arXiv subjects

Giulio Biroli

Publications and source records attributed to Giulio Biroli.

At least 55 records · Page 3Linked to original sources

arXiv2023

Large-deviation analysis of rare resonances for the Many-Body localization transition

A central theoretical issue at the core of the current research on many-body localization (MBL) consists in characterizing the statistics of rare long-range resonances in many-body eigenstates. This is of paramount importance to understand: (i) the critical properties of the MBL transition and the mechanism for its destabilization through quantum avalanches; (ii) the unusual transport and anomalously slow out-of-equilibrium relaxation when the transition is approached from the metallic side. In order to study and characterize such long-range rare resonances, we develop a large-deviations approach based on an analogy with the physics of directed polymers in random media, and in particular with their freezing glass transition on infinite-dimensional graphs. The basic idea is to enlarge the parameter space by adding an auxiliary parameter (which plays the role of the inverse temperature in the directed polymer formulation) which allows us to fine-tune the effect of anomalously large outliers in the far-tails of the probability distributions of the transmission amplitudes between far-away many-body configurations in the Hilbert space. We first benchmark our approach onto two non-interacting paradigmatic toy models, namely the single-particle Anderson model on the (loop-less) Cayley tree and the Rosenzweig-Porter random matrix ensemble, and then apply it to the study of a class of disordered quantum spin chains in a transverse field. This analysis shows the existence of a broad disorder range in which rare, long-distance resonances, that may form only for a few specific realizations of the disorder and a few specific choice of the random initial state, destabilize the MBL phase, while the genuine MBL transition is shifted to much larger values of the disorder than originally thought.

cond-mat.dis-nn↗

arXiv2023

Ductile-to-brittle transition and yielding in soft amorphous materials: perspectives and open questions

Soft amorphous materials are viscoelastic solids ubiquitously found around us, from clays and cementitious pastes to emulsions and physical gels encountered in food or biomedical engineering. Under an external deformation, these materials undergo a noteworthy transition from a solid to a liquid state that reshapes the material microstructure. This yielding transition was the main theme of a workshop held from January 9 to 13, 2023 at the Lorentz Center in Leiden. The manuscript presented here offers a critical perspective on the subject, synthesizing insights from the various brainstorming sessions and informal discussions that unfolded during this week of vibrant exchange of ideas. The result of these exchanges takes the form of a series of open questions that represent outstanding experimental, numerical, and theoretical challenges to be tackled in the near future.

cond-mat.soft↗

arXiv2023

Mean-Field Analysis of the Glassy Dynamics of an Elastoplastic Model of Super-Cooled Liquids

We present a mean-field theory of a coarse-grained model of a super-cooled liquid in which relaxation occurs via local plastic rearrangements. Local relaxation can be induced by thermal fluctuations or by the long-range elastic consequences of other rearrangements. We extract the temperature dependence of both the relaxation time and the lengthscale of dynamical correlations. We find two dynamical regimes. First, a regime in which the characteristic time and length scales diverge as a power law at a critical temperature $T_c$. This regime is found by an approximation that neglects activated relaxation channels, which can be interpreted as akin to the one found by the mode-coupling transition of glasses. In reality, only a cross-over takes place at $T_c$. The residual plastic activity leads to a second regime characterised by an Arrhenius law below $T_c$. In this case, we show that the lengthscale governing dynamical correlations diverges as a power law as $T\to 0$, and is logarithmically related to the relaxation time.

cond-mat.soft↗

arXiv2023

Curvature-driven pathways interpolating between stationary points: the case of the pure spherical 3-spin model

This paper focuses on characterizing the energy profile along pathways connecting different regions of configuration space in the context of a prototypical glass model, the pure spherical $p$-spin model with $p=3$. The study investigates pairs of stationary points (local minima or rank-1 saddles), analyzing the energy profile along geodesic paths and comparing them with "perturbed" pathways correlated to the landscape curvature. The goal is to assess the extent to which information from the local Hessian matrices around stationary points can identify paths with lower energy barriers. Surprisingly, unlike findings in other systems, the direction of softest local curvature is not a reliable predictor of low-energy paths, except in the case in which the direction of softest curvature corresponds to an isolated mode of the Hessian. However, other information encoded in the local Hessian does allow the identification of pathways associated with lower energy barriers. We conclude commenting on implications for the system's activated dynamics.

cond-mat.dis-nn↗

arXiv2023

High-Dimensional Non-Convex Landscapes and Gradient Descent Dynamics

In these lecture notes we present different methods and concepts developed in statistical physics to analyze gradient descent dynamics in high-dimensional non-convex landscapes. Our aim is to show how approaches developed in physics, mainly statistical physics of disordered systems, can be used to tackle open questions on high-dimensional dynamics in Machine Learning.

cond-mat.dis-nn↗

arXiv2023

On the Impact of Overparameterization on the Training of a Shallow Neural Network in High Dimensions

