  The jamming transition is a nonequilibrium critical phenomenon, which governs
characteristic mechanical properties of jammed soft materials, such as pastes,
emulsions, and granular matters. Both experiments and theory of jammed soft
materials have revealed that the complex modulus measured by conventional
macrorheology exhibits a characteristic frequency dependence. Microrheology is
a new type of method to obtain the complex modulus, which transforms the
microscopic motion of probes to the complex modulus through the generalized
Stokes relation (GSR). Although microrheology has been applied to jammed soft
materials, its theoretical understanding is limited. In particular, the
validity of the GSR near the jamming transition is far from obvious since there
is a diverging length scale $l_c$, which characterizes the heterogeneous
response of jammed particles. Here, we study the microrheology of jammed
particles by theory and numerical simulation. First, we develop a linear
response formalism to calculate the response function of the probe particle,
which is transformed to the complex modulus via the GSR. Then, we apply our
formalism to a numerical model of jammed particles and find that the storage
and loss modulus follow characteristic scaling laws near the jamming
transition. Importantly, the observed scaling law coincides with that in
macrorheology, which indicates that the GSR holds even near the jamming
transition. We rationalize this equivalence by asymptotic analysis of the
obtained formalism and numerical analysis on the displacement field of jammed
particles under a local perturbation.
