We discuss the statistical mechanics of granular matter and derive several significant results. First, we show that, contrary to common belief, the volume and stress ensembles are interdependent, necessitating the use of both. We use the combined ensemble to calculate explicitly expectation values of structural and stress-related quantities for two-dimensional systems. We thence demonstrate that structural properties may depend on the angoricity tensor and that stress-based quantities may depend on the compactivity. This calls into question previous statistical mechanical analyses of static granular systems and related derivations of expectation values. Second, we establish the existence of an intriguing equipartition principle-the total volume is shared equally amongst both structural and stress-related degrees of freedom. Third, we derive an expression for the compactivity that makes it possible to quantify it from macroscopic measurements.

We first show that the currently accepted statistical mechanics for granular matter is flawed. The reason is that it is based on the volume function, which depends only on a minute fraction of all the structural degrees of freedom and is unaffected by most of the configurational microstates. Consequently, the commonly used partition function underestimates the entropy severely. We then propose a new formulation, replacing the volume function with a connectivity function that depends on all the structural degrees of freedom and accounts correctly for the entire entropy. We discuss the advantages of the new formalism and derive explicit results for two-and three-dimensional systems. We test the formalism by calculating the entropy of an experimental two-dimensional system, as a function of system size, and showing that it is an extensive variable. The field of granular physics is in urgent need of equations of state, the traditional provider of which is statistical mechanics (SM). Yet, although a granular statistical mechanical formalism was introduced a quarter of a century ago [1][2][3], no such equations have been derived yet. Granular SM is entropy-based. Part of the entropy is structural [1-3] and corresponds to the different spatial arrangements of the grains, with each structural configuration regarded as a microstate. These microstates depend on N s d structural degrees of freedom (DOFs) in d dimensions, with N s the number of contact position vectors (see below). The volume sub-ensemble is based on a volume function W, which is analogous to the Hamiltonian in thermal SM. Namely, the probability that the system be at a structural microstate with volume V is presumed to be e −V /X0 , in analogy to the Boltzmann factor e −E/k B T . The factor X 0 = ∂ W /∂S, called the compactivity, is the analog of the temperature in thermal SM [1][2][3]. Every grain configuration can support an ensemble of different boundary forces, each giving rise to a different internal stress microstate [4][5][6][7][8][9]. The boundary forces, g m (m = 1, ..., M ) are the DOFs that determine the stress microstates. The combined partition function iswhere σ ij is the stress tensor, F ij = V σ ij is the force moment tensor, and X ij = ∂ F ij /∂S is the angoricity tensor [4,8]. The identity of the structural DOFs, r, is discussed below. The two sub-ensembles are not independent [8] and the total entropy, S, is the logarithm of the total number of microstates, both structural and stress. Numerical and experimental tests of the formalism abound [10][11][12][13][14][15] and some inconsistencies were observed [16]. In particular, that the compactivity does not equilibrate in some systems [17].Here, we first show that this stems from a fundamental problem with the formulation of the volume ensemble -the volume function, W in (1), is flawed in that it is independent of most of the structural microstates that it is supposed to describe. Consequently, it fails to account correctly for the entire entropy. We then propose an improved formulation tha...

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Abstract. This paper first reviews a recent report [1], showing that the volume and stress ensembles, often used separately in the statistical mechanics of stable granular systems, are interdependent and must both be used to compute expectation values. A reformulation of the combined partition function in 2D allows us to calculate exactly a number of structural and stressrelated expectation values. It is shown that structural measureables may depend on the angoricity and stress-based quantities on the compactivity, demonstrating that the compactivity and angoricity are not conjugate variables of volume and force moment, as commonly believed. We review a derivation of an equipartition principle, which makes it possible to determine the compactivity experimentally. We conclude by deriving an equation of state.