Systematic expansion in the order parameter for replica theory of the dynamical glass transition

Abstract

It has been shown recently that predictions from Mode-Coupling Theory for the glass transition of hard-spheres become increasingly bad when dimensionality increases, whereas replica theory predicts a correct scaling. Nevertheless if one focuses on the regime around the dynamical transition in three dimensions, Mode-Coupling results are far more convincing than replica theory predictions. It seems thus necessary to reconcile the two theoretic approaches in order to obtain a theory that interpolates between low-dimensional, Mode-Coupling results, and "mean-field" results from replica theory. Even though quantitative results for the dynamical transition issued from replica theory are not accurate in low dimensions, two different approximation schemes --small cage expansion and replicated Hyper-Netted-Chain (RHNC)-- provide the correct qualitative picture for the transition, namely a discontinuous jump of a static order parameter from zero to a finite value. The purpose of this work is to develop a systematic expansion around the RHNC result in powers of the static order parameter, and to calculate the first correction in this expansion. Interestingly, this correction involves the static three-body correlations of the liquid. More importantly, we separately demonstrate that higher order terms in the expansion are quantitatively relevant at the transition, and that the usual mode-coupling kernel, involving two-body direct correlation functions of the liquid, cannot be recovered from static computations.

19 pages, 8 figures, contribution to a J. Chem. Phys. special issue on the Glass Transition

References in corpus (26)

• Theoretical perspective on the glass transition and amorphous materials
• Mean field theory of hard sphere glasses and jamming
• The structural glass transition and the entropy of the metastable states
• Thermodynamics of glasses: a first principle computation
• Mode-Coupling Theory (MCT) Lecture Notes
• A first principle computation of the thermodynamics of glasses
• Exact theory of dense amorphous hard spheres in high dimension. I. The free energy
• The ideal glass transition of Hard Spheres
• A tentative Replica Study of the Glass Transition
• Statistical Physics of Structural Glasses
• Locally preferred structures and many-body static correlations in viscous liquids
• Dimensional Study of the Caging Order Parameter at the Glass Transition
• Glass transition and effective potential in the hypernetted chain approximation
• Dynamical field theory for glass-forming liquids, self-consistent resummations and time-reversal symmetry
• Quantitative field theory of the glass transition
• On the critical slowing down exponents of mode coupling theory
• Mode-Coupling as a Landau Theory of the Glass Transition
• Do current-density nonlinearities cut off the glass transition?
• Generic replica symmetric field-theory for short range Ising spin glasses
• One Dimensional Kondo Lattice Model Studied by the Density Matrix Renormalization Group Method
• Dynamic glass transition: bridging the gap between mode-coupling theory and the replica approach
• Dynamics of interacting Brownian particles: a diagrammatic formulation
• Mode coupling theory in the FDR-preserving field theory of interacting Brownian particles
• Static replica approach to critical correlations in glassy systems
• Smoluchowski dynamics and the ergodic-nonergodic transition
• Field theoretic formulation of a mode-coupling equation for colloids

Cited by in corpus (10)

• Exact theory of dense amorphous hard spheres in high dimension. II. The high density regime and the Gardner transition
• Static triplet correlations in glass-forming liquids: A molecular dynamics study
• Static replica approach to critical correlations in glassy systems
• Brownian dynamics: from glassy to trivial
• A self-consistent field theory of density correlations in classical fluids
• Breakdown of a renormalized perturbation expansion around mode-coupling theory of the glass transition
• Replica Field Theory for a Generalized Franz--Parisi Potential of Inhomogeneous Glassy Systems: New Closure and the Associated Self-Consistent Equation
• Universal Longtime Dynamics in Dense Simple Fluids
• Free-energy functional of instantaneous correlation field in liquids: field-theoretic derivation of the closures
• An alternative, dynamic density functional-like theory for time-dependent density fluctuations in glass-forming fluids