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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We identify two orthogonal sources of structural entropy in rattler-free granular systems: affine, involving structural changes that only deform the contact network, and topological, corresponding to different topologies of the contact network. We show that a recently developed connectivity-based granular statistical mechanics separates the two naturally by identifying the structural degrees of freedom with spanning trees on the graph of the contact network. We extend the connectivity-based formalism to include constraints on, and correlations between, degrees of freedom as interactions between branches of the spanning tree. We then use the statistical mechanics formalism to calculate the partition function generally and the different entropies in the high-angoricity limit. We also calculate the degeneracy of the affine entropy and a number of expectation values. From the latter, we derive an equipartition principle and an equation of state relating the macroscopic volume and boundary stress to the analog of the temperature, the contactivity.

Solid-oxide fuel cells produce electric current from energy released by a spontaneous electrochemical reaction. The efficiency of these devices depends crucially on the microstructure of their electrodes and, in particular, on the three-phase boundary (TPB) length, along which the energyproducing reaction occurs. We present a systematic maximisation of the TPB length as a function of four readily-controllable microstructural parameters, for any given mean hydraulic radius, which is a conventional measure of the permeability to gas flow. We identify the maximising parameters and show that the TPB length can be increased by a factor of over 300% compared to current common practices. We support this result by calculating the TPB of several numerically simulated structures. We also compare four models for a single intergranular contact in the sintered electrode and show that the model commonly used in the literature is oversimplified and unphysical. We then propose two alternatives.

The continuous-time random walk (CTRW) model is useful for alleviating the computational burden of simulating diffusion in actual media. In principle, isotropic CTRW only requires knowledge of the step-size, P l , and waiting-time, P t , distributions of the random walk in the medium and it then generates presumably equivalent walks in free space, which are much faster. Here we test the usefulness of CTRW to modelling diffusion of finite-size particles in porous medium generated by loose granular packs. This is done by first simulating the diffusion process in a model porous medium of mean coordination number, which corresponds to marginal rigidity (the loosest possible structure), computing the resulting distributions P l and P t as functions of the particle size, and then using these as input for a free space CTRW. The CTRW walks are then compared to the ones simulated in the actual media. In particular, we study the normal-to-anomalous transition of the diffusion as a function of increasing particle size. We find that, given the same P l and P t for the simulation and the CTRW, the latter predicts incorrectly the size at which the transition occurs. We show that the discrepancy is related to the dependence of the effective connectivity of the porous media on the diffusing particle size, which is not captured simply by these distributions. We propose a correcting modification to the CTRW model-adding anisotropy-and show that it yields good agreement with the simulated diffusion process. We also present a method to obtain P l and P t directly from the porous sample, without having to simulate an actual diffusion process. This extends the use of CTRW, with all its advantages, to modelling diffusion processes of finite-size particles in such confined geometries.