Geometric conjecture about phase transitions

As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A recent proposal called the topological hypothesis suggests that the existence of a phase transition could perhaps be inferred from changes to the topology of the accessible part of the configuration space. This paper instead suggests that such a topological change is often associated with a dramatic change in the configuration space geometry, and that the geometric change is the actual driver of the phase transition. More precisely, a geometric change that brings about a discontinuity in the mixing time required for an initial probability distribution on the configuration space to reach the steady state is conjectured to be related to the onset of a phase transition in the thermodynamic limit. This conjecture is tested by evaluating the diffusion diameter and $\epsilon$-mixing time of the configuration spaces of hard-disk and hard-sphere systems of increasing size. Explicit geometries are constructed for the configuration spaces of these systems and numerical evidence suggests that a discontinuity in the $\epsilon$-mixing time coincides with the solid-fluid phase transition in the thermodynamic limit.

Figure 1

The left panel shows the evolution of a model configuration space with points colored (shaded) by their $y$ coordinate. The filtration value is an upper bound for the $y$ coordinate of any point on the system's trajectory. The middle panel shows the average $x$ coordinate $\langle x\rangle$ calculated using the ergodic hypothesis and the topological hypothesis as a function of filtration value, assuming that the system starts at the lower left corner of the configuration space. The right panel shows the diffusion diameter and $\epsilon$-mixing time of the configuration space as functions of filtration value and that they are sensitive to changes in the configuration space geometry. The diffusion diameter, but not the $\epsilon$-mixing time, is also sensitive to the bottleneck. The five dotted lines show the filtration values used in the left panel.

Figure 2

A two-dimensional torus $T^2$ (left) and a three-dimensional torus $T^3$ (right) are shown with some of their periodic images. The unit vector ${\mathit{a}}_1$ points in the $x$ direction in both cases. All the unit vectors pass through the centers of the corresponding faces.

Figure 3

The equivalence classes of the critical points of the configuration space of four hard disks. The tautological function values are reported below the figures. The critical points are ordered from index-0 (bottom) to index-3 (top).

Figure 4

The configuration space quotient by rigid translations $\mathrm{\Lambda }\left(2\right)/\mathcal{T}$ (left) and the configuration space quotient by rigid translations, disk label permutations, and point group symmetries $\mathrm{\Lambda }\left(2\right)/\{\mathcal{T},\mathcal{P},\mathcal{L}\}$ (right) for two hard disks. Color (shading) represents the tautological function values and arrows indicate the locations of the critical points.

Figure 5

Number of points used in the graph representations for hard disks and spheres.

Figure 6

The quotient space $\mathrm{\Lambda }\left(2\right)/\{\mathcal{T},\mathcal{P},\mathcal{L}\}$ of two points on $T^2$ represented as an $\alpha$-complex (left) and a graph (right), with color (shading) indicating the value of the tautological function. The red (outlined) path connecting two points is a straight line on the left, but is slightly longer on the right.

Figure 7

Distributions of tautological function values for point clouds on $\mathrm{\Lambda }\left(4\right)$ sampled by three different procedures, with the dashed lines indicating the radii of previously identified critical points. Sampling points uniformly gives the distribution on the left. The distribution in the middle uses importance sampling and is more uniform. The distribution on the right is for a point cloud that both uses importance sampling and samples additional points in the neighborhoods of the critical points.

Figure 8

$\epsilon$-mixing times of the quotient spaces $\mathrm{\Gamma }(2,\eta )/\mathcal{T}$ and $\mathrm{\Gamma }(2,\eta )/\{\mathcal{T},\mathcal{P},\mathcal{L}\}$ for hard disks (top) and hard spheres (bottom). Dashed lines represent the packing fractions of the critical points.

Figure 9

Diffusion diameters and $\epsilon$-mixing times of the spaces $\mathrm{\Gamma }/\{\mathcal{T},\mathcal{P},\mathcal{L}\}$ for $n=3\cdots 7$ hard disks as functions of packing fraction. Dashed lines shows the packing fractions of the critical points.

Figure 10

Diffusion diameters and $\epsilon$-mixing times of the spaces $\mathrm{\Gamma }/\{\mathcal{T},\mathcal{P},\mathcal{L}\}$ for $n=3\cdots 6$ hard spheres as functions of packing fraction. Dashed lines shows the packing fractions of the critical points.

Figure 11

The largest discontinuous jumps (black squares) observed in Figs. 9 (top) and 10 (bottom) along with the packing fractions of all known critical points, colored by their indices: index-0 in purple (dark circles), index-1 in blue (medium circles), all others in green (light circles).

Figure 12

The sorted true distances $d_{\mathrm{\Lambda }/\mathcal{T}}$ calculated for a fixed generic configuration and the $7!\times 12$ symmetric versions of a second generic configuration for 7 hard disks.

Figure 13

The same configurations as in Fig. 12 are sorted by the approximate values of $d_{\mathrm{\Lambda }/\mathcal{T}}[\mathbf{x},S\left(\mathbf{y}\right)]$ for $m=1$, 10, 50, and 100 and are plotted with the true values of $d_{\mathrm{\Lambda }/\mathcal{T}}[\mathbf{x},S\left(\mathbf{y}\right)]$ on the vertical axis.

Figure 14

The $t\mathrm{th}$ powers of the eigenvalues of $\mathbf{P}$ for $\mathrm{\Lambda }\left(4\right)/\{\mathcal{T},\mathcal{P},\mathcal{L}\}$ at times $t=1$, 5, and 10. The values generally decay rapidly and are often numerically indistinguishable from zero for sufficiently large $t$.

Figure 15

Individual mixing times $t_{\epsilon }\left({\mathit{q}}_0\right)$ of the example system shown in Fig. 1. The average mixing time ${\langle t\rangle }_{\epsilon }$ is around $15000$.