David Aristoff

Mathematical Analysis of Temperature Accelerated Dynamics

We give a mathematical framework for temperature accelerated dynamics (TAD), an algorithm proposed by So rensen and Voter in [J. Chem. Phys., 112 (2000), pp. 9599--9606] to efficiently generate metastable stochastic dynamics. Using the notion of quasi-stationary distributions, we propose some modifications to TAD. Then considering the modified algorithm in an idealized setting, we show how TAD can be made mathematically rigorous.

On the phase transition curve in a directed exponential random graph model

Abstract We consider a family of directed exponential random graph models parametrized by edges and outward stars. Much of the important statistical content of such models is given by the normalization constant of the models, and, in particular, an appropriately scaled limit of the normalization, which is called the free energy. We derive precise asymptotics for the normalization constant for finite graphs. We use this to derive a formula for the free energy. The limit is analytic everywhere except along a curve corresponding to a first-order phase transition. We examine unusual behavior of the model along the phase transition curve.

Asymptotic structure and singularities in constrained directed graphs

We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward p-stars. Our models are close cousins of exponential random graph models, in which edges and certain other subgraph densities are controlled by parameters. We find that large graphs have either uniform or bipodal structure. When edge density (resp. p-star density) is fixed and p-star density (resp. edge density) is controlled by a parameter, we find phase transitions corresponding to a change from uniform to bipodal structure. When both edge and p-star density are fixed, we find only bipodal structures and no phase transition.

Random Loose Packing in Granular Matter

We introduce and simulate a two-dimensional Edwards-style model of granular matter at vanishing pressure. The model incorporates some of the effects of gravity and friction, and exhibits a random loose packing density whose standard deviation vanishes with increasing system size, a phenomenon that should be verifiable for real granular matter.

Random close packing in a granular model

We introduce a two-dimensional lattice model of granular matter. Using a combination of proof and simulation we demonstrate an order/disorder phase transition in the model, to which we associate the granular phenomenon of random close packing. We use Peierls contours to prove that the model is sensitive to boundary conditions at high density and Markov chain Monte Carlo simulation to show it is insensitive at low density.

The Parallel Replica Method for Simulating Long Trajectories of Markov Chains

The parallel replica dynamics, originally developed by A.F. Voter, eciently simulates very long trajectories of metastable Langevin dynamics. We present an analogous algorithm for discrete time Markov processes. Such Markov processes naturally arise, for example, from the time discretization of a continuous time stochastic dynamics. Appealing to properties of quasistationary distributions, we show that our algorithm reproduces exactly (in some limiting regime) the law of the original trajectory, coarsened over the metastable states.

First order phase transition in a model of quasicrystals

We introduce a family of two-dimensional lattice models of quasicrystals, using a range of square hard cores together with a soft interaction based on an aperiodic tiling set. Along a low temperature isotherm we find, by Monte Carlo simulation, a first order phase transition between disordered and quasicrystalline phases.

Dilatancy Transition in a Granular Model

We introduce a model of granular matter and use a volume/strain ensemble to analyze infinitesimal shearing. Monte Carlo simulation suggests the model exhibits a second order phase transition associated with the onset of dilatancy.

The parallel replica method for computing equilibrium averages of Markov chains

Abstract An algorithm is proposed for computing equilibrium averages of Markov chains which suffer from metastability – the tendency to remain in one or more subsets of state space for long time intervals. The algorithm, called the parallel replica method (or ParRep), uses many parallel processors to explore these subsets more efficiently. Numerical simulations on a simple model demonstrate consistency of the method. A proof of consistency is given in an idealized setting. The parallel replica method can be considered a generalization of A. F. Voters parallel replica dynamics, originally developed to efficiently simulate metastable Langevin stochastic dynamics.

Percolation of hard disks

Random arrangements of points in the plane, interacting only through a simple hard-core exclusion, are considered. An intensity parameter controls the average density of arrangements, in analogy with the Poisson point process. It is proved that, at high intensity, an infinite connected cluster of excluded volume appears almost surely.