Paper detail

Numerical calculation of granular entropy

We present numerical simulations that allow us to compute the number of ways in which $N$ particles can pack into a given volume $V$. Our technique modifies the method of Xu et al. (Phys. Rev. Lett. 106, 245502 (2011)) and outperforms existing direct enumeration methods by more than 200 orders of magnitude. We use our approach to study the system size dependence of the number of distinct packings of a system of up to 128 poly-disperse soft disks. We show that, even though granular particles are distinguishable, we have to include a factor $1/N!$ to ensure that the entropy does not change when exchanging particles between systems in the same macroscopic state. Our simulations provide strong evidence that the packing entropy, when properly defined, is extensive. As different packings are created with unequal probabilities, it is natural to express the packing entropy as $S=-\sum_i p_i\ln p_i -\ln N!$, where $p_i$ denotes the probability to generate the $i$-th packing. We can compute this quantity reliably and it is also extensive. The granular entropy thus (re)defined, whilst distinct from the one proposed by Edwards assumed (J. Phys.: Condens.Matter 2, SA63(1990)), does have all the properties Edwards assumed.