Complete theoretical description of the dimensional dependence of the Stokes–Einstein relation and its violation in glass-forming hard spheres

The Stokes–Einstein relation (SER) ties the self-diffusion coefficient of a particle to the inverse viscosity of its host fluid, but in supercooled liquids approaching the glass transition it breaks down: diffusion decouples from viscous relaxation. Charbonneau, Charbonneau, Jin, Parisi and Zamponi (2013) mapped this breakdown, quantified by the scaling $D \propto \tau_\alpha(k)^{-1+\omega(k)}$, across spatial dimensions $d = 3$ to $10$ in hard-sphere glass formers. They found that the violation exponent vanishes roughly linearly, $\omega(k^*) \approx 0.056(8-d)$, around an upper critical dimension $d_u = 8$ — consistent in form with the Biroli–Bouchaud mode-coupling prediction but off by a factor of about two in prefactor ($\approx 0.12(8-d)$) — and that a small violation may persist above $d = 8$. They concluded that the complete theoretical description of the dimensional dependence of the SER violation 'remains an open problem'. The problem is to deliver a quantitatively tested theory of how hopping, facilitation and dynamical heterogeneity generate the fat displacement tails that drive the violation, of how they modify mean-field behavior even above $d_u = 8$, and of why the measured prefactor disagrees with the prediction. Two further open issues of the same source — a microscopic SER-like interpolation between the Enskog kinetic and continuum hydrodynamic limits, and the origin of the renormalized finite-size length scale $\ell_{\mathrm{eff}} > \ell_{\mathrm{hydro}}$ — are background for the dimensional description. The problem remains open.

Background

The Stokes–Einstein relation (SER) has its roots in Einstein's 1905 theory of Brownian motion: a spherical particle of hydrodynamic radius $A$ moving through a fluid of shear viscosity $\eta_S$ at inverse temperature $\beta$ has self-diffusion coefficient $D$ with $\beta D \eta_S = 1/(6\pi A)$ for stick boundary conditions and $\beta D \eta_S = 1/(4\pi A)$ for slip conditions in $d = 3$. Charbonneau, Charbonneau, Jin, Parisi and Zamponi (2013) generalized both the SER and its leading finite-size correction to arbitrary spatial dimension $d$ and tested the generalizations in hard-sphere (HS) fluids — the minimal model of a glass former (a liquid that falls out of equilibrium into an amorphous rigid state upon cooling or compression; crystallization is suppressed by using sphere mixtures) — in $d = 3$ to $10$.

In deeply supercooled liquids the SER breaks down: the product $D \eta_S$ grows by orders of magnitude even though both transport coefficients slow down, i.e. diffusion decouples from viscous relaxation. Because viscosity is expensive to measure in sluggish regimes, the violation is usually quantified through the relaxation time $\tau_\alpha(k)$ of the self-intermediate scattering function $F_s(k,t)$ — the decorrelation of single-particle density at wavevector $k$, whose final (alpha) decay measures structural relaxation — via the scaling

$D \propto \tau_\alpha(k)^{-1+\omega(k)},$

where $\omega(k) > 0$ marks a violation and $\omega = 0$ the SER. Both $D$ and $\tau_\alpha(k)$ are functionals of the self van Hove function $G_s(r,t)$, the distribution of single-particle displacements, so a violation implies that $G_s(r,t_\alpha)$ develops a fat tail: a minority of particles move very far while the typical particle remains caged by its neighbors. This breakdown had been established numerically in $d = 2$–$4$ by Sengupta, Karmakar, Dasgupta and Sastry (2013) and is one of the classic signatures of dynamical heterogeneity — spatially correlated fluctuations of local mobility, measured for instance by the four-point susceptibility $\chi_4$.

The main theoretical reference point is mode-coupling theory (MCT), a mean-field-type theory in which slow dynamics emerges from the feedback of density fluctuations on themselves and which predicts an ideal dynamical transition at a density $\phi_d$. Biroli and Bouchaud (2007) argued that the SER violation is caused by the critical dynamical fluctuations of MCT, whose upper critical dimension (the dimension above which fluctuations no longer change the critical behavior and mean-field theory becomes exact) is $d_u = 8$; they concluded that below $d_u$ the violation exponent should vanish linearly, and using the MCT exponent $\gamma \approx 2.1$ for the relaxation-time divergence their estimate is $\omega \approx 0.12(8-d)$. Modern mean-field theory — the exact solution of glass dynamics in $d \to \infty$ (also called dynamical mean-field theory, DMFT), and the random first-order transition (RFOT) theory built on it, in which relaxation proceeds by activated barriers over a mosaic of amorphous states — provides the background that any finite-$d$ theory must correct.

