Edwards' statistical mechanics for jammed granular matter replaces energy by volume as the conserved extensive quantity and postulates that all jammed configurations at fixed volume VV and grain number NN are equiprobable, leading to a fundamental equation dV=X dS+γ dNdV = X\,dS + \gamma\,dN with compactivity XX (a temperature-like variable) and granular potential γ\gamma (the analogue of the chemical potential). Gramlich, Zarif and Bowles realized this program quantitatively in an exactly solvable quasi-one-dimensional model, showing that subsystems exchanging volume and grains equilibrate when XX and γ/X\gamma/X are equal on both sides and that γ\gamma differences can be measured from small-subsystem fluctuations. The open problem is whether the granular potential can be placed on the same fundamental footing as the chemical potential outside this idealized setting: in ensembles that include stress degrees of freedom (the stress and combined volume–stress ensembles, where angoricity appears), under protocol-dependent dynamics such as tapping and shearing, and in two- and three-dimensional assemblies where surface, edge and mesoscopic volume correlations complicate measurement. Recent work has begun empirical measurement of γ\gamma in 3D packings and has sharpened the ensemble-validity question, but no formulation of mass-exchange equilibrium in stress-inclusive ensembles and no theory of protocol-dependent sampling exist; the problem remains open.

Background

Granular materials such as sand, powders and seeds are athermal: the energy barriers between mechanically stable (jammed) configurations vastly exceed thermal fluctuations, so a static pile is trapped in one of an enormous number of jammed configurations. Edwards and Oakeshott proposed a statistical mechanics for such systems in which volume, rather than energy, plays the role of the conserved extensive quantity: the Edwards volume ensemble postulates that all jammed configurations with the same volume VV and the same number of grains NN are equally probable. Maximizing the resulting entropy S(V,N)S(V,N) gives a fundamental equation of state

dV=X dS+γ dN,dV = X\,dS + \gamma\,dN,

where the compactivity X=(∂V/∂S)NX = (\partial V/\partial S)_N is a temperature-like variable measuring the packing's capacity to be compressed, and the granular potential γ\gamma is the intensive variable conjugate to grain number — the analogue of the chemical potential in molecular thermodynamics. Whether the equiprobability postulate itself, and XX as a legitimate temperature, hold for real granular systems is a long-standing controversy: experiments on tapped packings support compactivity as a valid temperature satisfying a zeroth law (Yuan et al. 2021), cyclically sheared systems show equivalence between the Edwards temperature and a fluctuation-dissipation temperature (Zeng et al. 2022), yet other experiments find strong protocol dependence of the state variables (Bililign et al. 2019). The modern status of the whole framework is reviewed by Baule, Morone, Herrmann and Makse (2018).

Several generalized ensembles compete with or extend the volume ensemble. In the stress ensemble of Henkes, O'Hern and Chakraborty (2007), the force-moment tensor (the sum over grains of the outer product of the grain-centre-to-contact-point vector with the contact force, a tensorial measure of internal stress) plays the role of energy, and its conjugate intensive variable is called the angoricity; Blumenfeld, Jordan and Edwards (2012) analyzed the interdependence and equipartition of the combined volume–stress ensemble. In the force-moment-tile description of stressed packings, the tile-area variable conjugate to force-tile areas is called the keramicity; Bililign, Kollmer and Daniels (2019) showed experimentally that it can act as a temperature-like quantity while exposing pronounced protocol dependence. Recent simulations of three-dimensional frictional hard-sphere packings find that the angoricity ensemble yields good equations of state while the Voronoi-volume (compactivity) ensemble does not describe those systems well (Baranau and Tallarek 2024), so which ensemble and which intensive variables apply appears to be system-dependent.

Within the pure Edwards volume ensemble, Gramlich, Zarif and Bowles (Soft Matter 2023) gave the first exact, quantitative realization of the granular potential, using a quasi-one-dimensional model of jammed chains of grains whose set of jammed states can be enumerated exactly. They proved that two subsystems able to exchange both volume and grains come to equilibrium when the compactivities X1=X2X_1 = X_2 and the ratios γ1/X1=γ2/X2\gamma_1/X_1 = \gamma_2/X_2 are equal, with mass flowing from high γ/X\gamma/X to low γ/X\gamma/X; that the entropic effect of adding a grain changes sign with compactivity; and that differences in γ/X\gamma/X can be measured from the fluctuations of a small subsystem embedded in a larger equilibrated system, adapting a small-system (Hill-type nanothermodynamic) fluctuation method developed for chemical potentials in molecular fluids (Bråten, Wilhelmsen and Schnell 2021). All of these results, however, hold inside the Edwards volume ensemble, in a quasi-one-dimensional model whose exact solvability removes precisely the difficulties — stress fluctuations, protocol-dependent sampling, and boundary and correlation effects — that a general theory must face.

Problem Statement

The granular potential γ\gamma — the intensive variable conjugate to grain number in the fundamental equation dV=X dS+γ dNdV = X\,dS + \gamma\,dN of the Edwards volume ensemble — has been shown to govern mass-exchange equilibrium (equality of XX and of γ/X\gamma/X between coexisting subsystems) and to be measurable by small-subsystem fluctuations, but only within an exactly solvable quasi-one-dimensional model that assumes equiprobability of jammed states and the validity of the compactivity XX. The problem is to determine whether the granular potential can be placed on the same fundamental footing as the chemical potential in thermal systems, beyond this idealization. Specifically: (1) formulate the statistical ensemble, the fundamental equation and the equilibrium criteria for systems that exchange both particles and volume when stress degrees of freedom are included — in the stress ensemble and the combined volume–stress ensemble, where the force-moment tensor enters with angoricity (and, in the tile description, keramicity) as additional intensive variables — and determine whether γ\gamma survives as a state variable or must be modified or supplemented in those ensembles; (2) establish how protocol-dependent dynamics (tapping, shearing) that sample jammed states non-equiprobably affect the existence, value and measurability of γ\gamma and XX, i.e., whether γ\gamma is a state function or a path-dependent quantity under realistic granular dynamics; (3) extend the small-system measurement method to two- and three-dimensional granular assemblies, controlling the surface, edge, subsystem-shape and mesoscopic volume-correlation effects that block a straightforward extrapolation to the thermodynamic limit; and (4) identify the regimes — systems, densities and preparation protocols — in which γ\gamma and XX are meaningful empirical state variables, and those in which alternative ensembles or intensive parameters (angoricity, keramicity) are required. These four facets serve one overarching objective: a validated, internally consistent granular thermodynamics with a chemical-potential analogue capable of predicting mass transport, mixing and phase coexistence in jammed matter. Each of the four facets is open to a constructive resolution or a negative one. Facets (1) and (2) additionally admit a purely negative settlement: a rigorous demonstration that no protocol-independent state variable conjugate to grain number exists beyond the Edwards volume ensemble settles those two facets only for the ensembles or protocols it covers, and settles them unconditionally only when it covers all such ensembles; it does not by itself settle facet (3) or facet (4).

References

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