Pinning Susceptibility Near the Jamming Transition

The study of jamming in the presence of pinned obstacles is of both practical and theoretical interest. In simulations of soft, bidisperse disks and spheres, we pin a small fraction, $n_f$ of particles prior to the equilibration process. The presence of pinned particles is known to lower the critical packing fraction, $\phi_J$, for jamming. Further, around this threshold there is a peak in a quantity which we have termed the ``pinning susceptibility'': $\chi_P = \lim_{n_f \rightarrow 0} \frac{\partial P_J (\phi, n_f)}{\partial n_f}$. In the thermodynamic limit, we have posited that $\chi_P \propto |\Delta \phi |^{-\gamma_P} $. Finite-size scaling calculations, involving careful fits of $P_J$ to logistic sigmoidal functions, yield a value for the critical exponent, $\gamma_P$. This new exponent is proposed to be independent of inter-particle potential. Its dependence on dimensionality (2 vs. 3 dimensions) will be discussed.