We study the training dynamics of a shallow neural network with quadratic activation functions and quadratic cost in a teacher-student setup. In line with previous works on the same neural architecture, the optimization is performed following the gradient flow on the population risk, where the average over data points is replaced by the expectation over their distribution, assumed to be Gaussian.We first derive convergence properties for the gradient flow and quantify the overparameterization that is necessary to achieve a strong signal recovery. Then, assuming that the teachers and the students at initialization form independent orthonormal families, we derive a high-dimensional limit for the flow and show that the minimal overparameterization is sufficient for strong recovery. We verify by numerical experiments that these results hold for more general initializations.

math.OC↗

arXiv2023

Dynamic heterogeneity at the experimental glass transition predicted by transferable machine learning

We develop a transferable machine learning model which predicts structural relaxation from amorphous supercooled liquid structures. The trained networks are able to predict dynamic heterogeneity across a broad range of temperatures and time scales with excellent accuracy and transferability. We use the network transferability to predict dynamic heterogeneity down to the experimental glass transition temperature, $T_g$, where structural relaxation cannot be analyzed using molecular dynamics simulations. The results indicate that the strength, the geometry and the characteristic length scale of the dynamic heterogeneity evolve much more slowly near $T_g$ compared to their evolution at higher temperatures. Our results show that machine learning techniques can provide physical insights on the nature of the glass transition that cannot be gained using conventional simulation techniques.

cond-mat.soft↗

arXiv2023

Shear-Induced Phase Behavior and Topological Defects in Two-Dimensional Crystals

We investigate through numerical simulations how a two-dimensional crystal yields and flows under an applied shear. We focus over a range that allows us to both address the response in the limit of an infinitesimal shear rate and describe the phase behavior of the system at a finite shear rate. In doing so, we carefully discuss the role of the topological defects and of the finite-size effects. We map out the whole phase diagram of the flowing steady state in the plane formed by temperature and shear rate. Shear-induced melting of the two-dimensional crystal is found to proceed in two steps: first, the solid loses long-range bond-orientational order and flows, even for an infinitesimal shear rate (in the thermodynamic limit). The resulting flowing hexatic phase then melts to a flowing, rather isotropic, liquid at a finite shear rate that depends on temperature. Finally, at a high shear rate, a third regime corresponding to a strongly anisotropic string-like flowing phase appears.

cond-mat.soft↗

arXiv2023

Predicting dynamic heterogeneity in glass-forming liquids by physics-inspired machine learning

We introduce GlassMLP, a machine learning framework using physics-inspired structural input to predict the long-time dynamics in deeply supercooled liquids. We apply this deep neural network to atomistic models in 2D and 3D. Its performance is better than the state of the art while being more parsimonious in terms of training data and fitting parameters. GlassMLP quantitatively predicts four-point dynamic correlations and the geometry of dynamic heterogeneity. Transferability across system sizes allows us to efficiently probe the temperature evolution of spatial dynamic correlations, revealing a profound change with temperature in the geometry of rearranging regions.

cond-mat.soft↗

arXiv2023

Far-from-equilibrium criticality in the Random Field Ising Model with Eshelby Interactions

We study a quasi-statically driven random field Ising model (RFIM) at zero temperature with interactions mediated by the long-range anisotropic Eshelby kernel. Analogously to amorphous solids at their yielding transition, and differently from ferromagnetic and dipolar RFIMs, the model shows a discontinuous magnetization jump associated with the appearance of a band-like structure for weak disorder and a continuous magnetization growth, yet punctuated by avalanches, for strong disorder. Through a finite-size scaling analysis in 2 and 3 dimensions we find that the two regimes are separated by a finite-disorder critical point which we characterize. We discuss similarities and differences between the present model and models of sheared amorphous solids.

cond-mat.dis-nn↗

arXiv2023

Generative diffusion in very large dimensions

Generative models based on diffusion have become the state of the art in the last few years, notably for image generation. Here, we analyse them in the high-dimensional limit, where data are formed by a very large number of variables. We use methods from statistical physics and focus on two well-controlled high-dimensional cases: a Gaussian model and the Curie-Weiss model of ferromagnetism. In the latter case, we highlight the mechanism of symmetry breaking in the inverse diffusion, and point out that, in order to reconstruct the relative asymmetry of the two low-temperature states, and thus to obtain the correct probability weights, one needs a database with a number of points much larger than the dimension of each data point. We characterize the scaling laws in the number of data and in the number of dimensions for an efficient generation.

cond-mat.dis-nn↗

arXiv2023

Scaling Description of Dynamical Heterogeneity and Avalanches of Relaxation in Glass-Forming Liquids

We provide a theoretical description of dynamical heterogeneities in glass-forming liquids, based on the premise that relaxation occurs via local rearrangements coupled by elasticity. In our framework, the growth of the dynamical correlation length $ξ$ and of the correlation volume $χ_4$ are controlled by a zero-temperature fixed point. We connect this critical behavior to the properties of the distribution of local energy barriers at zero temperature. Our description makes a direct connection between dynamical heterogeneities and avalanche-type relaxation associated to dynamic facilitation, allowing us to relate the size distribution of heterogeneities to their time evolution. Within an avalanche, a local region relaxes multiple times, the more the larger is the avalanche. This property, related to the nature of the zero-temperature fixed point, directly leads to decoupling of particle diffusion and relaxation time (the so-called Stokes-Einstein violation). Our most salient predictions are tested and confirmed by numerical simulations of scalar and tensorial thermal elasto-plastic models.

