First Passage Time for Many Particle Diffusion in Space-Time Random Environments

The first passage time for a single diffusing particle has been studied extensively, but the first passage time of a system of many diffusing particles, as is often the case in physical systems, has received little attention until recently. We consider two models for many particle diffusion -- one treats each particle as independent simple random walkers while the other treats them as coupled to a common space-time random forcing field that biases particles nearby in space and time in similar ways. The first passage time of a single diffusing particle under both of these models show the same statistics and scaling behavior. However, for many particle diffusions, the first passage time among all particles (the `extreme first passage time') is very different between the two models, effected in the latter case by the randomness of the common forcing field. We develop an asymptotic (in the number of particles and location where first passage is being probed) theoretical framework to separate out the impact of the random environment with that of sampling trajectories within it. We identify a new power-law describing the impact to the extreme first passage time variance of the environment. Through numerical simulations we verify that the predictions from this asymptotic theory hold even for systems with widely varying numbers of particles, all the way down to 100 particles. This shows that measurements of the extreme first passage time for many-particle diffusions provide an indirect measurement of the underlying environment in which the diffusion is occurring

Universal microstructure and mechanical stability of jammed packings

Jammed packings' mechanical properties depend sensitively on their detailed local structure. Here we provide a complete characterization of the pair correlation close to contact and of the force distribution of jammed frictionless spheres. In particular we discover a set of new scaling relations that connect the behavior of particles bearing small forces and those bearing no force but that are almost in contact. By performing systematic investigations for spatial dimensions d=3-10, in a wide density range and using different preparation protocols, we show that these scalings are indeed universal. We therefore establish clear milestones for the emergence of a complete microscopic theory of jamming. This description is also crucial for high-precision force experiments in granular systems.Comment: 11 pages, 7 figure

Universal non-Debye scaling in the density of states of amorphous solids

At the jamming transition, amorphous packings are known to display anomalous vibrational modes with a density of states (DOS) that remains constant at low frequency. The scaling of the DOS at higher densities remains, however, unclear. One might expect to find simple Debye scaling, but recent results from effective medium theory and the exact solution of mean-field models both predict an anomalous, non-Debye scaling. Being mean-field solutions, however, these solutions are only strictly applicable to the limit of infinite spatial dimension, and it is unclear what value they have for finite-dimensional systems. Here, we study packings of soft spheres in dimensions 3 through 7 and find, far from jamming, a universal non-Debye scaling of the DOS that is consistent with the mean-field predictions. We also consider how the soft mode participation ratio converges to the mean-field prediction as dimension increases.Comment: 5 pages, 4 figure