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Glass and Jamming Transitions: From Exact Results to Finite-Dimensional Descriptions
Patrick Charbonneau, Jorge Kurchan, Giorgio Parisi, Pierfrancesco Urbani, Francesco Zamponi
TL;DR
The paper addresses the challenge of developing a first-principles theory for amorphous materials. It reviews the exact infinite-dimensional description of hard-sphere glasses and compares it with finite-dimensional simulations, finding dimensional robustness in some predictions while identifying sensitive features and open questions.
Problem
Amorphous materials lack a natural reference for conventional perturbative descriptions, making a first-principles theoretical account extremely challenging.
Method
The review synthesizes exact d →∞ results for hard spheres around dynamical, Gardner, and jamming transitions and compares them with finite-dimensional numerical and experimental findings.
Results
Many features of amorphous materials are qualitatively and sometimes quantitatively independent of spatial dimension, while critical scaling of χ4 and ξd remains unconvincingly tested.
Takeaways & Limitations
The infinite-dimensional framework provides clear definitions for glassy-state quantities and supports a dimensional-robust description of several hard-sphere glass phenomena.
Takeaways & Limitations
The physical origin of jamming's dimensional robustness lacks a fully satisfying explanation, and finite-dimensional analysis of the transitions remains a work in progress.
Abstract
from arXiv · showhide
Despite decades of work, gaining a first-principle understanding of amorphous materials remains an extremely challenging problem. However, recent theoretical breakthroughs have led to the formulation of an exact solution in the mean-field limit of infinite spatial dimension, and numerical simulations have remarkably confirmed the dimensional robustness of some of the predictions. This review describes these latest advances. More specifically, we consider the dynamical and thermodynamic descriptions of hard spheres around the dynamical, Gardner and jamming transitions. Comparing mean-field predictions with the finite-dimensional simulations, we identify robust aspects of the description and uncover its more sensitive features. We conclude with a brief overview of ongoing research.
1. INTRODUCTION
Amorphous materials lack a natural reference state for conventional theoretical descriptions, motivating an infinite-dimensional approach to hard-sphere glasses. This review presents that framework, assesses its robustness across dimensions, and compares exact predictions with finite-dimensional results.
- Motivation: Amorphous materials remain theoretically difficult because low-density and ideal-lattice reference strategies both fail for their strongly interacting, structurally disordered states.Constructing a first-principles theory therefore remains a major challenge in theoretical condensed matter physics.
- The infinite-dimensional approach: Dimensional expansion addresses this challenge by solving the problem at d →∞ and treating 1/d as a small parameter for comparison with d = 3.The approach was developed as an alternative when no natural perturbative reference system is available.
- The infinite-dimensional approach: The large-d approach also gives precise definitions to otherwise phenomenological concepts, including complexity and activated processes.Complexity counts metastable states with lifetimes exceeding a chosen t∗, while activated processes may involve times t ∼e^O(d).
- Finite-dimensional robustness: Finite-dimensional robustness is essential because an infinite-dimensional solution would have limited relevance if physical phenomena depended acutely on spatial details.The review contrasts this concern with dimension-specific ordered sphere packings.
- Scope and organization: The review focuses on hard-sphere glass formers and compares exact d →∞ theory with experimental and numerical results across liquid dynamics, amorphous-solid compression, jamming, and nonequilibrium processes.Its sections cover liquid dynamics, quasi-static compression, jamming endpoints, and out-of-equilibrium processes.
2. EQUILIBRIUM DYNAMICS AND MODE-COUPLING THEORY
The infinite-dimensional hard-sphere dynamics reduces interacting-particle motion to a self-consistent effective one-dimensional process, yielding precise predictions for glassy arrest and critical behavior. Several qualitative and quantitative predictions have finite-dimensional echoes, but the dynamical transition and some susceptibility scalings remain fragile or untested.
- Liquid dynamics: Increasing packing fraction slows liquid dynamics, with diffusivity decaying and structural relaxation time τα growing until the system loses ergodicity.This loss of ergodicity defines the experimental glass transition when crystallization is avoided.
- Mean-field dynamical equations: The d →∞ solution reduces N interacting particles to one effective degree of freedom in a self-consistent colored-noise potential.The memory kernel is determined by an averaged force-force correlation and differs from the simpler MCT closure.
