17 research outputs found
Out-of-equilibrium dynamical equations of infinite-dimensional particle systems. I. The isotropic case
We consider the Langevin dynamics of a many-body system of interacting
particles in dimensions, in a very general setting suitable to model
several out-of-equilibrium situations, such as liquid and glass rheology,
active self-propelled particles, and glassy aging dynamics. The pair
interaction potential is generic, and can be chosen to model colloids, atomic
liquids, and granular materials. In the limit , we show that the
dynamics can be exactly reduced to a single one-dimensional effective
stochastic equation, with an effective thermal bath described by kernels that
have to be determined self-consistently. We present two complementary
derivations, via a dynamical cavity method and via a path-integral approach.
From the effective stochastic equation, one can compute dynamical observables
such as pressure, shear stress, particle mean-square displacement, and the
associated response function. As an application of our results, we derive
dynamically the `state-following' equations that describe the response of a
glass to quasistatic perturbations, thus bypassing the use of replicas. The
article is written in a modular way, that allows the reader to skip the details
of the derivations and focus on the physical setting and the main results
Impact of jamming criticality on low-temperature anomalies in structural glasses
We present a novel mechanism for the anomalous behaviour of the specific heat
in low-temperature amorphous solids. The analytic solution of a mean-field
model belonging to the same universality class as high-dimensional glasses, the
spherical perceptron, suggests that there exists a crossover temperature above
which the specific heat scales linearly with temperature while below it a cubic
scaling is displayed. This relies on two crucial features of the phase diagram:
(i) The marginal stability of the free-energy landscape, which induces a
gapless phase responsible for the emergence of a power-law scaling (ii) The
vicinity of the classical jamming critical point, as the crossover temperature
gets lowered when approaching it. This scenario arises from a direct study of
the thermodynamics of the system in the quantum regime, where we show that,
contrary to crystals, the Debye approximation does not hold.Comment: 7 pages + 38 pages SI, 5 figure
Out-of-equilibrium dynamical equations of infinite-dimensional particle systems. II. The anisotropic case under shear strain
As an extension of the isotropic setting presented in the companion paper [J.
Phys. A 52, 144002 (2019)], we consider the Langevin dynamics of a many-body
system of pairwise interacting particles in dimensions, submitted to an
external shear strain. We show that the anisotropy introduced by the shear
strain can be simply addressed by moving into the co-shearing frame, leading to
simple dynamical mean field equations in the limit . The dynamics
is then controlled by a single one-dimensional effective stochastic process
which depends on three distinct strain-dependent kernels - self-consistently
determined by the process itself - encoding the effective restoring force,
friction and noise terms due to the particle interactions. From there one can
compute dynamical observables such as particle mean-square displacements and
shear stress fluctuations, and eventually aim at providing an exact benchmark for liquid and glass rheology. As an application of our
results, we derive dynamically the 'state-following' equations that describe
the static response of a glass to a finite shear strain until it yields.Comment: Typo corrected in Eq. (47
Density scaling of generalized Lennard-Jones fluids in different dimensions
Liquids displaying strong virial-potential energy correlations conform to an
approximate density scaling of their structural and dynamical observables. This
scaling property does not extend to the entire phase diagram, in general. The
validity of the scaling can be quantified by a correlation coefficient. In this
work a simple scheme to predict the correlation coefficient and the
density-scaling exponent is presented. Although this scheme is exact only in
the dilute gas regime or in high dimension d, a comparison with results from
molecular dynamics simulations in d = 1 to 4 shows that it reproduces well the
behavior of generalized Lennard-Jones systems in a large portion of the fluid
phase.Comment: Submission to SciPos
Théorie des liquides et verres en dimension infinie
