Supercooled liquids: Equivalence between mode-coupling theory and the replica approach

Abstract

We show that the replica approach to glassy dynamics provides, in spite of its static nature, a characterization of critical dynamics in the β regime of supercooled liquids that is equivalent to the one of mode-coupling-theory, both qualitatively and quantitatively. The nature and extent of this equivalence is discussed in connection to the main open problems of the current theory.

Quasi-equilibrium in glassy dynamics: a liquid theory approach

“…be derived the expression for the exponent parameter whose expression coincides with the one derived in [28]. We notice that in [29] MCT has been generalized to describe aging below T d .…”

Qualitative features at the glass crossover

“…On the low-temperature side, the replica method describes the properties of the (metastable) glassy states below T c [9,10] yielding fewer but similar quantitative predictions [11,12]. Random-first-order transition (RFOT) theory [13,14] builds on top of replica results by advocating entropic nucleation processes to restore ergodicity below T c and predicts a debated divergence of a correlation length below the calorimetric glass transition.…”

“…The standard approach to step a) in supercooled liquids is a replicated field theory, motivated by the discovery that a class of spin-glass models exhibits the MCT phenomenology [38,42]. Over the last decades, this connection has been substantiated by studies both within the original MCT context [43][44][45], and within approaches in the replica cloned-liquids framework that also start from first-principle microscopic calculations [46,47]. The connection between microscopic theory and effective field theory, step a), involves approximations.…”

Long-wavelength fluctuations lead to a model of the glass crossover

“…This theory arises naturally in the context of the socalled one-step-Replica-Symmetry-Breaking (1RSB) Spin-Glass (SG) models and its relevance for structural glasses is motivated by the discovery that these SG systems are controlled by the very same MCT equations ( 3) and (4) [23,24]. We refer the reader to [25] for an extensive presentation of the above replicated action in the context of structural glasses and to [26,27] for a discussion of the close relationship between static replica methods and MCT. The above static action makes sense only in the glassy phase τ > 0 where it can be extremized by the a RS field constant in space φ ab (x) = φ given by the solution of the equation of state:…”

“…λ) that are controlled by the constants that instead remain finite at T c . Besides λ and τ , the variance of the random temperature can be obtained by reading the coefficients m 2 and m 3 from the static replica formulation of MCT [26,27]. Less trivial is the computation of the coefficient of the Laplacian that should be estimated by means of the inhomogeneous MCT extension discussed in [33].…”