Pinning Susceptibility at the Jamming Transition

Jamming in the presence of fixed or pinned obstacles, representing quenched disorder, is a situation of both practical and theoretical interest. We study the jamming of soft, bidisperse discs in which a subset of discs are pinned while the remaining particles equilibrate around them at a given volume fraction. The obstacles provide a supporting structure for the jammed configuration which not only lowers the jamming threshold, $\phi_J$, but affects the coordination number and other parameters of interest as the critical point is approached. In the limit of low obstacle density, one can calculate a pinning susceptibility $\chi_P$, analogous to the magnetic susceptibility, with obstacle density playing the role of the magnetic field. The pinning susceptibility is thus expected to diverge in the thermodynamic limit as $\chi_P \propto |\phi-\phi_J|^{-\gamma_P}$. Finite-size scaling calculations allow us to confirm this and calculate the critical exponent, $\gamma_P$.