Certain scalar-tensor theories of gravity provide negative-energy, tachyonic modes to a fundamental scalar inside matter, giving rise to nonperturbative phenomena around compact stars. Studies of this and other tachyonic instabilities always average over local matter properties. We use elementary, flat space models to understand possible collective effects and the accuracy of the averaging procedure. In particular, we consider bodies made of elementary constituents which do not, in isolation, scalarize because their compactness C is too small, C less than or similar to C-crit. We show that when the individual constituents have compactness smaller but close to the threshold, one is able to scalarize composite bodies through collective effects, and the compactness of the composite body can be made arbitrarily small. On the other hand, our results suggest that when the fundamental building blocks have very low compactness, then scalarization of the composite body requires a global compactness C-global greater than or similar to C-crit. Thus, our results rule out scalarization of dilute bodies via collective effects.