In a recent commentary, J. M. Kosterlitz described how D. Thouless and he got motivated to investigate melting and suprafluidity in two dimensions [Kosterlitz JM (2016) J Phys Condens Matter 28:481001]. It was due to the lack of broken translational symmetry in two dimensions-doubting the existence of 2D crystalsand the first computer simulations foretelling 2D crystals (at least in tiny systems). The lack of broken symmetries proposed by D. Mermin and H. Wagner is caused by long wavelength density fluctuations. Those fluctuations do not only have structural impact, but additionally a dynamical one: They cause the Lindemann criterion to fail in 2D in the sense that the mean squared displacement of atoms is not limited. Comparing experimental data from 3D and 2D amorphous solids with 2D crystals, we disentangle Mermin-Wagner fluctuations from glassy structural relaxations. Furthermore, we demonstrate with computer simulations the logarithmic increase of displacements with system size: Periodicity is not a requirement for Mermin-Wagner fluctuations, which conserve the homogeneity of space on long scales.Mermin-Wagner fluctuations | 2D ensembles | glass transition | phase transition | confined geometry F or structural phase transitions, it is well known that the microscopic mechanisms breaking symmetry are not the same in two and in three dimensions. Whereas 3D systems typically show first-order transitions with phase equilibrium and latent heat, 2D crystals melt via two steps with an intermediate hexatic phase. Unlike in 3D, translational and orientational symmetry are not broken at the same temperature in 2D. The scenario is described within the Kosterlitz, Thouless, Halperin, Nelson, Young (KTHNY) theory (1-5), which was confirmed (e.g., in colloidal monolayers) (6, 7). However, for the glass transition, it is usually assumed that dimensionality does not play a role for the characteristics of the transition, and 2D and 3D systems are frequently used synonymously (8-12), whereas differences between the 2D and 3D glass transition are reported in ref. 13.In the present work, we compare data from colloidal crystals and glasses and show that Mermin-Wagner fluctuations, well known from 2D crystals, are also present in amorphous solids (14, 15). Mermin-Wagner fluctuations are usually discussed in the framework of long-range order (magnetic or structural). However, in the context of 2D crystals, they have also had an impact on dynamic quantities like mean squared displacements (MSDs). Long before 2D melting scenarios were discussed, there was an intense debate as to whether crystals and perfect longrange order (including magnetic order) can exist in 1D or 2D at all (16)(17)(18)(19). A beautiful heuristic argument was given by Peierls (17): Consider a 1D chain of particles with nearest neighbor interaction. The relative distance fluctuation between particle n and particle n + 1 at finite temperature may be ξ. Similar is the fluctuation between particle n + 1 and n + 2. The relative fluctuation between second nearest neig...

Investigations of strain correlations at the glass transition reveal unexpected phenomena. The shear strain fluctuations show an Eshelby-strain pattern ( ∼ cos (4θ)/r 2 ), characteristic for elastic response, even in liquids at long times [1]. We address this using a mode-coupling theory for the strain fluctuations in supercooled liquids and data from both, video microscopy of a two-dimensional colloidal glass former and simulations of Brownian hard disks. We show that long-ranged and longlived strain-signatures follow a scaling law valid close to the glass transition. For large enough viscosities, the Eshelby-strain pattern is visible even on time scales longer than the structural relaxation time τ and after the shear modulus has relaxed to zero.Glasses behave like isotropic elastic solids under external loads. Their strain fields are long-ranged as captured in elasticity theory. Considering appropriate boundary conditions, Eshelby obtained the elastic strain field surrounding an isolated local deformation [2], which is also the basis for modern theories of plasticity in disordered solids [3]: Plastic deformation proceeds via localized irreversible rearrangements coupled by elastic strain fields.While the relevance of strain in glass at low temperatures originates in the breaking of translational symmetry underlying solidification [4], the proper understanding of the evolution of strains at the crossover from metastable glass to supercooled liquid remains an open topic [5]. The concept of plastic events, which are elastically coupled, has emerged as one candidate rooted in the theory of solids which aims to capture the glass transition from the low temperature side. It suggests that 'supercooled liquids are solids that flow ' [6] and focuses on strain fluctuations and their correlations to probe plastic flow.Lemaître and colleagues found evidence for this concept in molecular dynamics simulations of twodimensional Lennard-Jones mixtures: They observed persistent long-ranged strain fluctuations in supercooled liquid states [1]. Because the time over which strains were accumulated exceeds the structural relaxation time τ of the liquid, observable elastic stresses have decayed. The observation of spatial dependences exhibiting a far-field Eshelby-strain pattern thus can not be a simple consequence of elasticity. It suggests that particle rearrangements in a fluid interact over large distances via strains likely in an underlying elastic structure, the 'inherent states' characterizing the potential energy landscape [7].Strain patterns have been observed experimentally, but not yet in quiescent supercooled liquids. An anisotropic decay of strain was found in a 3D colloidal hard sphere glass under steady shear [8], in 2D simulations [9], and in granular matter [10]. Eshelby-patterns are reported in 2D flowing emulsions [11] and in a 3D colloidal hard sphere glass, where they appear under shear and thermally induced in quiescent state [12]. They are also present in 2D soft hexagonal crystals with dipolar interaction [...

We present results from computer simulation and mode-coupling theory of the glass transition for the nonequilibrium relaxation of stresses in a colloidal glass former after the cessation of shear flow. In the ideal glass, persistent residual stresses are found that depend on the flow history. The partial decay of stresses from the steady state to this residual stress is governed by the previous shear rate. We rationalize this observation in a schematic model of mode-coupling theory. The results from Brownian-dynamics simulations of a glassy two-dimensional hard-disk system are in qualitative agreement with the predictions of the theory.

We review recent progress on a microscopic theoretical approach to describe the nonlinear response of glass-forming colloidal dispersions under strong external forcing leading to homogeneous and inhomogeneous flow. Using mode-coupling theory (MCT), constitutive equations for the rheology of viscoelastic shear-thinning fluids are obtained. These are, in suitably simplified form, employed in continuum fluid dynamics, solved by a hybrid-Lattice Boltzmann (LB) algorithm that was developed to deal with long-lasting memory effects. The combined microscopic theoretical and mesoscopic numerical approach captures a number of phenomena far from equilibrium, including the yielding of metastable states, process-dependent mechanical properties, and inhomogeneous pressure-driven channel flow.

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