Angoricity and compactivity describe the jamming transition in soft particulate matter

The application of concepts from equilibrium statistical mechanics to out-ofequilibrium systems has a long history of describing diverse systems ranging from glasses to granular materials. For dissipative jammed systems —particulate grains or droplets— a key concept is to replace the energy ensemble describing conservative systems by the volume-stress ensemble. Here, we test the applicability of the volume-stress ensemble to describe the jamming transition by comparing the jammed configurations obtained by dynamics with those averaged over the ensemble as a probe of ergodicity. Agreement between both methods suggests the idea of “thermalization” at a given angoricity and compactivity. We elucidate the thermodynamic order of the jamming transition by showing the absence of critical fluctuations in static observables like pressure and volume. The approach allows to calculate observables such as the entropy, volume, pressure, coordination number and distribution of forces to characterize the scaling laws near the jamming transition from a statistical mechanics viewpoint. Copyright c © EPLA, 2010 Introduction. – A granular system compresses into a mechanically stable configuration at a nonzero pressure in response to the application of an external strain [1–3]. This process is typically referred to as the jamming transition and occurs at a critical volume fraction φc [3]. The application of a subsequent external pressure with the concomitant particle rearrangements and compression results in a set of configurations characterized by the system volume V =NVg/φ (φ is the volume fraction of N particles of volume Vg) and applied external stress or pressure p (for simplicity we assume isotropic states). It has been long argued whether the jamming transition is a first-order transition at the discontinuity in the average coordination number, 〈Z〉, or a second-order transition with the power-law scaling of the system’s pressure as the system approaches jamming with φ−φc → 0 [4–7]. Previous work [8–10] has proposed to explain the jamming transition by a field theory in the pressure ensemble. Here, we use the idea of “thermalization” of an ensemble of mechanically stable granular materials at a given volume and pressure to study the jamming transition from a thermodynamic viewpoint. (a)E-mail: [email protected] For a fixed number of grains, there exist many jammed states [11] confined by the external pressure p in a volume V . In an effort to describe the nature of this nonequilibrium system from a statistical mechanics perspective, a pressure-volume ensemble [8,12–14] was introduced for jammed matter. In the canonical ensemble the probability of a state is given by exp[−W(∂S/∂V )− Γ(∂S/∂Γ)], where S is the entropy of the system, W is the volume function measuring the volume of the system as a function of the particle coordinates and Γ≡ pV is the boundary stress (or internal virial) [9] of the system. Just as ∂E/∂S = T is the temperature in equilibrium system, the temperature-like variables in jammed systems are the compactivity X = ∂V/∂S [12] and the angoricity A= ∂Γ/∂S [13,15–17]. In a recent paper [18] the compactivity was used to describe frictional hard spheres in the volume ensemble. Here, we test the validity of the statistical approach in the combined pressure-volume ensemble to describe deformable, frictionless particles, such as emulsion systems jammed under osmotic pressure near the jamming transition [19,20]. We demonstrate that the jamming transition can be probed thermodynamically by the angoricity A and the compactivity X.