Taming of free volume in statistical mechanics of the hard disks model

Abstract

We turn the long time puzzle of the free volume, known for its highly irregular form, into exact analytical formulae and develop statistical mechanics of the hard disk model. The free volume is exactly expressed in terms of the intersection areas of up to five exclusion circles, which can be computed analytically as functions of disk coordinates. In turn, the free volume determines the partition function and entropy. The partition function is shown to factorize into a product of free volumes and admits two exact limiting forms corresponding to gaslike and liquidlike regimes. From this construction, using Monte Carlo-generated disk coordinates, the entropy and pressure are obtained analytically and recover the known equation of state of hard disks in almost entire density range up to the close packing. At intermediate densities, the theory reveals a mixed liquid regime associated with defect formation preceding the hexagonal ordering. The intersection area of five disks emerges as a scalar measure of the local hexagonal order. The theory can be directly adopted for the hard sphere model.

Figures (10)

• Figure 1: Fragment of a system of $N$ HDs. The $N-1$ HDs are dark circles, and the connected $\sigma$ circles are light circles. The $n$ th disk and circle are shown by dashes. The inner white area is the free volume of $n$ th disk, the hatched fraction is its cavity, and the clear fraction is its private cell.
• Figure 2: Left panel. A system of $N=23$ HDs with dark cores. The chosen $k-1=9$ HDs have light concentric $\sigma$-circles. Right panel. The light $\sigma$ circles are those in the left panel and the empty area is the free volume $V_{10}\{x_{9}\}_{23}$ for any dark disk chosen as $10$ th disk. As here $\eta$ is below 0.5, the rest $N-k+1=14$ HDs are accommodated for any positions of the chosen 9 disks, the cavity dominates in the free volume and the GA is justified.
• Figure 3: The mean IAs $\mu _{2}$, $\mu _{3}$, and $\mu _{4}$ of two, three, and four $\sigma$-circles as functions of $\eta$. Inset: the mean IA $\mu _{5}$ of five $\sigma$ circles. The diameter $\sigma =1$.
• Figure 4: Theoretical pressure $P(\eta )$ (solid) superimposed on the simulation result $P_{exp}$ (crosses Kolafa and squares squares ). The vertical dash lines delimit, from left to right, the gas region, crossover (with a small mismatch a bit rightward of $\eta =0.55$) and mix-liquid region, the phase coexistence (empty), and liquid region.
• Figure 5: The pressures $P_{exp}$ (symbols), $P_{L}$ in the LA (solid), $P_{G}$ in the GA (dash), and pressure resulting from the ten virial terms (thin solid)virial10.