We show that soft spheres interacting with a linear ramp potential when overcompressed beyond the jamming point fall in an amorphous solid phase which is critical, mechanically marginally stable and share many features with the jamming point itself. In the whole phase, the relevant local minima of the potential energy landscape display an isostatic contact network of perfectly touching spheres whose statistics is controlled by an infinite lengthscale. Excitations around such energy minima are non-linear, system spanning, and characterized by a set of non-trivial critical exponents. We perform numerical simulations to measure their values and show that, while they coincide, within numerical precision, with the critical exponents appearing at jamming, the nature of the corresponding excitations is richer. Therefore, linear soft spheres appear as a novel class of finite dimensional systems that self-organize into new, critical, marginally stable, states.
The topography of the free energy landscape in phase space of a dense hard sphere system characterized by a discretized free energy functional of the Ramakrishnan-Yussouff form is investigated numerically using a ``microcanonical'' Monte Carlo procedure. We locate a considerable number of glassy local minima of the free energy and analyze the distributions of the free energy at a minimum and an appropriately defined phase-space ``distance'' between different minima. We find evidence for the existence of pairs of closely related glassy minima (``two-level systems''). We also investigate the way…
On the basis of long simulations of a binary mixture of soft spheres just below the glass transition, we make an exploratory study of the activated processes that contribute to the dynamics. We concentrate on statistical measures of the size of the activated processes.
We present a nonlocal Monte Carlo algorithm with particle swaps that greatly accelerates thermalization of soft sphere binary mixtures in the glassy region. Our first results show that thermalization of systems of hundreds of particles is achievable, and find behavior compatible with a thermodynamic glass transition.
The complex physics of glass forming systems is controlled by the structure of the low energy portions of their potential energy landscapes. Here, we report that a modified metadynamics algorithm efficiently explores and samples low energy regions of such high-dimensional landscapes. In the energy landscape for a model foam, our algorithm finds and descends meandering `canyons' in the landscape, which contain dense clusters of energy minima along their floors. Similar canyon structures in the energy landscapes of two model glass formers -- hard sphere fluids and the Kob-Andersen glass -- allow…
We propose damage spreading (DS) as a tool to investigate the topological features related to the ruggedness of the free energy landscape. We argue that DS measures the positiveness of the largest Lyapunov exponent associated to the basins of attraction visited by the system during its dynamical evolution. We discuss recent results obtained in the framework of mode-coupling theory and comment how to extend them to the study of realistic glasses. Preliminary results are presented for purely repulsive soft-sphere glasses.
In this paper we consider different models for soft modes in jammed hard spheres. We show how one can construct mean field models that can be solved analytically. The analytic solution of these models displays an excess of low energy soft modes that become more and more localized by decreasing the energy. A simple solution of these models is found in the infinite dimensional limit.
We present the study of the landscape structure of athermal soft spheres both as a function of the packing fraction and of the energy. We find that, on approaching the jamming transition, the number of different configurations available to the system has a steep increase and that a hierarchical organization of the landscape emerges. We use the knowledge of the structure of the landscape to predict the values of thermodynamic observables on the edge of the transition.
We show that a glass transition, signaled by a peak in the specific heat vs. temperature, can occur because a glassy system that shows no signs of aging progresses so slowly through the energy landscape that the time needed to obtain an accurate estimate of the thermodynamic averages exceeds the observation time. We find that below the glass transition temperature of a three dimensional binary mixture of soft spheres, the specific heat increases with measurement time spans orders of magnitude longer than previously recognized equilibration times.
Epithelial cell tissues have a slow relaxation dynamics resembling that of supercooled liquids. Yet, they also have distinguishing features. These include an extended short-time sub-diffusive transient, as observed in some experiments and recent studies of model systems, and a sub-Arrhenius dependence of the relaxation time on temperature, as reported in numerical studies. Here we demonstrate that the anomalous glassy dynamics of epithelial tissues originates from the emergence of a fractal-like energy landscape, particles becoming virtually free to diffuse in specific phase space directions u…
We study the effect of thermalization on the rigidity of a randomly packed soft repulsive sphere system around the jamming point by analyzing the shear-modulus using the cloned liquid theory with the 1 step replica symmetry breaking ansatz and molecular dynamics simulations. Contrarily to the usual harmonic picture for solids, we found that the thermalized jamming system is anomalously softer than at zero temperature such that the shear-modulus becomes as small as the pressure down to vanishingly low temperatures.
We propose a class of mean-field models for the isostatic transition of systems of soft spheres, in which the contact network is modeled as a random graph and each contact is associated to $d$ degrees of freedom. We study such models in the hypostatic, isostatic, and hyperstatic regimes. The density of states is evaluated by both the cavity method and exact diagonalization of the dynamical matrix. We show that the model correctly reproduces the main features of the density of states of real packings and, moreover, it predicts the presence of localized modes near the lower band edge. Finally, t…
Structure of solutions to continuous constraint satisfaction problems through the statistics of wedged and inscribed spheres
ICTP South American Institute for Fundamental Research（ICTP南美基础研究所）；Instituto de Física Teórica, Universidade Estadual Paulista “Júlio de Mesquita Filho”（圣保罗州立大学“朱利奥·德·梅斯基塔·费略”理论物理研究所）
Packing spheres efficiently in large dimension $d$ is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize a lower bound on packing density. Our results suggest that exponentially many (in the number of particles) distinct disordered sphere packings can be effectively constructed by this method, up to a packing fraction close to $7\, d\, 2^{-d}$. The latter is determined by solving the inverse problem of maximizing the dynamical glass transition over the space o…
For amorphous solids, it has been intensely debated whether the traditional view on solids, in terms of the ground state and harmonic low energy excitations on top of it, such as phonons, is still valid. Recent theoretical developments of amorphous solids revealed the possibility of unexpectedly complex free energy landscapes where the simple harmonic picture breaks down. Here we demonstrate that standard rheological techniques can be used as powerful tools to examine non-trivial consequences of such complex free energy landscapes. By extensive numerical simulations on a hard sphere glass unde…
We discuss an analysis of Constraint Satisfaction problems, such as Sphere Packing, K-SAT and Graph Coloring, in terms of an effective energy landscape. Several intriguing geometrical properties of the solution space become in this light familiar in terms of the well-studied ones of rugged (glassy) energy landscapes. A `benchmark' algorithm naturally suggested by this construction finds solutions in polynomial time up to a point beyond the `clustering' and in some cases even the `thermodynamic' transitions. This point has a simple geometric meaning and can be in principle determined with stand…
