Efficient Numerical Solution of Stochastic Differential Equations Using Exponential Timestepping

We present an exact timestepping method for Brownian motion that does not require Gaussian random variables to be generated. Time is incremented in steps that are exponentially-distributed random variables; boundaries can be explicitly accounted for at each timestep. The method is illustrated by numerical solution of a system of diffusing particles.     

Defect Formation in a Dynamic Transition

When a system that undergoes a continuous phase transition is swept through its critical point the initial symmetry is broken and domains are formed. Because of critical slowing down it is not possible to sweep adiabatically; the number of domains therefore depends on the rate of increase of the critical parameter. We give a summary of recent theoretical results for the number of defects produced as a function of how rapidly the transition point is passed. They are obtained from a simplified model, using a stochastic partial differential equation that is also solved numerically.     

Dynamics of kinks: nucleation, diffusion, and annihilation

We investigate the nucleation, annihilation, and dynamics of kinks in a classical (1+1)-dimensional straight phi(4) field theory at finite temperature. From large scale Langevin simulations, we establish that the nucleation rate is proportional to the square of the equilibrium density of kinks. We identify two annihilation time scales: one due to kink-antikink pair recombination after nucleation, the other from nonrecombinant annihilation. We introduce a mesoscopic model of diffusing kinks based on "paired" and "survivor" kinks and antikinks. Analytical predictions for the dynamical time scales, as well as the corresponding length scales, are in good agreement with the simulations.     

A parallel multi-configuration self-consistent field algorithm

A scalable multi-configuration self-consistent field (MCSCF) algorithm is described. The method for optimizing the orbital and configurational parameters is based upon the two-step Newton–Raphson approach with an augmented orbital Hessian matrix. A single copy of the two-electron integrals in the molecular orbital basis is distributed over the memory of all processors. Storage of the augmented Hessian is avoided by re-computing its elements as needed. A replicated data approach is used to parallelize the configuration interaction step. Scalability to 1024 processors is demonstrated.     

Stochastic Calculus: Application to Dynamic Bifurcations and Threshold Crossings

For the dynamic pitchfork bifurcation in the presence of white noise, the statistics of the last time at zero are calculated as a function of the noise level and the rate of change of the parameter . The threshold crossing problem used, for example, to model the firing of a single cortical neuron is considered, concentrating on quantities that may be experimentally measurable but have so far received little attention. Expressions for the statistics of pre-threshold excursions, occupation density, and last crossing time of zero are compared with results from numerical generation of paths.