Against this backdrop, Charbonneau et al. (2013) established the following. (i) The SER does not hold generically for self-diffusion of a solvent-sized particle at any density or dimension; the approximate SER plateau seen in $d = 3$ at intermediate densities is largely accidental. (ii) In the sluggish regime near (but below) the MCT transition density, the SER violation — equivalently, anomalous single-particle displacements — is strongly reduced with increasing $d$ and almost disappears around $d = 8$: at the wavevector $k^*$ of the first peak of the static structure factor, $\omega(k^*) \approx 0.056(8-d)$, qualitatively consistent with Biroli–Bouchaud but off by a factor $\approx 2$ in prefactor; moreover, a small $\omega$ seems to persist even for $d > 8$, hinting at additional mechanisms. (iii) The $k$-dependence of $\omega(k)$ is strong, and the plateau of $\omega(k)$ expected at asymptotically low $D$ is not yet reached in the accessible range. (iv) On the SER side proper, the hybrid boundary condition of Hynes, Kapral and Weinberg (HKW) — a microscopic treatment that bridges Enskog kinetic theory (the finite-density extension of Boltzmann's kinetic theory to hard spheres) and continuum hydrodynamics — overshoots $D$ at intermediate densities in all dimensions, and no quantitative microscopic SER-like relation interpolating between the two limits exists. (v) The leading finite-size correction to $D$ in periodic systems, $D(L) = D_0[1-(\ell/L)^{d-2}]$, involves an effective length $\ell_{\mathrm{eff}}$ that exceeds the hydrodynamic prediction $\ell_{\mathrm{hydro}}$ in the sluggish regime, for reasons that were only speculated upon (a renormalization by growing static and dynamical correlations, in the spirit of the RFOT computation of Xia and Wolynes). The paper's abstract states the resulting challenge: 'The SER violation with $d$ establishes a new benchmark for theory, and its complete description remains an open problem.'

Problem Statement

Charbonneau, Charbonneau, Jin, Parisi and Zamponi (2013) left the complete theoretical description of the Stokes–Einstein relation (SER) violation as a function of spatial dimension $d$ — in their words, 'a new benchmark for theory' whose 'complete description remains an open problem' — and identified the concrete issues such a description must settle. The problem is to deliver that description for simple, hard-sphere-like glass formers: a theory (or precisely specified framework), quantitatively tested against simulation, that resolves the following jointly linked questions.

(1) Exponent and prefactor. Predict the violation exponent $\omega(d)$, in particular the observed linear regime $\omega(k^*) \approx 0.056(8-d)$ below the upper critical dimension $d_u = 8$ ($k^*$ being the wavevector of the first peak of the static structure factor); explain the factor-of-two discrepancy with the Biroli–Bouchaud prediction $\omega \approx 0.12(8-d)$ derived from mode-coupling critical fluctuations with exponent $\gamma \approx 2.1$; and determine whether a small $\omega$ persists above $d = 8$, and why.

(2) Mechanism. Disentangle and quantify the separate contributions of local hopping (activated escape of a single particle from its cage of neighbors), dynamical facilitation (relaxation that can proceed only after neighboring regions have relaxed), and dynamical heterogeneity to the fat tail of the self van Hove function $G_s(r,t_\alpha)$ that drives the violation; in particular, how hopping and facilitation can be described within a field-theoretic/mean-field framework, and how they modify the mean-field behavior even above $d_u = 8$.

(3) Wavevector dependence. Determine the $k$-dependence of $\omega(k)$ and how its plateau sets in at asymptotically low diffusivities — a regime below the equilibration range of current simulations.

These three components are the concrete content of the single objective stated by the source, which concerns the complete dimensional description rather than a restricted case such as a single dimension or one mechanism in a controlled model with no quantified connection to hard spheres across dimensions.

The source separately raises two associated open issues, which are not part of that single objective and are recorded here as context: (i) a quantitative, microscopic SER-like relation for self-diffusion that smoothly interpolates between the Enskog kinetic limit at low density and the continuum hydrodynamic limit at high density, correcting the systematic overestimate of $D$ at intermediate densities produced by the Hynes–Kapral–Weinberg hybrid boundary condition in all dimensions; and (ii) an explanation of why the effective length $\ell_{\mathrm{eff}}$ in the leading finite-size correction $D(L) = D_0[1-(\ell/L)^{d-2}]$ exceeds the hydrodynamic value $\ell_{\mathrm{hydro}}$ in dense, sluggish systems, and of its dependence on state point and dimension. Progress on these associated issues is relevant background for the dimensional description.

