Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics

Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial non-ergodicity, depending on the constraint parameter and the dimensionality. But in the ergodic cases, the dynamics is shown to be intrinsically cooperative at high densities giving rise to glassy dynamics as observed in simulations. The cooperativity is characterized by two length scales whose behavior controls finite-size effects: these are essential for interpreting simulations. In contrast to hypercubic lattices, on Bethe lattices KA models undergo a dynamical (jamming) phase transition at a critical density: this is characterized by diverging time and length scales and a discontinuous jump in the long-time limit of the density autocorrelation function. By analyzing generalized Bethe lattices (with loops) that interpolate between hypercubic lattices and standard Bethe lattices, the crossover between the dynamical transition that exists on these lattices and its absence in the hypercubic lattice limit is explored. Contact with earlier results are made via analysis of the related Fredrickson-Andersen models, followed by brief discussions of universality, of other approaches to glass transitions, and of some issues relevant for experiments.Comment: 59 page

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Clemson Newsletter, 1989-1991

Information for the faculty and staff of Clemson Universityhttps://tigerprints.clemson.edu/clemson_newsletter/1021/thumbnail.jp

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Path Integrals in the Sky: Classical and Quantum Problems with Minimal Assumptions

Cosmology has, after the formulation of general relativity, been transformed from a branch of philosophy into an active field in physics. Notwithstanding the significant improvements in our understanding of our Universe, there are still many open questions on both its early and late time evolution. In this thesis, we investigate a range of problems in classical and quantum cosmology, using advanced mathematical tools, and making only minimal assumptions. In particular, we apply Picard-Lefschetz theory, catastrophe theory, infinite dimensional measure theory, and weak-value theory. To study the beginning of the Universe in quantum cosmology, we apply Picard-Lefschetz theory to the Lorentzian path integral for gravity. We analyze both the Hartle-Hawking no-boundary proposal and Vilenkin's tunneling proposal, and demonstrate that the Lorentzian path integral corresponding to the mini-superspace formulation of the two proposals is well-defined. However, when including fluctuations, we show that the path integral predicts the existence of large fluctuations. This indicates that the Universe cannot have had a smooth beginning in Euclidean de Sitter space. In response to these conclusions, the scientific community has made a series of adapted formulations of the no-boundary and tunneling proposals. We show that these new proposals suffer from similar issues. Second, we generalize the weak-value interpretation of quantum mechanics to relativistic systems. We apply this formalism to a relativistic quantum particle in a constant electric field. We analyze the evolution of the relativistic particle in both the classical and the quantum regime and evaluate the back-reaction of the Schwinger effect on the electric field in $1+1$-dimensional spacetime, using analytical methods. In addition, we develop a numerical method to evaluate both the wavefunction and the corresponding weak-values in more general electric and magnetic fields. We conclude the quantum part of this thesis with a chapter on Lorentzian path integrals. We propose a new definition of the real-time path integral in terms of Brownian motion on the Lefschetz thimble of the theory. We prove the existence of a $\sigma$-measure for the path integral of the non-relativistic free particle, the (inverted) harmonic oscillator, and the relativistic particle in a range of potentials. We also describe how this proposal extends to more general path integrals. In the classical part of this thesis, we analyze two problems in late-time cosmology. Multi-dimensional oscillatory integrals are prevalent in physics, but notoriously difficult to evaluate. We develop a new numerical method, based on multi-dimensional Picard-Lefschetz theory, for the evaluation of these integrals. The virtue of this method is that its efficiency increases when integrals become more oscillatory. The method is applied to interference patterns of lensed images near caustics described by catastrophe theory. This analysis can help us understand the lensing of astrophysical sources by plasma lenses, which is especially relevant in light of the proposed lensing mechanism for fast radio bursts. Finally, we analyze large-scale structure formation in terms of catastrophe theory. We show that the geometric structure of the three-dimensional cosmic-web is determined by both the eigenvalue and the eigenvector fields of the deformation tensor. We formulate caustic conditions, classifying caustics using properties of these fields. When applied to the Zel'dovich approximation of structure formation, the caustic conditions enable us to construct a caustic skeleton of the three-dimensional cosmic-web in terms of the initial conditions

Studying Turbulence Using Numerical Simulation Databases. 4: Proceedings of the 1992 Summer Program

Papers are presented under the following subject areas: small scales; turbulence physics; compressible flow and modeling; and reacting flows and combustion