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Length scales of dynamic heterogeneities in a network of fluctuating mechanical constraints.

We describe a random bond network with fluctuating mechanical constraints for which structural and dynamical length scales are explicitly defined and can be compared with the observable properties of the system. The constraints give well-defined individual structures, describing rigid clusters of particles, and there are distinct structural changes between different configurations. In addition, unambiguous unconstrained motions, including extended collective motions, can be defined within each configuration of constraints. By considering fluctuations of bonds in the network, we can analyze the relaxation of a simple model amorphous material subject to randomly arranged mechanical constraints. The lengths of the collective motions are obtained directly and compared with both the relaxation kinetics and the susceptibility χ(4). Above percolation, the latter quantity is shown to decrease with increasing degree of constraint, a behavior linked, in this paper, to the presence of overconstraint in the network at bond densities beyond the percolation point. Our model system, despite a propensity for highly collective motions, shows that an increasing timescale for relaxation can be accompanied by a decreasing χ(4).