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 history 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 find 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 vibrational modes of three-dimensional jammed packings of soft ellipsoids of revolution as a function of particle aspect ratio ε and packing fraction. At the jamming transition for ellipsoids, as distinct from the idealized case using spheres where ε = 1, there are many unconstrained and non-trivial rotational degrees of freedom. These constitute a set of zerofrequency modes that are gradually mobilized into a new rotational band as |ε−1| increases. Quite surprisingly, as this new band is separated from zero frequency by a gap, and lies below the onset frequency for translational vibrations, ω * , the presence of these new degrees of freedom leaves unaltered the basic scenario that the translational spectrum is determined only by the average contact number. Indeed, ω * depends solely on coordination as it does for compressed packings of spheres. We also discuss the regime of large |ε − 1|, where the two bands merge.