The geometry of jamming algorithms in the random Lorentz gas

Abstract

Deterministic optimization algorithms unequivocally partition a complex energy landscape in inherent structures (ISs) and their respective basins of attraction. Can these basins be defined solely through geometric principles? This question is paramount to understanding hard sphere jamming, a key model of disordered matter. We here address the issue by proposing a geometric class of gradient descent-like algorithms, which we use to study a system in the hard-sphere universality class, the random Lorentz gas. The statistics of the resulting ISs is found to be strictly inherited from those of Poisson-Voronoi tessellations. The landscape roughness is further found to give rise to a hierarchical organization of ISs, which various algorithms explore differently. In particular, greedy and reluctant schemes tend to favor ISs of markedly different densities. The resulting ISs nevertheless robustly exhibit a universal force distribution, thus confirming the geometric nature of the jamming universality class. Along the way, the physical origin of a dynamical Gardner transition is identified.

Figures (14)

• Figure 1: (b): Geometric (or entropic) landscape of the $d=2$ RLG with obstacles (black dots) and ISs (red stars). A sph is in contact with $d+1$ obstacles. It is unstable if these contacts are co-hemispheric (light blue) and stable otherwise (light red). Contact vectors for the (a) unstable and (c) stable sph in (b), along with the VA cone of possible expansion directions (purple) for the former. (d): Delaunay tessellation of the sample in (b). An unstable DS does not contain its circumcenter (blue), while a stable DS does (red), thus identifying an IS (stars). (e): Delaunay basins from Eq. \ref{['eq:DelB']} (different colors) of the sample in (b). These basins are generally composed of one stable DS and zero or more surrounding unstable DS. The DSs (black lines) and the Voronoi tessellation (orange lines) are provided as reference.
• Figure 2: Probability distribution of ISs with packing fraction $\hat{\varphi}_\mathrm{IS}$ in $d=2\ldots6$. The expression in Eq. \ref{['eq:cIS']} (full lines) agrees with the numerical enumeration of stable ISs (dotted lines). Note that the distribution peaks at $\mathrm{mode}[\hat{\varphi}_\mathrm{IS}]=(d-1)/d$, and that the expected jamming packing fraction is $\mathbb{E}[\hat{\varphi}_\mathrm{IS}]=1$ for all $d$. VA algorithms, by contrast, typically reach significantly larger values, as shown here for $d=2\dots 6$ (green band). (inset) The proportion of stable DSs (points) empirically scales as $1/2^{d-1}$ (dotted line), while the ratio between the expected volume of stable DSs and the expected volume of all DSs grows sublinearly in $d$.
• Figure 3: (a): Schematic of the VA-max algorithm for the $d=2$ RLG. The tracer starts at $\mathbf{x}_\mathrm{in}$, moves radially from $\mathbf{p}_0$ until it reaches the Voronoi line at $\mathbf{x}_\mathrm{P}$, and then follows it until reaching the first VV at $\mathbf{x}_\mathrm{VV}$. It subsequently follows the VA edge until $\mathbf{x}_\mathrm{IS}$, which corresponds to a vertex of the Voronoi tessellation. (b): In generic $d$, the VA cone at each VV can present up to $d-1$ VA edges, thus giving rise to the multi-path structure of the VA-edge algorithm. After the initial $d$-step projection up to $\hat{\varphi}_\mathrm{VV}$ (green), multifurcation along various VA edges results in different IS (red crosses). Path coalescence can also arise, but is rare. The VA-max (red) and the VA-min (purple) paths are the greediest and the most reluctant VA-edge algorithms, respectively. For this $d=5$ example, the abscissa is chosen so as to minimize path crossings.
• Figure 4: (a): Tree graphs of the Delaunay basins in Fig. \ref{['fig:Fig1']}e as described in the text. The fraction of basins with a single DS (circled in blue) decreases exponentially as $d$ increases. (b): Tree graph of a large Delaunay basin in $d=4$ illustrating the growing fractal-like nature of these basins with $d$. (c): Joint distribution of the logarithm of the Delaunay basin volume $\log(V_{\mathcal{D}_\mathrm{IS}})$ and the circumradius $r_\mathrm{IS}$ in $d=2\ldots5$ from direct Delaunay tessellation (SI Appendix). The volume of basins composed of one simplex is upper bounded by the volume of regular simplexes (blue dashed line). The non-compact dependence of the volume of Delaunay basins (green cloud) trends consistently with $\propto r_\mathrm{IS}^{d^2}$ (green dashed-dotted line), hinting at their fractal-like nature. Both blue and green clouds are pdfs with contour lines at $2^{k}$ with $k=-5\ldots1$. (d): Probability distribution of the logarithm of the Delaunay basin volume, conditioned on the packing fraction: the mode of the distribution, $\hat{\varphi}_\mathrm{IS} = (d-1)/d$, and a typical value obtained from the dynamics, $\hat{\varphi}_\mathrm{IS} = \langle \hat{\varphi}_\mathrm{IS} \rangle_\mathrm{CALiPPSO}$. Log-normal fits (dotted lines) are provided as reference.
• Figure 5: (a): Dimensional dependence of the jamming density reached from $\hat{\varphi}_\mathrm{in}=0$, $\langle\hat{\varphi}_\mathrm{IS}\rangle_\mathrm{alg}$, for VA-max, VA-min, force-min, and CALiPPSO. The empirical scaling $d^{-1/3}$ gives a (nearly) linear scaling in large $d$. Both VA-max, the greediest local algorithm, and CALiPPSO, its non-local equivalent, achieve similar densities. Their extrapolation (thin dotted lines) suggests that $\hat{\varphi}_\mathrm{J0}=\langle\hat{\varphi}_\mathrm{IS}\rangle_\mathrm{VA\!-\!max}=2.73(2)$ in the limit $d\rightarrow\infty$. For all $d>2$, VA-min and force-min give significantly denser results, with $\langle\hat{\varphi}_\mathrm{IS}\rangle_\mathrm{VA\!-\!min}=2.94(9)$ asymptotically. inset: The cumulative distribution of $\hat{\varphi}_\mathrm{IS}$ for VA-max (full lines) in $d=2\ldots64$ markedly sharpens as $d$ increases. CALiPPSO results (dotted lines) are almost indistinguishable on this scale. The expected value for the uniform measure over ISs (dashed vertical line) as well as $\hat{\varphi}_\mathrm{J0}$ in the limit $d\rightarrow\infty$ (dotted vertical line) are given as reference. (b): Dimensional dependence of $\langle\hat{\varphi}_\mathrm{IS}\rangle_\mathrm{VA-max}$ (full lines) and of the finite-$d$ estimate of $\langle\hat{\varphi}_\mathrm{VV}\rangle_\mathrm{VA-max}$ (dashed-dotted lines) reached from various $\hat{\varphi}_\mathrm{in}$. (The red curve is the same as in (a).) For $\hat{\varphi}_\mathrm{in}\gg\hat{\varphi}_\mathrm{d}$, $\hat{\varphi}_\mathrm{G}$ (symbols) agrees with $\langle\hat{\varphi}_\mathrm{VV}\rangle_\mathrm{VA-max}$ obtained by a linear extrapolation in $1/d$ (thin line), but not so for $\hat{\varphi}_\mathrm{in}\gtrsim\hat{\varphi}_\mathrm{d}$. Dotted lines indicate the VA-min curve for $\hat{\varphi}_\mathrm{in} = 3.2$ and $6.4$, closely matching the VA-max curve at these densities.