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where X is the compactivity of the system and S the entropy, related as: 

<img src='content_image/336433.jpg'>

Since thermodynamic pressure is not considered in our case, the volume W and enthalpy H are equivalent and the Helmholtz free energy F and Gibbs free energy G are identical, so that we refer to them only as the free energy. The differentials of the thermodynamic potentials are 

where µ is the chemical potential and N is the number of particles. 

<img src='content_image/336434.jpg'>

The free energy F , or Gibbs free energy G is equal to the product of N and µ , 

<img src='content_image/336435.jpg'>

The free energy and chemical potential are continuous in the first order phase transition. On the other hand, the derivative of the chemical potential, S = − ∂µ $_{∂X}$, is discontinuous in a first order phase transition. 

The compactivity X , as well as the entropy density s = S/N , can be calculated from the fluctuations of the Voronoi volumes, which is an analogy of the Edwards theory to the standard Boltzmann statistical mechanics. The definition of a Voronoi cell is a convex polygon whose interior consists of all points closer to a given particle than to any other. The Voronoi volume of a given particle, V$_{vor}$ , is the volume of such a Voronoi cell (see [7] for more details). 

We define the Voronoi fluctuations as σ 2 ≡ 〈 w 2 vor 〉 − 〈 w$_{vor}$ 〉 $^{2}$, where w$_{vor}$ = V$_{vor}$/V$_{g}$ is the reduced Voronoi volume of each particle and the average is done over all the particles in a packing. The quantities σ$_{1}$ ( φ$_{j}$ ) and σ$_{2}$ ( φ$_{j}$ ) denote the Voronoi fluctuations in the disordered and ordered phases, respectively, as a function of φ$_{j}$ . 

We use the Einstein fluctuation theorem for jammed matter which is obtained from the similar relation in equilibrium systems by replacing the energy by the volume fluctuations [23–25]: 

