on a helix, depending on the interparticle distance $\Phi = \Phi_1$. The potential is shown for $\rho=0.2$, $\rho=0.5$, $\rho=0.8$, and $\rho=1.1$. Additionally, the dashed lines show the upper and lower envelope of the potential energy for $\rho=0.2$ and $\rho = 1.1$, respectively. (b) An intersection of the many-body potential landscape $V(\Phi)$, where $\Phi = \Phi_1 = \Phi_2 = \ldots = \Phi_{N-1}$, i.e. for equidistant particles, for different $N$ and $\rho = 0.8$. The energetically lowest curve corresponds to $N=2$, the second lowest to $N=3$, etc. The black crosses mark the minimum of the outermost potential well of each curve.
• Figure 3: The total number of attained ECs, contingent upon the number of particles. The inset illustrates the number of ECs by $N$ on a double logarithmic scale, exhibiting that it follows a power law $(\approx N^{2.7})$.
• Figure 4: (a) Distribution of interparticle distances $\Phi_0^i$ of all ECs by the number of particles $N$. Each finite, red, horizontal line corresponds to one $\Phi_0^i$. Note that some values of $\Phi_0^i$ can occur multiple times within an EC or across different ECs, condensing to one horizontal line. Furthermore, similar $\Phi_0^i$ can lead to a high density of lines, resulting in an almost continuous range of $\Phi_0^i$ values. The occurring values clusterize, where the clusters are indexed as well as subdivided by the blue dashed lines. (b-g) Histogram of the values $\Phi_0^i$ showing the probability $P(\Phi_0^i)$ of their occurrence within defined intervals.
• Figure 5: (a) The probability on how many distinct clusters the interparticle distances $\Phi_0^i$ of an EC (Clusters/EC) are distributed in dependence of the number of particles $N$. For example, for the EC $\mathbf{\Phi}_0 \approx (1.35, 0.36)$, $C(\Phi_0^1) = 2$ and $C(\Phi_0^2) = 1$, so $\Phi_0^i$ is distributed on two Clusters in that EC. The probability is normalized for each $N$. (b-h) The probability on which specific clusters the $\Phi_0^i$ are distributed, given the value of Clusters/EC. Note that $C(\Phi_0^i)$ is cut at the value of 8 because $C(\Phi_0^i) = 9,10$ are only relevant for the trivial case of $N=2$. The probability is normalized for each value of Clusters/EC.