Moment isotropy and discrete rotational symmetry of two-dimensional lattice vectors.

We present a direct proof of a theorem linking the order of moment isotropy and degree of discrete rotational symmetry for a two-dimensional set of lattice vectors. This theorem has been proved previously based on properties of sinusoidal functions. The new proof is based instead on purely linear algebraic arguments.

Maximal potential energy transport: a variational principle for solidification problems.

We analyze numerically the mechanisms controlling the spacing of chimneys--channels devoid of solid--in two-dimensional mushy layers formed by solidifying a binary alloy. Chimneys are the principal conduits through which buoyancy effects material transport out of the mushy layer and into the liquid from which it formed. Experiments show a coarsening of chimney spacing; we pursue the hypothesis that the spacing adjusts to optimize material transport and hence maximize the rate of removal of potential energy stored in the mushy layer.

Extended Boltzmann kinetic equation for turbulent flows.

Complex fluid physics can be modeled using an extended kinetic (Boltzmann) equation in a more efficient way than using the continuum Navier-Stokes equations. Here, we explain this method for modeling fluid turbulence and show its effectiveness with the use of a computationally efficient implementation in terms of a discrete or "lattice" Boltzmann equation.

Moment expansions in spatial ecological models and moment closure through Gaussian approximation.

We describe the dynamics of competing species in terms of interactions between spatial moments. We close the moment hierarchy by employing a Gaussian approximation which assumes that fluctuations are independent and distributed normally about the mean values. The Gaussian approximation provides the lowest-order systematic correction to the mean-field approximation by incorporating the effect of fluctuations.

Nucleation and relaxation from meta-stability in spatial ecological models.

We study a model for competing species that explicitly accounts for effects due to discreteness, stochasticity and spatial extension of populations. If a species does better locally than the other by an amount epsilon, the global outcome depends on the initial densities (uniformly distributed in space), epsilon and the size of the system. The transition point moves to lower values of the initial density of the superior species with increasing system size.