Jamming, marginal stability and vibrations of amorphous solids

5 problems

• Does an Edwards-type ensemble describe granular packings away from jamming?

  Partially resolved

  Edwards proposed that all stable arrangements of grains at a given volume are equally likely, which would give packed grains a statistical mechanics like that of a gas. Computer counting found this true only exactly at the jamming point, and whether some modified ensemble works for real, frictional, tapped or sheared grains is unknown.

• Does a true Gardner transition exist in three-dimensional glasses?

  Contested

  Mean-field theory predicts that a glass compressed deep into its solid state undergoes a further transition, the Gardner transition, where each glassy state splits into many nearly equivalent sub-states. Whether this transition survives in real three-dimensional glasses or is only a smooth crossover is unknown.

• Why quasilocalized modes in glasses scale as $\omega$ to the fourth

  Partially resolved

  Computer glasses contain soft vibration patterns concentrated on a few dozen particles, and their number grows as the fourth power of frequency in every dimension tested. A finite-dimensional theory that derives this law is missing.

• Is random close packing a well-defined density?

  Contested

  Shaking or pouring identical balls into a box gives a disordered packing filling about 64 percent of the space, called random close packing. Whether this number is a true property of spheres or depends on how the packing was made is disputed.

• Are jamming exponents mean-field already in two and three dimensions?

  Partially resolved

  Theory solved exactly in infinitely many dimensions predicts specific numbers for how contacts and forces behave at the jamming point. Simulations find the same numbers in two and three dimensions, and nobody has shown why.