Memory formation in matter is a theme of broad intellectual relevance; it sits at the interdisciplinary crossroads of physics, biology, chemistry, and computer science. Memory connotes the ability to encode, access, and erase signatures of past histo
ry in the state of a system. Once the system has completely relaxed to thermal equilibrium, it is no longer able to recall aspects of its evolution. Memory of initial conditions or previous training protocols will be lost. Thus many forms of memory are intrinsically tied to far-from-equilibrium behavior and to transient response to a perturbation. This general behavior arises in diverse contexts in condensed matter physics and materials: phase change memory, shape memory, echoes, memory effects in glasses, return-point memory in disordered magnets, as well as related contexts in computer science. Yet, as opposed to the situation in biology, there is currently no common categorization and description of the memory behavior that appears to be prevalent throughout condensed-matter systems. Here we focus on material memories. We will describe the basic phenomenology of a few of the known behaviors that can be understood as constituting a memory. We hope that this will be a guide towards developing the unifying conceptual underpinnings for a broad understanding of memory effects that appear in materials.
We compare the elastic response of spring networks whose contact geometry is derived from real packings of frictionless discs, to networks obtained by randomly cutting bonds in a highly connected network derived from a well-compressed packing. We fin
d that the shear response of packing-derived networks, and both the shear and compression response of randomly cut networks, are all similar: the elastic moduli vanish linearly near jamming, and distributions characterizing the local geometry of the response scale with distance to jamming. Compression of packing-derived networks is exceptional: the elastic modulus remains constant and the geometrical distributions do not exhibit simple scaling. We conclude that the compression response of jammed packings is anomalous, rather than the shear response.
We study the localization of vibrational modes of frictionless granular media. We introduce a new method, motivated by earlier work on non-Hermitian quantum problems, which works well both in the localized regime where the localization length $xi$ is
much less than the linear size $L$ and in the regime $xi$ grater or of order $L$ when modes are extended throughout our finite system. Our very lowest frequency modes show quasi-localized resonances away from the jamming point; the spatial extent of these regions increases as the jamming point is approached, as expected theoretically. Throughout the remaining frequency range, our data show no signature of the nearness of the jamming point and collapse well when properly rescaled with the system size. Using Random Matrix Theory we derive the scaling relation $xi$ ~ $L^{d/2}$ for the regime $xi$ >> $L$ in $d$ dimensions.
