Stochastic level-set method for shape optimisation

We present a new method for stochastic shape optimisation of engineering structures. The method generalises an existing deterministic scheme, in which the structure is represented and evolved by a level-set method coupled with mathematical programming. The stochastic element of the algorithm is built on the methods of statistical mechanics and is designed so that the system explores a Boltzmann-Gibbs distribution of structures. In non-convex optimisation problems, the deterministic algorithm can get trapped in local optima: the stochastic generalisation enables sampling of multiple local optima, which aids the search for the globally-optimal structure. The method is demonstrated for several simple geometrical problems, and a proof-of-principle calculation is shown for a simple engineering structure.Comment: 17 pages, 10 fig

Dynamic facilitation explains democratic particle motion of metabasin transitions

Transitions between metabasins in supercooled liquids seem to occur through rapid "democratic" collective particle rearrangements. Here we show that this apparent homogeneous particle motion is a direct consequence of dynamic facilitation. We do so by studying metabasin transitions in facilitated spin models and constrained lattice gases. We find that metabasin transitions occur through a sequence of locally facilitated events taking place over a relatively short time frame. When observed on small enough spatial windows these events appear sudden and homogeneous. Our results indicate that metabasin transitions are essentially "non-democratic" in origin and yet another manifestation of dynamical heterogeneity in glass formers.Comment: 6 pages, 6 figure

Preparation and relaxation of very stable glassy states of a simulated liquid

We prepare metastable glassy states in a model glass-former made of Lennard-Jones particles by sampling biased ensembles of trajectories with low dynamical activity. These trajectories form an inactive dynamical phase whose `fast' vibrational degrees of freedom are maintained at thermal equilibrium by contact with a heat bath, while the `slow' structural degrees of freedom are located in deep valleys of the energy landscape. We examine the relaxation to equilibrium and the vibrational properties of these metastable states. The glassy states we prepare by our trajectory sampling method are very stable to thermal fluctuations and also more mechanically rigid than low-temperature equilibrated configurations.Comment: Minor revisions in light of referee comments. 5 pages, 4 fig

Excitations are localized and relaxation is hierarchical in glass-forming liquids

For several atomistic models of glass formers, at conditions below their glassy dynamics onset temperatures, ${T_\mathrm{o}}$, we use importance sampling of trajectory space to study the structure, statistics and dynamics of excitations responsible for structural relaxation. Excitations are detected in terms of persistent particle displacements of length $a$. At supercooled conditions, for $a$ of the order of or smaller than a particle diameter, we find that excitations are associated with correlated particle motions that are sparse and localized, occupying a volume with an average radius that is temperature independent and no larger than a few particle diameters. We show that the statistics and dynamics of these excitations are facilitated and hierarchical. Excitation energy scales grow logarithmically with $a$. Excitations at one point in space facilitate the birth and death of excitations at neighboring locations, and space-time excitation structures are microcosms of heterogeneous dynamics at larger scales. This nature of dynamics becomes increasingly dominant as temperature $T$ is lowered. We show that slowing of dynamics upon decreasing temperature below $T_\mathrm{o}$ is the result of a decreasing concentration of excitations and concomitant growing hierarchical length scales, and further that the structural relaxation time $\tau$ follows the parabolic law, $\log(\tau / \tau_\mathrm{o}) = J^2(1/T - 1/T_\mathrm{o})^2$, for $T<T_\mathrm{o}$, where $J$, $\tau_\mathrm{o}$ and $T_\mathrm{o}$ can be predicted quantitatively from dynamics at short time scales. Particle motion is facilitated and directional, and we show this becomes more apparent with decreasing $T$. We show that stringlike motion is a natural consequence of facilitated, hierarchical dynamics.Comment: 15 pages, 6 figures, + links to movies; To appear in Phys. Rev.

Fast Simulation of Facilitated Spin Models

We show how to apply the absorbing Markov chain Monte Carlo algorithm of Novotny to simulate kinetically constrained models of glasses. We consider in detail one-spin facilitated models, such as the East model and its generalizations to arbitrary dimensions. We investigate how to maximise the efficiency of the algorithms, and show that simulation times can be improved on standard continuous time Monte Carlo by several orders of magnitude. We illustrate the method with equilibrium and aging results. These include a study of relaxation times in the East model for dimensions d=1 to d=13, which provides further evidence that the hierarchical relaxation in this model is present in all dimensions. We discuss how the method can be applied to other kinetically constrained models.Comment: 8 pages, 4 figure