Strong reduction of the rigidity of repulsive contact systems at vanishingly low temperatures

Contrarily to ordinary solids, the amorphous solid states of repulsive contact systems such as colloids and emulsions may not be regarded simply as harmonic states \footnote{C. F. Schreck,T. Bertrand, C. S. O'Hern, and M. D. Shattuck, Phys. Rev. Lett. 107, 078301 (2011).}. We studied the rigidity, i.~e. the shear-modulus of such a class of systems at vanishingly low but finite temperatures using the cloned liquid approach \footnote{H. Yoshino and M. M\'{e}zard, Phys. Rev. Lett. {\bf 105}, 015504 (2010), H. Yoshino, J. Chem. Phys. {\bf 136}, 214108 (2012), H. Yoshino, arXiv:1210.6826 (2012).} and molecular dynamic simulations. Our result implies breakdown of the commutation of the thermodynamic limit $N \to \infty$ and zero temperature limit $T \to 0$ for the response to shear: we found the rigidity in the limit $T \to 0$ is significantly smaller and exhibit a different scaling compared with that at $T=0$. Interestingly the rigidity in the limit $T \to 0$ exhibits the same scaling as the pressure, as observed experimentally in emulsions\footnote{T. G. Mason, J. Bibette and D. A. Weitz, Phys Rev. Lett. {\bf 75}, 2051 (1995)}. Detailed numerical examination suggests that the strong stress relaxation is due to contact opening events activated at vanishingly small temperatures.