Frustration and slow dynamics of granular packings

Dynamical processes in granular media show fascinating behavior @1–4#. The particular role of disorder and fluctuations in granular dynamics has led several authors @1,5–7# to propose an analogy to frustrated statistical systems such as spin glasses @8#. Recently a frustrated lattice gas model has been introduced to describe static and dynamic properties of granular materials @9#. This microscopic model is based on an analogy with frustrated percolation @10#, expressed in terms of a Hamiltonian formalism in which disorder and frustration are key elements. In this statistical model, ‘‘vibrations’’ play the role of temperature in usual thermal systems @1,11–13#. Its quenched disorder and the consequent frustration try to describe the general physical mechanisms underlying the phenomenon of ‘‘geometrical’’ frustration known in granular media. The same model, without the gravitational contribution in the Hamiltonian, has been previously related to the physics of the glass transition in glass forming liquids @14#. Here we describe the combined effects of vibrations and gravity in this model, and compare them to experimental data. A well known experiment with granular systems is the compaction of sand. When a box filled with loose packed sand is shaken at low amplitude, density visibly increases @15#. If in addition, the density goes beyond a definite threshold, the mechanical properties of sand abruptly change and the granular structure cannot be sheared any longer without a volume increase. This phenomenon, very important in practical applications @16#, was observed by Reynolds @17#, and is referred to as the ‘‘Reynolds’’ or ‘‘dilatancy’’ transition. In the present model an analogy appears between the cooperativity effects underlying the Reynolds transition in granular media and the actual spin glass transition where a diverging length naturally exists @7,9#. This paper also analyzes the grain density relaxation in different dynamical situations, and relates our observations to corresponding phenomena in real experiments. We find that in the present model the logarithmic behavior, known from experimental measurements in sequences of taps @15#, is recovered, and further predictions are possible. We observe a different dynamical behavior for grain deposition in a single vibration process, where stretched exponentials are