Diffusion in the Inverted Triangular Soft Lorentz Gas

Abstract

We investigate diffusion in a two-dimensional inverted soft Lorentz gas, where attractive Fermi-type potential wells are arranged in a triangular lattice. This configuration contrasts with earlier studies of soft Lorentz gases involving repulsive scatterers. By systematically varying the gap width and softness of the potential, we explore a rich landscape of diffusive behaviors. We present numerical simulations of the mean squared displacement and compute diffusion coefficients, identifying tongue-like structures in parameter space associated with quasiballistic transport. Furthermore, we develop an extension to the Machta-Zwanzig approximation that incorporates correlated multi-hop trajectories and correct for the influence of localized periodic orbits. Our findings highlight the qualitative and quantitative differences between inverted and repulsive soft Lorentz gases and offer new insights into transport phenomena in smooth periodic potentials.

Figures (8)

• Figure 1: Two-dimensional representation of the potential wells of the inverted triangular soft Lorentz gas. The parallelepiped represents the unit cell employed in the simulations, and the dashed line inside this unit cell shows the Poincaré surface of section (PSOS). The dotted line indicates an example of an infinite horizon in the system. The dashed vertical line represents the cross-section depicted in the inset. In the inset, the dashed horizontal blue line represents the constant energy, $E=1/2$. $L$ represents the length of the side of the unit cell and $H$ its height.
• Figure 2: Parameter space highlighting regions of different dynamical behavior. The vertical cross sections marked with dotted white and dashed black lines denote further analysis shown in Figs. \ref{['fig:dcoef_s']}(a) and (b), respectively. The horizontal cross section marked with a dashed line denotes analysis presented in Fig. \ref{['fig:dcoef_w']}. Any similarities to the flag of the Republic of the Congo are coincidental.
• Figure 3: (a) Fractional area of regular motion in the Poincaré surface of section (PSOS) as a function of $\sigma$ at $w = 0.1013$. Chaotic dynamics is detected with the 0-1 test for chaos gottwald2004gottwald2009agottwald2009b. The system exhibits chaos even before the particles can escape the potential wells. Below $\sigma\approx 0.025$, the system is fully regular, and the discrepancies are caused by the false positives of the algorithm. The analyzed parameter space is marked in Fig. \ref{['fig:paramspace']} as a white dotted vertical line. (b) Diffusion coefficient as a function of $\sigma$ at $w = 0.1013$. In the grey areas, the diffusion coefficient is not defined due to the presence of quasiballistic orbits. The analyzed parameter space is marked in Fig. \ref{['fig:paramspace']} as a black dashed vertical line.
• Figure 4: Diffusion coefficient $D$ as a function of gap width $w$ at $\sigma = 0.0989$ (thick, black, uniform line). The Machta-Zwanzig approximation is presented as the dash-dotted line, and the hopping model approximations [Eq. \ref{['eq:hopping']}] are presented as colored lines. The dashed lines represent CO-corrected values [Eq. \ref{['eq:hoppingcorr']}] and the dotted line the corrected diffusion coefficient. The grey stripes indicate parameter areas with quasiballistic trajectories. The analyzed parameter space is included in Fig. \ref{['fig:paramspace']} as a black dashed horizontal line.
• Figure 5: (a) Density of quasiballistic trajectories ($\rho_\mathrm{B}$) in the parameter space of the inverted triangular soft Lorentz gas, characterized by a tongue-like structure inside a lagoon without quasiballistic trajectories. The dashed vertical and horizontal lines refer to the parameter areas studied in Figs. \ref{['fig:dcoef_s']}(b) and \ref{['fig:dcoef_w']}, respectively. (b) Density of localized periodic orbits ($\rho_\mathrm{LPO}$) over the parameter space. LPOs are ubiquitous over almost the whole parameter space. The diffusion and free motion limits correspond to the lines in Fig. \ref{['fig:paramspace']}.