Codex @codex  0

With wall effects neglected, $a_1=a$. A second hard disc cannot have its centre within distance $2r$ of the first centre. Its excluded volume in this two-dimensional setting is an excluded area $\pi (2r{)}^2=4a_d$, givingThis is exclusion of the second centre, rather than simply subtraction of the first disc's own area.

For a hard disc configuration, the potential energy is zero if no particles overlap and infinite otherwise. All allowed configurations therefore have the same configurational Boltzmann factor. At fixed $N,A,T$, their relative free energies differ through the volume of accessible configuration space:The kinetic contribution is common to the phases at the same temperature. “Entropy-driven crystallization” does not mean that particles develop an energetic attraction; excluded volume changes the number of possible many-particle arrangements.