The dynamics of liquids, regarded as strongly-interacting classical particle systems, remains a field where theoretical descriptions are limited. So far, there is no microscopic theory starting from first principles and using controlled approximations. At the thermodynamic level, static equilibrium properties are well understood in simple liquids only far from glassy regimes. Here we derive, from first principles, the dynamics of liquids and glasses using the limit of large spatial dimension, which provides a well-defined mean-field approximation with a clear small parameter. In parallel, we recover their thermodynamics through an analogy between dynamics and statics. This gives a unifying and consistent view of the phase diagram of these systems. We show that this mean-field solution to the structural glass problem is an example of the Random First-Order Transition scenario, as conjectured thirty years ago, based on the solution of mean-field spin glasses. These results allow to show that an approximate scale invariance of the system, relevant to finite-dimensional experiments and simulations, becomes exact in this limit.La dynamique des liquides, considĂ©rĂ©s comme des systĂšmes de particules classiques fortement couplĂ©es, reste un domaine oĂč les descriptions thĂ©oriques sont limitĂ©es. Pour lâinstant, il nâexiste pas de thĂ©orie microscopique partant des premiers principes et recourant Ă des approximations contrĂŽlĂ©es. Thermodynamiquement, les propriĂ©tĂ©s statiques dâĂ©quilibre sont bien comprises dans les liquides simples, Ă condition dâĂȘtre loin du rĂ©gime vitreux. Dans cette thĂšse, nous rĂ©solvons, en partant des Ă©quations microscopiques du mouvement, la dynamique des liquides et verres en exploitant la limite de dimension spatiale infinie, qui fournit une approximation de champ moyen bien dĂ©finie. En parallĂšle, nous retrouvons leur thermodynamique Ă travers une analogie entre la dynamique et la statique. Cela donne un point de vue Ă la fois unificateur et cohĂ©rent du diagramme de phase de ces systĂšmes. Nous montrons que cette solution de champ moyen au problĂšme de la transition vitreuse est un exemple du scĂ©nario de transition de premier ordre alĂ©atoire (RFOT), comme conjecturĂ© il y a maintenant trente ans, sur la base des solutions des modĂšles de verres de spin en champ moyen. Ces rĂ©sultats nous permettent de montrer quâune invariance dâĂ©chelle approchĂ©e du systĂšme, pertinente pour les expĂ©riences et les simulations en dimension finie, devient exacte dans cette limite
Theory of high-dimensional liquids and glasses
La dynamique des liquides, considĂ©rĂ©s comme des systĂšmes de particules classiques fortement couplĂ©es, reste un domaine oĂč les descriptions thĂ©oriques sont limitĂ©es. Pour lâinstant, il nâexiste pas de thĂ©orie microscopique partant des premiers principes et recourant Ă des approximations contrĂŽlĂ©es. Thermodynamiquement, les propriĂ©tĂ©s statiques dâĂ©quilibre sont bien comprises dans les liquides simples, Ă condition dâĂȘtre loin du rĂ©gime vitreux. Dans cette thĂšse, nous rĂ©solvons, en partant des Ă©quations microscopiques du mouvement, la dynamique des liquides et verres en exploitant la limite de dimension spatiale infinie, qui fournit une approximation de champ moyen bien dĂ©finie. En parallĂšle, nous retrouvons leur thermodynamique Ă travers une analogie entre la dynamique et la statique. Cela donne un point de vue Ă la fois unificateur et cohĂ©rent du diagramme de phase de ces systĂšmes. Nous montrons que cette solution de champ moyen au problĂšme de la transition vitreuse est un exemple du scĂ©nario de transition de premier ordre alĂ©atoire (RFOT), comme conjecturĂ© il y a maintenant trente ans, sur la base des solutions des modĂšles de verres de spin en champ moyen. Ces rĂ©sultats nous permettent de montrer quâune invariance dâĂ©chelle approchĂ©e du systĂšme, pertinente pour les expĂ©riences et les simulations en dimension finie, devient exacte dans cette limite.The dynamics of liquids, regarded as strongly-interacting classical particle systems, remains a field where theoretical descriptions are limited. So far, there is no microscopic theory starting from first principles and using controlled approximations. At the thermodynamic level, static equilibrium properties are well understood in simple liquids only far from glassy regimes. Here we derive, from first principles, the dynamics of liquids and glasses using the limit of large spatial dimension, which provides a well-defined mean-field approximation with a clear small parameter. In parallel, we recover their thermodynamics through an analogy between dynamics and statics. This gives a unifying and consistent view of the phase diagram of these systems. We show that this mean-field solution to the structural glass problem is an example of the Random First-Order Transition scenario, as conjectured thirty years ago, based on the solution of mean-field spin glasses. These results allow to show that an approximate scale invariance of the system, relevant to finite-dimensional experiments and simulations, becomes exact in this limit
Statics and dynamics of infinite-dimensional liquids and glasses: a parallel and compact derivation
International audienc
Generating dense packings of hard spheres by soft interaction design
International audienc
Generating dense packings of hard spheres by soft interaction design
Packing spheres efficiently in large dimension 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 . The latter is determined by solving the
inverse problem of maximizing the dynamical glass transition over the space of
the interaction potentials. Our method crucially exploits a recent exact
formulation of the thermodynamics and the dynamics of simple liquids in
infinite dimension