Monte Carlo rejection as a tool for measuring the energy landscape scaling of simple fluids

A simple modification of the Monte Carlo algortihm is proposed to explore the topography and the scaling of the energy landscape. We apply this idea to a simple hard-core fluid. The results for different packing fractions show a power law scaling of the landscape boundary, with a characteristic scale that separates the values of the scaling exponents. Finally, it is shown how the topology determines the freezing point of the system due to the increasing importance and complexity of the boundary.

Figure 1

Two disks in a rectangular box of length $L$ and width $\sigma$. Below the box, the corresponding configurational part of the phase space is shown. The allowed part of the landscape is the area indicated with the grey shadow. A grid of scale $\delta$ is indicated by dotted lines. Boundary points are indicated by open circles. Closed circles are states in the forbidden part of the phase space. Notice that in this problem, ergodicity is always broken, since the allowed parts of the landscape are not connected.

Figure 2

Rejection ratio as a function of the scale $\delta$ (measured in units of $\sigma$) for $L=50\sigma$ (squares), $30\sigma$ (circles), $15\sigma$ (diamonds), $10\sigma$ (triangles up), and $4\sigma$ (triangles down).

Figure 3

Slopes of the fitting lines that appear in Fig. 2 as a function of $L$. The solid line is the function $m\left(L\right)$.

Figure 4

Parameter $M_B\left(\delta \right)⟨p_B\left(\delta \right)⟩$ as a function of the scale $\delta$, for different packing fractions. From top to bottom, $\varphi =0.74$ (squares), $\varphi =0.71(\times )$, $\varphi =0.39$ (triangles), $\varphi =0.23$ (stars), $\varphi =0.12$ (diamonds), $\varphi =0.08$ (squares), and $\varphi =0.04$ (filled circles). The lines were obtained using a power law fit.

Figure 5

Parameter $M_B\left(\delta \right)⟨p_B\left(\delta \right)⟩$ close to freezing as a function of $\delta$, at $\varphi =0.74$ (squares) and $\varphi =0.63$ (circles).

Figure 6

Exponent $D_f$ as a function of the ratio between the packing fraction and the packing fraction at freezing $\left({\varphi }_0\right)$. The size of the squares is proportional to the maximal error, and the line is a visual guide to the eye.