Whether a tapped granular medium is fully characterized macroscopically by its packing fraction and the shaking intensity — or instead retains memory of its preparation history and therefore requires additional state variables — is a foundational question for the statistical mechanics of granular matter, posed in this form by the Chicago compaction experiments and restated in the lecture-notes chapter by Josserand, Lagrée and Lhuillier. The short-term existence of memory is experimentally established: at equal packing fraction and equal tap intensity, subsequent relaxation depends on how the sample was prepared, so 'hidden' variables beyond density exist. What remains open is the general closure: the macroscopic state variables beyond (ρ,Γ)(\rho,\Gamma) — candidates discussed in the literature include volume/compactivity, stress/angoricity and structural variables — together with the probability measure over jammed configurations induced by tapping, that is, how the tapping dynamics samples the jammed states, needed for a description consistent with both transient relaxation and stationary fluctuations. The evidence base is actively split between validations of the Edwards volume ensemble in stationary regimes and demonstrations of non-ergodicity and protocol dependence at low tap intensity, so the problem is open with a precise surviving core.

Background

Granular media — assemblies of macroscopic, athermally dissipative particles such as sand or glass beads — are conventionally driven by 'tapping': the container is subjected to discrete vertical pulses, each characterized by the dimensionless intensity Γ\Gamma (the peak container acceleration divided by gravitational acceleration gg). A loose packing then densifies ('compacts') extremely slowly: the packing fraction ρ\rho (solid volume divided by total volume) grows approximately logarithmically in the number of taps over thousands of taps, approaching a steady-state value that decreases as Γ\Gamma increases (Knight et al. 1995). Near the steady state, the spectrum of density fluctuations is strongly non-Lorentzian — it cannot be described by a single relaxation time (a Lorentzian spectrum would be the Fourier signature of one exponential decay), and the fastest and slowest timescales differ by orders of magnitude (Nowak et al. 1998). When the tap intensity is first raised and then lowered, the system moves onto a 'reversible branch' on which the density is a single-valued function of Γ\Gamma, while the initial 'irreversible branch' retains information about how the sample was prepared.

This phenomenology invites a statistical-mechanical description. Edwards and Oakeshott (1989) proposed that the mechanically stable ('jammed') configurations visited by a gently driven granular medium could be treated as equally probable at fixed total volume, by analogy with the microcanonical ensemble of thermal statistical mechanics, with a temperature-like variable conjugate to the volume, the compactivity X=∂V/∂SX=\partial V/\partial S, where SS is the entropy counting jammed configurations. Later generalizations enlarge the ensemble with stress variables — notably the angoricity, conjugate to the force-moment tensor (the volume-averaged first moment of the interparticle contact forces, whose isotropic part gives the stress) — reflecting the fact that frictional packings of the same density can carry different contact-force networks. The central question behind the present problem is whether the pair (ρ,Γ)(\rho,\Gamma) — or, in ensemble language, the volume alone — is a complete macroscopic specification of a tapped granular medium, or whether history memory forces extra state variables into any granular thermodynamics. Josserand et al. (2000) showed experimentally that at short times the answer is 'extra variables': samples driven to the same density ρ0\rho_0 by different tapping intensities and then tapped identically evolve differently, which the authors linked directly to the non-Lorentzian, multi-timescale relaxation spectrum. Whether, and in which regimes, the long-time stationary states are nevertheless describable by volume statistics alone (or by a small enlarged set of variables) is what remains contested.

Problem Statement

For a granular medium compacted by a sequence of vertical taps of controlled intensity Γ\Gamma (the peak container acceleration normalized by gravity): does the packing fraction ρ\rho, together with Γ\Gamma, uniquely determine the subsequent relaxation toward — and the fluctuations about — the steady state? Or does the medium retain memory of its preparation and driving history, so that a complete macroscopic description requires additional state variables? The short-term existence of such memory is taken as given (see above); the open question is the closure of the macroscopic description, which comprises three jointly posed parts: first, whether (ρ,Γ)(\rho,\Gamma) — or, in ensemble language, the volume alone — is a complete macroscopic specification of a tapped granular medium; second, where it is not, the macroscopic state variables beyond (ρ,Γ)(\rho,\Gamma) (candidates including the Edwards compactivity conjugate to the volume, the angoricity conjugate to the force-moment tensor, and structural variables such as the coordination number, the average number of interparticle contacts per particle, or the geometric family, a continuous set of mechanically stable packings sharing one contact network — these candidates are examples, not required items); and third, the probability measure over mechanically stable (jammed) configurations that the tapping dynamics induces — that is, how the tapping dynamics samples the jammed states. The sought closure is the macroscopic description consistent with the documented preparation-history memory experiments and with the observed stationary states and their fluctuations, and whether an Edwards-type granular statistical mechanics, or a well-defined generalization of it, follows is part of the closure question. The regimes (tap intensity, interparticle friction, system dimension) in which the single-volume-variable description holds and those in which it breaks down remain important context, since the evidence base is actively split between them; establishing the closure in a restricted but well-characterized regime resolves the problem, without needing to cover the other regimes.

References

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