Diverging length scale of the inhomogeneous mode-coupling theory: a numerical investigation

Abstract

Biroli et al.'s extension of the standard mode-coupling theory to inhomogeneous equilibrium states [Phys. Rev. Lett. 97, 195701 (2006)] allowed them to identify a characteristic length scale that diverges upon approaching the mode-coupling transition. We present a numerical investigation of this length scale. To this end we derive and numerically solve equations of motion for coefficients in the small q expansion of the dynamic susceptibility ${\chi }_{\mathbf{q}}(\mathbf{k};t)$ that describes the change of the system's dynamics due to an external inhomogeneous potential. We study the dependence of the characteristic length scale on time, wave-vector, and on the distance from the mode-coupling transition. We verify scaling predictions of Biroli et al. In addition, we find that the numerical value of the diverging length scale qualitatively agrees with lengths obtained from four-point correlation functions. We show that the diverging length scale has very weak k dependence, which contrasts with very strong $k$ dependence of the $q\to 0$ limit of the susceptibility, ${\chi }_{\mathbf{q}=0}(\mathbf{k};t)$. Finally, we compare the diverging length obtained from the small q expansion to that resulting from an isotropic approximation applied to the equation of motion for the dynamic susceptibility ${\chi }_{\mathbf{q}}(\mathbf{k};t)$.

References in corpus (7)

• Inhomogeneous Mode-Coupling Theory and Growing Dynamic Length in Supercooled Liquids
• Growing length and time scales in glass forming liquids
• Spatial correlations in the dynamics of glassforming liquids: Experimental determination of their temperature dependence
• Spontaneous and induced dynamic fluctuations in glass-formers I: General results and dependence on ensemble and dynamics
• Critical scaling and heterogeneous superdiffusion across the jamming/rigidity transition of a granular glass
• Lower bound on the four-point dynamical susceptibility: Direct experimental test on a granular packing
• Three-point susceptibilities ${\chi }_n(k;t)$ and ${\chi }_n^s(k;t)$: mode-coupling approximation

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• Static replica approach to critical correlations in glassy systems
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• Unveiling the anatomy of mode-coupling theory
• Dynamic Length Scales in Glass-Forming Liquids: A Inhomogeneous Molecular Dynamics Simulation Approach
• Connecting Relaxation Time to a Dynamical Length Scale in Athermal Active Glass Formers
• Theory of simple glasses
• Competing Relaxation Channels in Continuously Polydisperse Fluids: A Mode-Coupling Study
• Theory of the $\beta$-Relaxation Beyond Mode-Coupling Theory: A Microscopic Treatment