Rigorous Inequalities Between Length and Time Scales in Glassy Systems

By Andrea Montanari, Guilhem Semerjian. 2006

186 citations recorded by the source corpus.

Glassy systems are characterized by an extremely sluggish dynamics without any simple sign of long range order. It is a debated question whether a correct description of such phenomenon requires the emergence of a large correlation length. We prove rigorous bounds between length and time scales implying the growth of a properly defined length when the relaxation time increases. Our results are valid in a rather general setting, which covers finite-dimensional and mean field systems.As an illustration, we discuss the Glauber (heat bath) dynamics of p-spin glass models on random regular graphs. We present the first proof that a model of this type undergoes a purely dynamical phase transition not accompanied by any thermodynamic singularity.

Subfields: Condensed Matter Physics, Economics and Econometrics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/e73bfdcf1d777643764fd159f9417fa628d8bbf3

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Citations in the corpus

Revealing Hidden Structural Order Controlling Both Fast and Slow Glassy Dynamics in Supercooled Liquids (2018, background): “Here it is worth noting that, although not direct, there is a rigorous inequality between the time scale τα and the length scale ξ [26]: τα ≤ τ0 expðconst × ξdÞ, where τ0 is the microscopic time scale, d is the dimensionality of the system, and ξ is the longest correlation length in the system.”
The role of local structure in dynamical arrest (2014, background): “Montanari and Semmerjian imagined a cavity in which re-arrangements are possible, but which is pinned by immobile particles outside in much the same way as the RFOT ideas outlined above [366].”
Theoretical perspective on the glass transition and amorphous materials (2010, background): “Recently, the so-called ‘pointto-set’ correlation lengthscale was defined both in the context of RFOT theory as a practical measure of the mosaic lengthscale (Bouchaud and Biroli, 2004), and in more general settings (Montanari and Semerjian, 2006b; Mézard and Montanari, 2006).”
Comparing dynamics: deep neural networks versus glassy systems (2018, background): “There is a second regime of time-scales, that diverge exponentially with the number of degrees of freedom (Montanari & Semerjian, 2006; Ben Arous & Jagannath, 2017), in which barrier crossing does take place.”
Statistical physics of inference: thresholds and algorithms (2015, background): “number of Monte Carlo steps per variable in the thermodynamic limit) diverges as the dynamical temperature is approached [170] and is exponentially large in the system size between the dynamical and Kauzmann temperature.”
Mean-field theory of hard sphere glasses and jamming (2008, background): “The interested reader should look to the original literature (Bouchaud and Biroli, 2004; Cavagna et al., 2007; Franz, 2005; Montanari and Semerjian, 2006; Xia and Wolynes, 2001a,b).”
Non-monotonic temperature evolution of dynamic correlations in glass-forming liquids (2011, methodology): “One possibility to tackle this problem involves using point-to-set correlations, which are an elegant, general method to capture the multi-point static correlations which might characterize the non-trivial structure of viscous liquid”