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The Marginally Rigid State is a candidate paradigm for what makes granular material a state of matter distinct from both liquid and solid. Coordination number is identified as a discriminating characteristic, and for rough-surfaced particles we show that the low values predicted are indeed approached in simple two dimensional experiments. We show calculations of the stress transmission suggesting that this is governed by local linear equations of constraint between the stress components. These constraints can in turn be related to the generalised forces conjugate to the motion of grains rolling over each other. The lack of a spatially coherent way of imposing a sign convention on these motions is a problem for up-scaling the equations, which leads us to attempt a renormalisation group calculation. Finally we discuss how perturbations propagate through such systems, suggesting a distinction between the behaviour of rough and of smooth grains.
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The ensemble averaged power scattered in and out of lossless chaotic cavities decays as a power law in time for large times. In the case of a pulse with a finite duration, the power scattered from a single realization of a cavity closely tracks the power law ensemble decay initially, but eventually transitions to an exponential decay. In this paper, we explore the nature of this transition in the case of coupling to a single port. We find that for a given pulse shape, the properties of the transition are universal if time is properly normalized. We define the crossover time to be the time at which the deviations from the mean of the reflected power in individual realizations become comparable to the mean reflected power. We demonstrate numerically that, for randomly chosen cavity realizations and given pulse shapes, the probability distribution function of reflected power depends only on time, normalized to this crossover time.
The Generalized Central Limit Theorem is a remarkable generalization of the Central Limit Theorem, showing that the sum of a large number of independent, identically-distributed (i.i.d) random variables with infinite variance may converge under appropriate scaling to a distribution belonging to a special family known as Levy stable distributions. Similarly, the maximum of i.i.d. variables may converge to a distribution belonging to one of three universality classes (Gumbel, Weibull and Frechet). Here, we rederive these known results following a mathematically non-rigorous yet highly transparent renormalization-group-like approach that captures both of these universal results following a nearly identical procedure.
We endow a system of interacting particles with two distinct, local, Markovian and reversible microscopic dynamics. Using common field-theoretic techniques used to investigate the presence of a glass transition, we find that while the first, standard, dynamical rules lead to glassy behavior, the other one leads to a simple exponential relaxation towards equilibrium. This finding questions the intrinsic link that exists between the underlying, thermodynamical, energy landscape, and the dynamical rules with which this landscape is explored by the system. Our peculiar choice of dynam- ical rules offers the possibility of a direct connection with replica theory, and our findings therefore call for a clarification of the interplay between replica theory and the underlying dynamics of the system.
We show how a finite number of conservation laws can globally `shatter Hilbert space into exponentially many dynamically disconnected subsectors, leading to an unexpected dynamics with features reminiscent of both many body localization and quantum scars. A crisp example of this phenomenon is provided by a `fractonic model of quantum dynamics constrained to conserve both charge and dipole moment. We show how the Hilbert space of the fractonic model dynamically fractures into disconnected emergent subsectors within a particular charge and dipole symmetry sector. This shattering can occur in arbitrary spatial dimensions. A large number of the emergent subsectors, exponentially many in system volume, have dimension one and exhibit strictly localized quantum dynamics---even in the absence of spatial disorder and in the presence of temporal noise. Other emergent subsectors display non-trivial dynamics and may be constructed by embedding finite sized non-trivial blocks into the localized subspace. While `fractonic models provide a particularly clean realization, the shattering phenomenon is more general, as we discuss. We also discuss how the key phenomena may be readily observed in near term ultracold atom experiments. In experimental realizations, the conservation laws are approximate rather than exact, so the localization only survives up to a prethermal timescale that we estimate. We comment on the implications of these results for recent predictions of Bloch/Stark many-body localization.
Discrete amorphous materials are best described in terms of arbitrary networks which can be embedded in three dimensional space. Investigating the thermodynamic equilibrium as well as non-equilibrium behavior of such materials around second order phase transitions call for special techniques. We set up a renormalization group scheme by expanding an arbitrary scalar field living on the nodes of an arbitrary network, in terms of the eigenvectors of the normalized graph Laplacian. The renormalization transformation involves, as usual, the integration over the more rapidly varying components of the field, corresponding to eigenvectors with larger eigenvalues, and then rescaling. The critical exponents depend on the particular graph through the spectral density of the eigenvalues.
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