In this thesis, we study energy transport and fluctuations in simple models of fragile matter : a unique state of matter that has a vanishingly small window of linear response since one or both of... Show moreIn this thesis, we study energy transport and fluctuations in simple models of fragile matter : a unique state of matter that has a vanishingly small window of linear response since one or both of its elastic moduli (shear and bulk) are nearly zero. As a consequence, even the tiniest perturbations travel as nonlinear waves. In addition, most models of fragile matter have an amorphous structure. It is the interaction of the non-linear waves with the underlying disorder and the resulting fluctuations, that constitutes the unifying theme explored in this thesis. There are at least two seemingly distinct sources of fragility: a local source stemming from the strongly non-linear interaction potential between particles so that one can not expand around a potential minimum to define a spring constant, and a second, global source, whereby the collective response of the sample can be considered weakly linear. As a model of the first kind, we consider a two dimensional packing of soft frictionless elastic disks that are just touching their nearest neighbours. The interaction potential between elastic disks is given by the nonlinear Hertz law that has no harmonic part. Consequently, for a packing in this state, the bulk modulus is vanishingly small and the smallest compressions imparted at the edges leads to nonlinear solitary like waves. As a model of the second kind, we consider a two dimensional random network of harmonic springs where each node has on average around four nearest neighbours. Here, despite the contact interaction being harmonic, the network has a vanishingly small shear modulus. Consequently, even the tiniest shear strains elicit non-linear waves. There are many important similarities and differences between the nature of non-linear waves and the role played by disorder in the two models described above, which we are gradually beginning to understand. Show less
