Non-Affine Displacements Below Jamming under Athermal Quasi-Static Compression

Critical properties of frictionless spherical particles below jamming are studied using extensive numerical simulations, paying particular attention to the non-affine part of the displacements during the athermal quasi-static compression. It is shown that the squared norm of the non-affine displacement exhibits the power-law divergence toward the jamming transition point, and that the participation ratio of the displacements decreases near the transition point, meaning that the non-affine displacements become more localized. This can be interpreted from the development of the power-law tail of the distribution of the non-affine displacements. The critical exponents evaluated in two and three dimensions are different from each other, suggesting that the upper critical dimension below jamming is $d_{\rm uc}>2$, in contrast with above jamming where $d_{\rm uc}\leq 2$.