References

1. Benoit Charbonneau, Patrick Charbonneau, Yuliang Jin, Giorgio Parisi, Francesco Zamponi, Dimensional dependence of the Stokes–Einstein relation and its violation, J. Chem. Phys. 139, 164502 (2013). DOI: 10.1063/1.4825177, https://doi.org/10.1063/1.4825177 (preprint: arXiv:1210.6073, https://arxiv.org/abs/1210.6073)
2. Giulio Biroli, Jean-Philippe Bouchaud, Critical fluctuations and breakdown of the Stokes–Einstein relation in the mode-coupling theory of glasses, J. Phys.: Condens. Matter 19, 205101 (2007). DOI: 10.1088/0953-8984/19/20/205101, https://doi.org/10.1088/0953-8984/19/20/205101 (preprint: arXiv:cond-mat/0609705, https://arxiv.org/abs/cond-mat/0609705)
3. Shiladitya Sengupta, Smarajit Karmakar, Chandan Dasgupta, Srikanth Sastry, Breakdown of the Stokes–Einstein relation in two, three, and four dimensions, J. Chem. Phys. 138, 12A548 (2013). DOI: 10.1063/1.4792356, https://doi.org/10.1063/1.4792356 (preprint: arXiv:1211.0686, https://arxiv.org/abs/1211.0686)
4. James T. Hynes, Raymond Kapral, Michael Weinberg, Molecular theory of translational diffusion: Microscopic generalization of the normal velocity boundary condition, J. Chem. Phys. 70, 1456–1466 (1979). DOI: 10.1063/1.437584, https://doi.org/10.1063/1.437584
5. Xiaoyu Xia, Peter G. Wolynes, Diffusion and the Mesoscopic Hydrodynamics of Supercooled Liquids, J. Phys. Chem. B 105, 6570–6573 (2001). DOI: 10.1021/jp004616m, https://doi.org/10.1021/jp004616m
6. Patrick Charbonneau, Yuliang Jin, Giorgio Parisi, Francesco Zamponi, Hopping and the Stokes–Einstein relation breakdown in simple glass formers, Proc. Natl. Acad. Sci. USA 111, 15025–15030 (2014). DOI: 10.1073/pnas.1417182111, https://doi.org/10.1073/pnas.1417182111 (preprint: arXiv:1407.5677, https://arxiv.org/abs/1407.5677)
7. Soree Kim, Dayton G. Thorpe, Chanwoo Noh, Juan P. Garrahan, David Chandler, YounJoon Jung, Study of the upper-critical dimension of the East model through the breakdown of the Stokes–Einstein relation, J. Chem. Phys. 147, 084504 (2017). DOI: 10.1063/1.4999791, https://doi.org/10.1063/1.4999791 (preprint: arXiv:1705.00095, https://arxiv.org/abs/1705.00095)
8. Benoit Charbonneau, Patrick Charbonneau, Grzegorz Szamel, A microscopic model of the Stokes–Einstein relation in arbitrary dimension, J. Chem. Phys. 148, 224503 (2018). DOI: 10.1063/1.5029464, https://doi.org/10.1063/1.5029464 (preprint: arXiv:1803.02229, https://arxiv.org/abs/1803.02229)
9. Patrick Charbonneau, Jorge Kurchan, Giorgio Parisi, Pierfrancesco Urbani, Francesco Zamponi, Glass and Jamming Transitions: From Exact Results to Finite-Dimensional Descriptions, Annu. Rev. Condens. Matter Phys. 8, 265–288 (2017). DOI: 10.1146/annurev-conmatphys-031016-025334, https://doi.org/10.1146/annurev-conmatphys-031016-025334 (preprint: arXiv:1605.03008, https://arxiv.org/abs/1605.03008)
10. Monoj Adhikari, Smarajit Karmakar, Srikanth Sastry, Spatial Dimensionality Dependence of Heterogeneity, Breakdown of the Stokes–Einstein Relation, and Fragility of a Model Glass-Forming Liquid, J. Phys. Chem. B 125, 10232–10239 (2021). DOI: 10.1021/acs.jpcb.1c03887, https://doi.org/10.1021/acs.jpcb.1c03887 (preprint: arXiv:2108.08550, https://arxiv.org/abs/2108.08550)
11. Patrick Charbonneau, Yi Hu, Joyjit Kundu, Peter K. Morse, The dimensional evolution of structure and dynamics in hard sphere liquids, J. Chem. Phys. 156, 134502 (2022). DOI: 10.1063/5.0080805, https://doi.org/10.1063/5.0080805
12. Ali Tahaei, Giulio Biroli, Misaki Ozawa, Marko Popović, Matthieu Wyart, Scaling Description of Dynamical Heterogeneity and Avalanches of Relaxation in Glass-Forming Liquids, Phys. Rev. X 13, 031034 (2023). DOI: 10.1103/PhysRevX.13.031034, https://doi.org/10.1103/PhysRevX.13.031034
13. Corentin C. L. Laudicina, Patrick Charbonneau, Yi Hu, Liesbeth M. C. Janssen, Peter K. Morse, Ilian Pihlajamaa, Grzegorz Szamel, Simple Fluctuations in Simple Glass Formers, J. Phys. Chem. B 128, 12237–12249 (2024). DOI: 10.1021/acs.jpcb.4c05447, https://doi.org/10.1021/acs.jpcb.4c05447 (preprint: arXiv:2408.06933, https://arxiv.org/abs/2408.06933)
14. T. Oppelstrup, B. Sadigh, S. Sastry, M. Dzugutov, The origin of the Stokes–Einstein relation in simple dense liquids, J. Chem. Phys. 163, 164117 (2025). DOI: 10.1063/5.0288544, https://doi.org/10.1063/5.0288544
15. Massimo Pica Ciamarra, Jeppe C. Dyre, Edan Lerner, Matthieu Wyart, Theories of the Glass Transition Based on Local Excitations, arXiv:2603.05209 (2026). https://arxiv.org/abs/2603.05209
16. Albert Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys. 322, 549–560 (1905). DOI: 10.1002/andp.19053220806, https://doi.org/10.1002/andp.19053220806