Packing a multidisperse system of hard disks in a circular environment

We consider the problem of finding the densest closed packing of hard disks with proposed different radii in a circular environment, such that the radius of the circumcircle is minimal. With our approach, we are able to find denser packings for various problem instances than known from the literature. Both for the dynamics of the simulation and for the optimum values of the radii of the circumcircles, we find various scaling laws.

Figure 1

Best solution found for the multidisperse packing problem with 50 disks: the circumcircle has a radius $R=220.6004187\dots$. The numbers within the larger disks denote their radius values.

Figure 2

(Color online) Linear decrease of the mean energy $⟨\mathcal{H}⟩$ with decreasing temperature $T$ at high temperatures (left) and specific heat $C$ vs temperature $T$ (right) for various system sizes.

Figure 3

(Color online) Expectation value $⟨\mathcal{P}⟩$ of the sum of the penalties (left) and corresponding susceptibility $\chi$ vs temperature $T$ (right) for various system sizes.

Figure 4

(Color online) Overall acceptance rate $A$ vs temperature $T$ for various system sizes.

Figure 5

Minimum, mean (with error bars), and maximum quality of results achieved for various system sizes $N$ with the maximum amount of $\zeta$ shifting a disk in the $x$ and $y$ direction: the results are averaged over 200 optimization runs each.

Figure 6

(Color online) Results for the expectation value $⟨\mathcal{H}⟩$ of the energy, the specific heat $C$, the expectation value $⟨\mathcal{P}⟩$ of the penalty sum, and the susceptibility $\chi$ for various values of $\zeta$ and a system size of $N=50$ disks.

Figure 7

Probability distribution of the number $e$ of edges denoting the neighborhood relations between the various disks.

Figure 8

Average number $⟨n\left(i\right)⟩$ of neighbors of disk $i$ with radius $r_i=i$: the fit curve for large $r_i$ is given by the power law $⟨n\left(i\right)⟩=0.48r_i^{0.71}$.

Figure 9

(Color online) Probability that the disks $i$ and $j$ with radii $r_i=i$ and $r_j=j$ are neighboring each other.

Figure 10

(Color online) Mean and maximum distance between disks $i$ and $j$ with radii $r_i=i$ and $r_j=j$ according to the metric $\Delta$ induced by the neighborhood relations.

Figure 11

(Color online) Average Euclidean distance between disks $i$ and $j$ with radii $r_i=i$ and $r_j=j$: $⟨d(i,j)⟩$ denotes the average distance of the midpoints of the two corresponding disks, from which the sum of their radii is subtracted.

Figure 12

Probability with which a specific value of our unnormalized overlap values $q_{\sigma \tau }$ is found for a pair of configurations $(\sigma ,\tau )$ with $\tau \ne \sigma$.

Figure 13

Average quality ${⟨\mathcal{H}⟩}_t$ of solutions for the problem instance with $N=50$ disks achieved after calculation time $t$ (measured in sweeps per temperature step) vs the inverse of the logarithm of the calculation time: the data shown were produced with the same simulation parameters as in Sec. 3, but with 726, 1000, 2500, 5000, 10 000, 22 000, 44 000, 88 000, 145 244, 200 000, 400 000, and 800 000 sweeps per temperature step, respectively. For each data point, we averaged the results over 200 optimization runs. A fit through all mean points gives the fit function $220.6+15∕\ln \left(t\right)$.