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Hopping and the Stokes-Einstein relation breakdown in simple glass formers

Patrick Charbonneau, Yuliang Jin, Giorgio Parisi, Francesco Zamponi

arXiv:1407.5677v2cond-mat.dis-nncond-mat.stat-mech

TL;DR

Finite-dimensional glass formers depart from the ideal mean-field picture, and the role of hopping in that departure is difficult to isolate. The paper combines cavity reconstruction, theory, and simulations in the MK model to characterize hopping. It finds hopping breaks the Stokes–Einstein relation and supersedes the dynamical transition, while leaving part of the critical regime intact.

  • Problem

    Finite-dimensional glass formers exhibit VTF viscosity scaling and Stokes–Einstein breakdown unlike the ideal mean-field scenario, complicating assessment of mean-field theory for experiments.

    Source: Introduction - · pp. 1–2

  • Method

    The paper isolates hopping in the MK model using cavity reconstruction to characterize cages and hopping, alongside molecular dynamics.

    Source: Introduction - · pp. 2–3, Caging - · p. 4

  • Results

    Hopping breaks the Stokes–Einstein relation and prevents full arrest above the fitted transition, while the critical regime remains intact below the crossover.

    Source: Hopping · pp. 5–6

  • Takeaways & Limitations

    In the MK model, single-particle hopping suffices for Stokes–Einstein breakdown, while facilitation likely amplifies the effect in hard spheres.

    Source: Hopping · p. 6

    In the MK model, VTF fits have no associated Adam–Gibbs relation, and their ϕ0 has no thermodynamic meaning or connection to a Kauzmann transition.

    Source: Hopping · p. 7

Abstract

from arXiv · show

One of the most actively debated issues in the study of the glass transition is whether a mean-field description is a reasonable starting point for understanding experimental glass formers. Although the mean-field theory of the glass transition -- like that of other statistical systems -- is exact when the spatial dimension $d\rightarrow\infty$, the evolution of systems properties with $d$ may not be smooth. Finite-dimensional effects could dramatically change what happens in physical dimensions, $d=2,3$. For standard phase transitions finite-dimensional effects are typically captured by renormalization group methods, but for glasses the corrections are much more subtle and only partially understood. Here, we investigate hopping between localized cages formed by neighboring particles in a model that allows to cleanly isolate that effect. By bringing together results from replica theory, cavity reconstruction, void percolation, and molecular dynamics, we obtain insights into how hopping induces a breakdown of the Stokes--Einstein relation and modifies the mean-field scenario in experimental systems. Although hopping is found to supersede the dynamical glass transition, it nonetheless leaves a sizable part of the critical regime untouched. By providing a constructive framework for identifying and quantifying the role of hopping, we thus take an important step towards describing dynamic facilitation in the framework of the mean-field theory of glasses.

Introduction -

Finite-dimensional glass formers show VTF viscosity scaling and Stokes–Einstein breakdown unlike the ideal mean-field scenario, while several finite-dimensional processes complicate quantitative interpretation. The paper uses the Mari–Kurchan model to isolate hopping and assess its effect on caging and transport.

Introduction -

  • Glass theory is difficult because standard low-density and harmonic expansions fail for dense amorphous materials lacking reference equilibrium particle positions.The paper identifies a need for a framework that can describe their dynamics.Source: Introduction - · p. 1
  • Finite-dimensional systems show VTF viscosity scaling and Stokes–Einstein breakdown, unlike the ideal mean-field scenario, leaving the relevance of that picture for experiments debated.The ideal scenario predicts a sharp transition, divergent viscosity, and the Stokes–Einstein relation.Source: Introduction - · pp. 1–2
  • Finite-dimensional corrections include critical fluctuations, hopping and facilitation, crystal nucleation, and finite lifetimes of metastable glass states.These processes can alter the mean-field transition and transport behavior.Source: Introduction - · p. 2
  • The Mari–Kurchan model isolates hopping in a finite-dimensional mean-field setting, using cavity reconstruction to compute cage-size distributions and separate hopping processes.Its quenched disorder eliminates crystallization and suppresses coherent activated glass-glass nucleation, while finite neighbor counts leave hopping corrections possible.Source: Introduction - · p. 2
  • The model’s simple liquid structure and accessible equilibrated configurations above the dynamical transition enable systematic study of caging across that transition.Planting enables equilibrated configurations at all densities, and cavity reconstruction supports caging studies beyond ϕd.Source: Introduction - · pp. 2–3

Caging -

In the MK model, finite-dimensional caging departs from ideal mean-field predictions: fitted transition densities and Stokes–Einstein breakdown reflect hopping, which also prevents full arrest above the fitted transition. Cavity reconstruction characterizes the cages and hopping processes behind these effects.