cond-mat.soft↗

arXiv2023

Quenched complexity of equilibria for asymmetric Generalized Lotka-Volterra equations

We consider the Generalized Lotka-Volterra system of equations with all-to-all, random asymmetric interactions describing high-dimensional, very diverse and well-mixed ecosystems. We analyze the multiple equilibria phase of the model and compute its quenched complexity, i.e., the expected value of the logarithm of the number of equilibria of the dynamical equations. We discuss the resulting distribution of equilibria as a function of their diversity, stability and average abundance. We obtain the quenched complexity by means of the replicated Kac-Rice formalism, and compare the results with the same quantity obtained within the annealed approximation, as well as with the results of the cavity calculation and, in the limit of symmetric interactions, of standard methods to compute the complexity developed in the context of glasses.

cond-mat.dis-nn↗

arXiv2023

Generalized Lotka-Volterra equations with random, non-reciprocal interactions: the typical number of equilibria

We compute the typical number of equilibria of the Generalized Lotka-Volterra equations describing species-rich ecosystems with random, non-reciprocal interactions using the replicated Kac-Rice method. We characterize the multiple-equilibria phase by determining the average abundance and similaritybetween equilibria as a function of their diversity (i.e. of the number of coexisting species) and of the variability of the interactions. We show that linearly unstable equilibria are dominant, and that the typical number of equilibria differs with respect to the average number.

cond-mat.dis-nn↗

arXiv2023

Artificial selection of communities drives the emergence of structured interactions

Species-rich communities, such as the microbiota or microbial ecosystems, provide key functions for human health and climatic resilience. Increasing effort is being dedicated to design experimental protocols for selecting community-level functions of interest. These experiments typically involve selection acting on populations of communities, each of which is composed of multiple species. If numerical simulations started to explore the evolutionary dynamics of this complex, multi-scale system, a comprehensive theoretical understanding of the process of artificial selection of communities is still lacking. Here, we propose a general model for the evolutionary dynamics of communities composed of a large number of interacting species, described by disordered generalised Lotka-Volterra equations. Our analytical and numerical results reveal that selection for scalar community functions leads to the emergence, along an evolutionary trajectory, of a low-dimensional structure in an initially featureless interaction matrix. Such structure reflects the combination of the properties of the ancestral community and of the selective pressure. Our analysis determines how the speed of adaptation scales with the system parameters and the abundance distribution of the evolved communities. Artificial selection for larger total abundance is thus shown to drive increased levels of mutualism and interaction diversity. Inference of the interaction matrix is proposed as a method to assess the emergence of structured interactions from experimentally accessible measures.

q-bio.PE↗

arXiv2023

Landscape Complexity for the Empirical Risk of Generalized Linear Models

We present a method to obtain the average and the typical value of the number of critical points of the empirical risk landscape for generalized linear estimation problems and variants. This represents a substantial extension of previous applications of the Kac-Rice method since it allows to analyze the critical points of high dimensional non-Gaussian random functions. Under a technical hypothesis, we obtain a rigorous explicit variational formula for the annealed complexity, which is the logarithm of the average number of critical points at fixed value of the empirical risk. This result is simplified, and extended, using the non-rigorous Kac-Rice replicated method from theoretical physics. In this way we find an explicit variational formula for the quenched complexity, which is generally different from its annealed counterpart, and allows to obtain the number of critical points for typical instances up to exponential accuracy.

stat.ML↗

arXiv2022

Finite-disorder critical point in the yielding transition of elasto-plastic models

Upon loading, amorphous solids can exhibit brittle yielding, with the abrupt formation of macroscopic shear bands leading to fracture, or ductile yielding, with a multitude of plastic events leading to homogeneous flow. It has been recently proposed, and subsequently questioned, that the two regimes are separated by a sharp critical point, as a function of some control parameter characterizing the intrinsic disorder strength and the degree of stability of the solid. In order to resolve this issue, we have performed extensive numerical simulations of athermally driven elasto-plastic models with long-range and anisotropic realistic interaction kernels in two and three dimensions. Our results provide clear evidence for a finite-disorder critical point separating brittle and ductile yielding, and we provide an estimate of the critical exponents in 2D and 3D.

cond-mat.soft↗

arXiv2022

Elasticity, Facilitation and Dynamic Heterogeneity in Glass-Forming liquids

We study the role of elasticity-induced facilitation on the dynamics of glass-forming liquids by a coarse-grained two-dimensional model in which local relaxation events, taking place by thermal activation, can trigger new relaxations by long-range elastically-mediated interactions. By simulations and an analytical theory, we show that the model reproduces the main salient facts associated with dynamic heterogeneity and offers a mechanism to explain the emergence of dynamical correlations at the glass transition. We also discuss how it can be generalized and combined with current theories.

cond-mat.dis-nn↗