- Dynamical transition: At the dynamical transition bϕd, the plateau length diverges, the long-time displacement remains finite at ∆EA, and memory persists through MEA > 0.Diffusivity vanishes and viscosity diverges, while bDbηS/T ∼bϕ remains finite.
- Critical exponents and susceptibility: For hard spheres in d →∞, the exponent parameter is λ = 0.70698..., with ηS ∼τα ∼bD−1 ∼|ϕ −ϕd|−γ.The exponent γ is related to the critical exponents by γ = 1/(2a) + 1/(2b).
- Limitations and open questions: The d →∞ dynamical transition does not formally exist at finite d because activated processes give configurations finite lifetimes, and χ4 and ξd critical scaling remains insufficiently tested.The asymptotic regime for bϕd remains distant even in d = 12, while extracting χ4 and ξd in d > 3 is computationally demanding.
- Liquid dynamics in finite dimensions: Finite-dimensional simulations reproduce two-step relaxation, exponent scaling, and growing dynamical susceptibility in d = 3, while higher dimensions extend the observed τα power-law regime.The scaling range grows from barely more than a decade in d = 3 to nearly three decades in d = 8.
3. FOLLOWING GLASSES UNDER SLOW COMPRESSION
The review translates glass dynamics under density changes into metastable-state thermodynamics, using the Franz-Parisi potential to characterize trapping, equations of state, and transitions. It then compares infinite-dimensional predictions with finite-dimensional preparation and dynamics, finding robust hysteresis but unresolved status for the Gardner crossover.
- 3.1. Metastable states: In d →∞, vanishing diffusivity traps the dynamics near an initial configuration, defining a metastable state despite finite-dimensional escape.The long-time displacement remains finite in the infinite-dimensional limit, whereas finite-dimensional residence is only metastable.
- 3.2. The Franz-Parisi restricted free energy and the order parameters: The Franz-Parisi potential characterizes metastability through Δr: a finite-distance local minimum indicates trapping, while its absence indicates a liquid.The auxiliary order parameter Δ is selected by free-energy minimization, whereas Δr has the direct interpretation as the distance between configurations.
- 3.2. The Franz-Parisi restricted free energy and the order parameters: Differentiating the Franz-Parisi construction yields glass equations of state and predicts hysteresis during decompression.For fixed preparation density, the pressure is obtained by following the adiabatic evolution of metastable states.
- 3.2. The Franz-Parisi restricted free energy and the order parameters: At equilibrium-glass density, the glass pressure analytically continues the liquid pressure beyond bϕd while the system remains frozen in metastable states.This identifies the continued liquid line with glassy states sharing the pressure that the equilibrium liquid would have at the same density.
- 3.4. Gardner transition: At the Gardner transition, the metastable basin becomes a hierarchically organized metabasin, accompanied by divergent τβ, χ, and ξG.The reported scalings are τβ ∼ |bϕ − bϕG|^-a and χ ∼ |bϕG − bϕ|^-1.
- 3.6. Gardner transition and marginal glass in finite dimensions: Finite-dimensional simulations show a sharper Gardner-related crossover at larger bϕg, but larger systems and longer timescales are needed to establish a true thermodynamic transition.Alternative preparation methods can access equilibrium glasses and validate robust hysteresis features, while the Gardner transition remains less settled.
4. JAMMING
The jamming endpoint of compressed glass states occurs at diverging pressure within the marginal glass phase, while finite-dimensional studies find robust, largely universal critical behavior despite protocol-dependent jammed densities. Mean-field predictions agree closely with simulations, but rattlers, bucklers, and localized excitations expose effects beyond the d →∞ solution.
- Mean-field jamming: Jamming occurs at p →∞ within the marginal glass phase, after a Gardner transition from a normal glass at finite pressure.Different compressions produce different jammed configurations, and the jamming density depends on the glass preparation density.
- Mean-field jamming: Jamming critical behavior is universal despite strong protocol dependence, with force and gap distributions following power laws.The exact solution predicts Z ∼ dN and irrational force and gap exponents near jamming.
- Vibrational modes: The d →∞ solution predicts excess low-frequency vibrational modes with a constant low-frequency density of states, whose extended structure does not resemble plane waves.These modes form a Boson peak relative to the Debye model.
- Finite-dimensional robustness: Finite-dimensional simulations confirm isostaticity, observing Z = dN(1 − fratt(d)) + O(d), with rattlers producing an exponentially small correction.Boundary conditions contribute an O(d) correction, while rattlers are excluded from structural analyses because their fraction vanishes exponentially with dimension.