Caging -

  • The mean-square displacement separates into ballistic, caging, and long-time diffusive regimes, with cage size set by the square root of the plateau value.The plateau emerges progressively from an inflection point to a full caging regime as density rises from ˜ϕonset.Source: Caging - · p. 3
  • The fitted ˜ϕd differs from the Gaussian-theory ϕd, and a steady drift persists above ˜ϕd, showing the apparent transition is not full dynamical arrest.The two transition estimates move closer with increasing dimension.Source: Caging - · p. 3
  • The Stokes–Einstein relation holds from caging onset to ϕSER, then breaks down with ω ≈ 0.22; ϕSER approaches ϕd and ˜ϕd as dimension increases.The crossover time τD(ϕSER) also increases with dimension, while the MSD plateau’s steady drift above ˜ϕd diminishes.Source: Caging - · p. 3, Figure 1 · p. 3
  • Cavity reconstruction yields similar cage-size distributions for Gaussian and ball cage shapes, reasonably approximated by a gamma distribution above ϕd.The mean cage size agrees quantitatively with analytical predictions, including a square-root singularity near ϕd.Source: Caging - · p. 4, Figure 2 · p. 4
  • Cavity reconstruction’s single-cage fixed point requires removing hopping segments; connected cages can percolate, so bounded particle motion requires density above the void-percolation threshold ϕp.The method’s single-cage assumption is therefore tied to void percolation.Source: Caging - · p. 4
  • Hopping quickly dominates as density rises above ϕd, and its time distribution is power-law-like while the characteristic hopping time grows roughly exponentially with density.MD and single-particle cavity estimates of hopping-time distributions largely coincide, indicating weaker facilitation than in standard finite-dimensional hard spheres.Source: Caging - · p. 4, Figure 3 · p. 5

Hopping

In the MK model, hopping cuts off the critical dynamical slowdown and accounts for Stokes–Einstein relation breakdown, while leaving an earlier critical-like regime intact. The section connects this behavior to cage-network structure, its dimensional dependence, and comparisons with hard spheres.

Hopping

  • Hopping lets particles escape cages and diffuse, cutting off the critical divergence of τD; MK dynamical data fit a VTF form better than a critical power law.The fitted ϕ0 lies between ϕd and ϕp and has no direct static interpretation.Source: Hopping · p. 5, Figure 4 · p. 5
  • Cage networks span the system for ϕd < ϕ < ϕp, become finite above ϕp, and allow particles to explore multiple cages despite bounded single-particle MSD.Perfect single-cage trapping occurs only as ϕ → ∞ in finite dimensions; near ϕd, hopping detection becomes sensitive to the algorithm's threshold.Source: Hopping · p. 5
  • As dimension grows, the fitted and iRFOT transition densities converge, while the cage-network topology limits escape channels and entropically suppresses hopping.The channel cross-section is expected to stay roughly constant; the fraction of cage surface covered by channels vanishes as dimension increases.Source: Hopping · p. 6, Figure 5 · p. 6
  • After hopping is identified, the SER holds in the critical iRFOT regime, while deviations from its extrapolated scaling coincide with SER breakdown attributed to hopping.As dimension increases, ϕSER approaches ϕd, consistent with convergence toward the idealized mean-field behavior.Source: Hopping · p. 6
  • For hard spheres, hopping near ϕSER is consistent with MK-like hopping, but structural voids enhance correlated hopping and cooperativity at higher densities.The authors associate the reduction in measured ω with a delayed onset of hopping as dimension increases.Source: Hopping · p. 6

1. The model

  • The MK model uses quenched random shifts in pair distances, which suppress crystallization and make hopping less correlated than in hard spheres.The shifts also make the model suitable for cavity analysis; unlike hard spheres, its neighboring particles can overlap with one another.Source: §1 · pp. 8–9

a. Molecular dynamics simulations

  • The molecular-dynamics simulations use event-driven dynamics for MK in d = 2–6 with N = 4000, and hard-sphere data in d = 4–8 with N = 8000.The d = 3 hard-sphere simulations use a binary mixture to prevent crystallization.Source: a. Molecular dynamics simulations · p. 10

b. Planting

Planting makes it possible to study equilibrated MK liquids at densities that are difficult to reach by conventional equilibration, while the section also defines the model’s cage-size observables. Its equilibrium validity depends on liquid-phase equivalence of annealed and quenched averages.

