Visualizing free-energy landscapes for four hard disks

We present a simple model system with four hard disks moving in a circular region for which free-energy landscapes can be directly calculated and visualized in two and three dimensions. We construct several energy landscapes for our system, and we explore the strengths and limitations of each in terms of understanding system dynamics, in particular the relationship between state transitions and free-energy barriers. We also demonstrate the importance of distinguishing between system dynamics in real space and those in landscape coordinates, and we show that care must be taken to appropriately combine dynamics with barrier properties to understand the transition rates. This simple model provides an intuitive way to understand free-energy landscapes, and it illustrates the benefits that free-energy landscapes can have over potential energy landscapes.

Figure 1

Sketch of a potential energy surface as a function of two coordinates. The solid lines are contours of constant $U$. The dashed lines separate local minimal (solid circles) and connect at local maxima. Saddle points are marked with $\times$. From Stillinger and Weber [12]. Reprinted with permission from AAAS.

Figure 2

Sketch of a potential energy surface as a function of one coordinate. This particular sketch represents different possible characters of the phase space, including a crystalline state (left) and amorphous states (middle and right). Reprinted from F. H. Stillinger [13], with the permission of AIP Publishing.

Figure 3

Illustration of our model system, with four hard disks confined to a two-dimensional circular region. The disks have radius equal to 1, and the circular region has a radius $3+\epsilon$; in this sketch, $\epsilon =0.3$.

Figure 4

The six distinct “equilibrium” states, labeled by the order of the disks in a clockwise direction. The bottom row of states are mirror images of the top row. At any given moment, it is unlikely that the four particles are exactly arranged in a square, but rather these states should be understood as the situations in which the order of the four particles is well defined. Likewise, rotations of these states are considered equivalent; only the order matters. For example, the state shown in Fig. 3 is (1243).

Figure 5

(a) Real-space trajectories during a transition with system size $\epsilon =0.18$ and using diffusive dynamics. The letters correspond to the snapshots of the system shown in panels (b)–(e). In this transition $\left(1243\right)\to \left(1324\right)$, the outlined disk moves through the middle. The total time pictured is $10{\tau }_D$, and steps are drawn spaced by $\mathrm{\Delta }t=0.01{\tau }_D$. (f) Free-energy landscape in the variables $(c_1,c_2,c_3)$. (g) Free-energy landscape in the variables $(d_1,d_2)$. (h) Free-energy landscape in the variables $(u_1,u_2)$. In panels (f), (g), and (h), the positions corresponding to snapshots (b)–(e) are marked.

Figure 6

(a) Real-space trajectories during a transition with system size $\epsilon =0.20$ and using ballistic dynamics. Letters correspond to the snapshots shown in panels (b)–(e). In this transition $\left(3421\right)\to \left(1234\right)$, the outlined disk moves through the middle. The total time pictured is $4{\tau }_v$. (f) Free-energy landscape in the variables $(c_1,c_2,c_3)$. (g) Free-energy landscape in the variables $(d_1,d_2)$. (h) Free-energy landscape in the variables $(u_1,u_2)$. In panels (f), (g), and (h), the positions corresponding to snapshots (b)–(e) are marked. Note that the phase space trajectory in (f) has been rotated; in particular, this is a different perspective from that shown in Fig. 5.

Figure 7

Trajectories of the coordinate $c_1$ as a function of time, for system sizes $\epsilon$ as indicated, using diffusive dynamics. Transitions indicate one of the disks passing through the middle of the system. Note that $c_1$ is zero for four of the six states shown in Fig. 4, so some transitions that keep $c_1=0$ are not apparent in the data, but would be apparent in plots of $c_2$ and $c_3$.

Figure 8

The switching timescale $\tau$ as a function of the system size $\epsilon$ for diffusive dynamics (circles, in units of ${\tau }_D$) and ballistic dynamics (triangles, in units of ${\tau }_v$). As $\epsilon \to 0$, the timescale grows as ${\epsilon }^{-9/2}$ as indicated by the solid line. The uncertainties are 95% confidence intervals; where not shown, the uncertainties are smaller than the symbol size.

Figure 9

Free-energy landscape in the variables $c_1,c_2,c_3$ for system size $\epsilon =0.5$, projected onto a unit sphere. The blue lines indicate the axes ($c_1$, etc.) and the red points are contours at $2k_{B}T$ (the square-shaped contour) and $4k_{B}T$ (the stretched contour). The black regions pierced by the axes correspond to the lowest points in the landscape (states with highest entropy).

Figure 10

(a) Free-energy landscape in $(d_1,d_2)$ coordinates for $\epsilon =0.4$. The landscape is bounded by the whitest points. Within the landscape, darker regions are lower (more probable). The red contours are at $1k_{B}T, 3k_{B}T$, and $5k_{B}T$. The center of the landscape is a broad local maximum with a height of $4.9k_{B}T$. (b) Map of the effective diffusivity at each point in the landscape. The diffusivity in the center is 1.22 times greater than the diffusivity in the three darker spots. The diffusivity near the long straight outer edges of the landscape is 1.48 times greater than the diffusivity in the darker spots. (c) Free-energy landscape in the $(u_1,u_2)$ coordinates for $\epsilon =0.4$. The red contours are at $1k_{B}T, 3k_{B}T, 5k_{B}T$, and $7k_{B}T$. The center of the landscape has a height of $7.3k_{B}T$. (d) Map of the effective diffusivity at each point in the landscape. The diffusivity in the center is 2.8 times greater than the diffusivity at the dark edges.

Figure 11

The blue circles are $ln\left(\tau \right)$, the log of the switching timescale. The straight line is the same as that shown in Fig. 8. The diamonds, squares, and triangles indicate the free-energy barrier height for the $u, c$, and $d$ free-energy landscape variables, respectively. Uncertainties for all points are smaller than the symbol size.

Figure 12

Free-energy landscapes with $\epsilon =0.2$ for (a) the $(d_1,d_2)$ variables and (b) the $(u_1,u_2)$ variables. Equivalent points in the two phase spaces are marked by the red regions (left side of both images) and the green regions (right side of both images).