- Finite-dimensional robustness: The small-gap exponent is γ = 0.40(4) for d = 2 to 12, independently of preparation protocol, and agrees with the d →∞ prediction.This anomalous power-law scaling concerns small interparticle voids in jammed hard-sphere configurations.
- Finite-dimensional robustness: Separating bucklers from the rest of the force network yields θe = 0.40(4) and θℓ = 0.18(2), consistent with mean-field and mechanical-marginality predictions.Bucklers generate quasi-localized excitations, and their exponentially vanishing fraction makes them nonperturbative in dimension.
5. SAMPLING GLASSY STATES
The review develops generalized measures for sampling hard-sphere glassy states at fixed density and pressure, then compares Edwards and adiabatic-compression ensembles. The resulting phase diagram distinguishes equilibrium, Gardner, threshold, and jammed regimes, while showing both universal and protocol-sensitive features.
- 5.1. Effective temperature: A generalized partition measure samples glassy states with an effective temperature parameter m, recovering equilibrium weighting at m = 1.For m ≠ 1, states receive weights different from their equilibrium values; configurational entropy identifies the dominant states.
- 5.1. Effective temperature: At fixed density and m, dominant glass states share a pressure p(bϕ, m), allowing inversion to construct a measure essentially uniform over states with fixed bϕ and p.The replica method provides the configurational entropy and the associated phase diagram.
- 5.2. Equilibrium sampling: The equilibrium line follows the liquid pressure, while the configurational entropy vanishes at bϕK ∼ log d, marking the Kauzmann transition in infinite dimension.The relevance of the infinite-dimensional solution does not depend on whether an ideal thermodynamically stable glass exists in finite dimension.
- 5.3. The Gardner line: The Gardner line separates stable glasses from marginally stable glass states, with distinct glass-state groups sampled at each point in the (bϕ, p) plane.Moving through this plane does not adiabatically follow one fixed group of states.
- 5.4. Jamming line: the Edwards ensemble: The Edwards ensemble uniformly weights stable jammed states at fixed density, and its jammed states are entirely within the marginal glass phase.The critical exponents obtained from this ensemble match those from adiabatic state following, motivating a conjecture of universal jamming critical properties.
- 5.4. Jamming line: the Edwards ensemble: Adiabatic compression samples exponentially fewer jammed packings than the Edwards ensemble and reaches a smaller density range.Its configurational entropy is systematically below the Edwards configurational entropy.
- 5.5. The threshold and aging dynamics after a crunch: Threshold states form two marginally stable branches: normal glasses on the first and hierarchically subdivided states on the second.Unlike deep Gardner states, threshold metabasins themselves have nearly flat connecting directions that enable aging.
- 5.6. Out-of-equilibrium glasses in finite dimension: The structural properties of jammed configurations are protocol-invariant, but final density depends on algorithmic details and rattler variations remain poorly understood.Experiments and simulations have not matched the Edwards measure even under constrained preparation conditions, and incorporating proposed explanations into the exact solution remains unfinished.
6. ONGOING AND FUTURE DIRECTIONS
The review identifies unresolved questions about finite-dimensional rheology, interactions beyond hard spheres, and the theoretical origin and formal development of jamming universality. These gaps concern both validation of predictions and the scope of the mean-field description.
- Rheology: Finite-dimensional rheology remains insufficiently assessed, including predictions that Gardner transitions precede yielding and marginality coincides with system-spanning avalanches.Direct experimental or computational validation of these predictions has not yet been obtained.
- Beyond hard spheres: The hard-sphere predictions are expected to apply broadly across liquid interactions, but low-temperature glass behavior may be less universal.In soft spheres, marginality eventually disappears when compression proceeds sufficiently far above jamming.
- Open theoretical questions: The dimensional robustness of jamming remains puzzling because its physical origin lacks a fully satisfying explanation, while finite-dimensional analyses and renormalization-group treatments remain incomplete.The infinite-dimensional solution also suggests packing bounds tighter than currently known rigorous results.
DISCLOSURE STATEMENT
The review reports no affiliations, memberships, funding, or financial holdings that might be perceived as affecting its objectivity.
- The authors report no affiliations, memberships, funding, or financial holdings perceived as affecting the review’s objectivity.