b. Planting

  • Planting switches the order of sampling particle positions and constraints, making the annealed average a practical alternative to slowly equilibrating fixed-disorder samples.The procedure draws ideal-gas positions, then independently samples compatible pair constraints before evolving the system by molecular dynamics.Source: b. Planting · p. 10
  • Annealed and quenched averages coincide when disorder-induced partition-function fluctuations are weak, as in the liquid, but this condition fails in glass states away from the equilibrium liquid line.For the MK model, the liquid entropy exceeds the finite glass entropy as N →∞, supporting the annealed average.Source: b. Planting · p. 11
  • Planted simulations reproduce the liquid equation of state and maintain stable pressure over time, checking that the initial configurations represent equilibrium.The figure also shows planted states at ϕ = 1.78 with the correct equilibrium pressure from t = 0.Source: b. Planting · p. 11, Figure 7 · p. 11

a. Mean square displacement (MSD) and cage sizes

  • The MSD has ballistic, caging, and diffusive regimes, and its plateau defines the mean cage size when it becomes stable.The individual cage size ∆i can be measured in simulations and is twice the variance of particle positions within a cage; Gaussian and ball cage forms are used in the theoretical methods.Source: a. Mean square displacement (MSD) and… · p. 12

b. Characteristic timescales

The paper defines characteristic times by the motion they track and by rescaling collapses. The tailored relaxation-time measure uses typical particle displacement because the MK model lacks a well-defined structure-factor peak.

b. Characteristic timescales

  • τ0 is the microscopic reference time defined by the velocity autocorrelation reaching 1/e.The characteristic timescale τh denotes the typical time for a caged particle to escape.Source: b. Characteristic timescales · pp. 12–13
  • τD = σ^2/D is determined from MSD collapse in the caging and diffusive regimes, allowing estimates near ϕd even when diffusion is unreachable.τα is proportional to viscosity in very sluggish fluids, and its breakdown of the Stokes–Einstein relation is consistent with an unexpectedly large proportion of fast particles.Source: b. Characteristic timescales · pp. 12–13
  • Because the MK model has no well-defined structure-factor peak, τα is estimated from collapse of the typical displacement in the MSD caging regime.The typical displacement tracks the majority of particles and is measured using z = 0.1, close to the z →0 limit.Source: b. Characteristic timescales · p. 13
  • τα is consistent with the viscosity timescale in sluggish fluids, while the SER breakdown is consistent with a larger-than-expected proportion of fast particles.The definitions of τα and τD weight slower and faster particles, respectively.Source: b. Characteristic timescales · p. 13

c. Characteristic densities

The section defines characteristic densities for the onset of caging, hopping-related SER breakdown, and dynamical arrest. It distinguishes the theoretically defined ϕd from the fit-dependent effective threshold ˜ϕd and reports the dimensional scope of the data.

c. Characteristic densities

  • The MSD-based onset density ˜ϕonset marks the lower limit of caging and likely underestimates the onset inferred from finite configurational entropy.The onset is associated with an MSD inflection point and non-Fickian behavior.Source: c. Characteristic densities · p. 13
  • Above ϕSER, hopping is faster than the MCT timescale τD, and both MCT scaling and the SER are violated.Below ϕSER, hopping is irrelevant or indistinguishable from regular liquid dynamics, and MCT scaling relations hold.Source: c. Characteristic densities · p. 13
  • The fitted effective threshold ˜ϕd is systematically higher than ϕd and depends on the density range used for the power-law fit.The theoretical ϕd is determined by replica calculations for MK; for HS it is fitted from diffusivity in the regime where hopping is negligible.Source: c. Characteristic densities · p. 14
  • Below ϕp, particles can diffuse by successive hops through the percolating cage network, so ϕp is an upper bound on ϕd.The infinite-time MSD is bounded only above ϕp.Source: c. Characteristic densities · p. 14
  • For the MK model, the Kauzmann density ϕK diverges in the thermodynamic limit because its configurational entropy includes a term proportional to ln N.The VTF-fit parameter ˜ϕ0 is distinct from ϕK = ∞.Source: c. Characteristic densities · p. 14
  • The reported data cover MK in d = 2–6 and HS in d = 3–8; no SER violation is detected for HS in d = 8 within the accessible dynamical regime.The table caption identifies the tabulated quantities as characteristic densities and MCT exponents, but does not provide their cell values.Source: c. Characteristic densities · p. 15, Table I · p. 15

a. Calculation of the mean caging order parameter: the replica method

The replica calculation adapts prior hard-sphere treatments to the MK model, where neglecting higher-order structural correlations is exact. It obtains the cage size from a stationary replicated entropy and identifies the dynamical transition from the resulting cage equation.

