Abstract
We investigate the dimensional dependence of dynamical fluctuations related to dynamic heterogeneity in supercooled liquid systems using kinetically constrained models. The d-dimensional spin-facilitated East model with embedded probe particles is used as a representative super-Arrhenius glass forming system. We examine the existence of an upper critical dimension in this model by considering decoupling of transport rates through an effective fractional Stokes-Einstein relation, D∼τ-1+ω, with D and τ the diffusion constant of the probe particle and the relaxation time of the model liquid, respectively, and where ω>0 encodes the breakdown of the standard Stokes-Einstein relation. To the extent that decoupling indicates non-mean-field behavior, our simulations suggest that the East model has an upper critical dimension at least above d = 10 and argue that it may actually be infinite. This result is due to the existence of hierarchical dynamics in the East model in any finite dimension. We discuss the relevance of these results for studies of decoupling in high dimensional atomistic models.
Highlights
The East model and its higher dimensional generalizations [1,2,3,4] describe the cooperative relaxation dynamics of glass formers through a simple facilitation mechanism
The theoretical perspective on the glass transition that emerges from the study of the East model and other kinetically constrained models (KCMs), sometimes called dynamic facilitation (DF) theory, is one of fluctuation dominance in the dynamics with a very limited role played by the thermodynamics of glass formers
A recent computational study of hard sphere dynamics in large dimensions [25] tested this prediction by considering the violation of the Stokes-Einstein relation, with numerical results that seemed compatible with an absence of transport decoupling - and mean-field behavior - for dimensions d ≥ 8
Summary
The East model and its higher dimensional generalizations [1,2,3,4] describe the cooperative relaxation dynamics of glass formers through a simple facilitation mechanism. Particle displacements that persist for a significant period of time require correlated motion of several particles inside of what is termed an “excitation.” The concentration of these excitations drops with the temperature, allowing fewer particles to make persistent moves at lower temperature [19] This perspective contrasts with theoretical approaches based on mean-field theory, in particular that of the random first-order transition (RFOT) perspective (see [20, 21] for reviews). We do so by considering the relation between structural relaxation time τ and diffusion rate D, which in the normal liquid state obeys the mean-field like Stokes-Einstein relation (SER), D ∼ τ −1 Departure from this relation, termed transport “decoupling” [27], is a manifestation of fluctuating, non mean-field, dynamics. VI we conclude by connecting our results to the observations in atomistic simulations of Ref. [25]
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