a. Calculation of the mean caging order parameter: the replica method

  • The replica treatment of the replicated entropy applies to MK because its liquid structure makes neglect of third- and higher-order correlations exact.The derivation relies only on the liquid pair correlation function.Source: a. Calculation of the mean caging… · p. 15
  • In odd dimensions, the integral used to evaluate the Gaussian-cage convolution can be computed analytically, facilitating numerical evaluation.The convolution uses the d-dimensional Gaussian cage and a modified Bessel function.Source: a. Calculation of the mean caging… · p. 15
  • The cage parameter A is obtained by setting the derivative of the replicated entropy with respect to A to zero.The liquid cage radius follows by solving the equation in the m →1 limit, with mean cage size ¯∆ = 2dA.Source: a. Calculation of the mean caging… · p. 16
  • The dynamical transition ϕd occurs when the maximum of the cage-equation function reaches its threshold value.Above ϕd, the equation has two solutions, of which only the smaller is a stable physical solution.Source: a. Calculation of the mean caging… · p. 16

b. Calculation of the cage size distribution: the cavity method

The cavity method represents glassy cages through a distribution of localized cavity fields and reconstructs that distribution self-consistently. Gaussian or ball cage-shape ansätze make the calculation tractable, while excluding hopping from sampled cages enables convergence beyond the dynamical transition.

d. Reconstruction equations –

  • The reconstruction equations use the equilibrium choice m = 1 to map the 1RSB cavity equations onto fields localized around a point.The free parameter m selects glassy states by free volume or internal entropy; equilibrium Gibbs-Boltzmann weighting requires m = 1.Source: b. Calculation of the cage size… · p. 17

e. Ansatz on the cage shape –

  • A fixed spherical cage-shape ansatz, chosen as Gaussian or ball, reduces the cavity fields to a distribution of cage sizes.The cage centers are assumed uniformly distributed, and the cage sizes follow Pf(A).Source: b. Calculation of the cage size… · p. 17

f. Reconstruction procedure –

  • Repeatedly sampling neighboring cages and updating the new cavity field yields a self-consistent Pf(A), from which Pf(∆) follows.The procedure iterates until the cage-size distribution converges; the mean-square-displacement distribution is then computed from it.Source: b. Calculation of the cage size… · p. 18

g. Numerical details –

  • Removing hopping trajectories from the cage-size statistics allows the cavity solution to converge above ϕd, whereas unfiltered reconstruction diverges near ϕd.Anew is included only when no hopping is detected during the local Monte Carlo trajectory.Source: b. Calculation of the cage size… · p. 19, Figure 9 · p. 19

c. Comparing theoretical predictions with simulations

Replica and cavity predictions are generally consistent with molecular-dynamics cage measurements, with hopping complicating comparisons near ϕd. MCT scaling describes the full time regime only below ϕSER; above it, agreement is confined to early β-relaxation.

c. Comparing theoretical predictions with simulations

  • Replica and cavity calculations agree with MD cage measurements, but hopping makes cage sizes ambiguous and produces deviations near ϕd.The consistency holds for replica calculations with Gaussian cages and a δ-function cage-size distribution, and cavity calculations with Gaussian or ball ansätze.Source: c. Comparing theoretical predictions… · pp. 19–20, Figure 10 · p. 20

2. Caging dynamics: mode-coupling theory (MCT) and beyond

  • MCT scalings match the full time regime only below ϕSER; above it, agreement remains only in early β-relaxation as hopping mixes with MCT dynamics.The comparison concerns general schematic MCT scaling laws, not the traditional MCT kernel, which is unsuitable for the MK model.Source: §2 · p. 20

a. Testing the mode-coupling theory

The tests show that MCT scaling is reliable only below ϕSER, with later-time deviations above that density. The analysis compares predicted MSD relaxation regimes against simulations using parameters obtained from replica and diffusivity fits.

Testing MCT

  • MCT divides MSD relaxation into early β-approach to the plateau, late β-departure, and subsequent diffusion.The early regime scales as t^-a, the late regime as t^b, and diffusion is linear in time.Source: a. Testing the mode-coupling theory · p. 21
  • The tests fix ϕd and ∆d from replica theory, derive exponents from diffusivity fits, then compare MSD time dependence below and above ϕd.The characteristic exponents a and b are obtained from γ through the MCT exponent relations.Source: a. Testing the mode-coupling theory · p. 21
  • Above ϕSER, MCT scaling fails: diffusivity relaxation departs from its predicted power law, and only early β-relaxation remains consistent.The fitted apparent power law over the full density range is not reliable because MCT fails above ϕSER.Source: a. Testing the mode-coupling theory · p. 21

b. Breakdown of the Stokes-Einstein relation

At ϕSER, SER breakdown coincides with MCT-scaling violations and a hopping time comparable to the diffusion time. The cage network also undergoes a void-percolation transition, separating system-spanning hopping paths from local clusters.

b. Breakdown of the Stokes-Einstein relation

  • At ϕSER, SER breaks down with ω = 0.22, MCT scalings fail, and the hopping time becomes comparable to the diffusion time.The exponent ω is reported as invariant with dimension d.Source: b. Breakdown of the Stokes-Einstein… · p. 22

3. Percolation of the cage network

  • Below ϕp, cage connections span the system; above ϕp, only local clusters remain and particles are strictly confined.The transition maps to void percolation in the same universality class as regular percolation; the d = 3 estimate is ϕp = 2.4.Source: §3 · p. 22, Figure 12 · p. 23

a. Mapping the glass transition to a void percolation transition

  • With frozen neighbors, hopping paths become paths through leftover void after rescaling neighbor sizes to 2σ; allowing cage motion gives a best-case upper bound.The mapping assumes ball-shaped cages, since Gaussian cages retain a finite, though vanishingly small, probability of hopping even when mapped as closed.Source: a. Mapping the glass transition to a… · p. 22

b. Methodology for determining the void percolation threshold

The method maps void space into a Voronoi-derived network and estimates its percolation threshold with finite-size scaling adapted to density-dependent cage polydispersity. The measured mean cluster-size scaling agrees with the lattice-percolation exponent.

b. Methodology for determining the void percolation threshold

  • Void-space connectivity is tested on a network whose nodes are Voronoi vertices and whose links are unblocked Voronoi-polyhedron edges.A path from the network center to its boundary indicates percolation.Source: b. Methodology for determining the… · p. 23
  • Polydisperse cage spheres are handled by mapping radical Voronoi tessellation in d dimensions to ordinary Voronoi tessellation in d + 1 dimensions.The construction maps back to the radical Voronoi cells of the original configuration.Source: b. Methodology for determining the… · p. 24
  • Because cage-size polydispersity varies with ϕ, the method rescales spheres with a density-dependent factor K and uses finite-size scaling to obtain Kp(ϕ).For each density, the threshold is found from 1000 realizations across system sizes; ϕp is where Kp(ϕp) = 1.Source: b. Methodology for determining the… · p. 24
  • The mean cluster size scaling agrees with γp = 1.8, supporting a shared universality class with lattice percolation.The mean cluster size of cages connected to the planted central cage is examined at the threshold.Source: b. Methodology for determining the… · pp. 24–25
  • The procedure obtains cage-size distributions from the cavity method, plants compatible configurations, and repeats threshold searches across system sizes and densities.Percolation is tested after rescaling the particle sizes and constructing the Voronoi network.Source: b. Methodology for determining the… · p. 25

Appendix C: Hopping

  • The hopping appendix describes event detection and phenomenology, while leaving theoretical investigations for future study..Source: Appendix C · p. 25

1. Detecting hopping

Hopping is detected by splitting particle trajectories and comparing the two segments against a particle-specific cage-size threshold. The procedure identifies distinct cage escapes in the trajectory time series.

1. Detecting hopping

  • The detector finds the time t* maximizing the mean-square distance between the trajectory's two time-separated subsets.It first saves trajectories, determines cage sizes, and splits each trajectory at candidate times.Source: §1 · p. 25
  • A hop is detected when the maximum split-trajectory displacement δi exceeds particle i's cage size Δi, after which the test is repeated recursively on each segment.The algorithm relaxes the assumption that all particles have equal-sized cages.Source: §1 · pp. 25–26
  • In the d = 2 example at ϕ = 2.40, the trajectory shows two cages and detected hops at t = 3948.0 and t = 7863.6.The hopping times are marked by dotted lines in the time series.Source: Figure 13 · p. 26

2. Hopping dynamics

Detected hopping times follow a power-law distribution with exponent below one. The characteristic hopping timescale grows roughly exponentially with density, suggesting it has no diverging density.

2. Hopping dynamics

  • Detected hopping times follow a power-law distribution with exponent μ < 1.The authors express its cumulative distribution function as Gh(t) = (t/τh)^(1−μ).Source: §2 · p. 26
  • The characteristic hopping timescale τh is roughly an exponential function of ϕ and depends on density.τh is the time needed for all particles to hop.Source: §2 · p. 26
  • The observed density dependence suggests that τh has no diverging density..Source: §2 · p. 26
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