is increasingly concentrated near a saddle point of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on © ¤ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter

Tracer dispersion has been simulated in three-dimensional models of regular and random sphere packings for a range of Peclet numbers. A random-walk particle-tracking (PT) method was used to simulate tracer movement within pore-scale flow fields computed with the lattice-Boltzmann (LB) method. The simulation results illustrate the time evolution of dispersion, and they corroborate a number of theoretical and empirical results for the scaling of asymptotic longitudinal and transverse dispersion with Peclet number. Comparisons with nuclear magnetic resonance (NMR) spectroscopy experiments show agreement on transient, as well as asymptotic, dispersion rates. These results support both NMR findings that longitudinal dispersion rates are significantly lower than reported in earlier experimental literature, as well as the fact that asymptotic rates are observed in relatively short times by techniques that employ a uniform initial distribution of tracers, like NMR.

The problem of noise-induced escape from a metastable state arises in physics, chemistry, biology, systems engineering, and other areas. The problem is well understood when the underlying dynamics of the system obey detailed balance. When this assumption fails many of the results of classical transition-rate theory no longer apply, and no general method exists for computing the weak-noise asymptotics of fundamental quantities such as the mean escape time. In this paper we present a general technique for analysing the weak-noise limit of a wide range of stochastically perturbed continuous-time nonlinear dynamical systems. We simplify the original problem, which involves solving a partial differential equation, into one in which only ordinary differential equations need be solved. This allows us to resolve some old issues for the case when detailed balance holds. When it does not hold, we show how the formula for the mean escape time asymptotics depends on the dynamics of the system along the most probable escape path. We also present new results on short-time behavior and discuss the possibility of focusing along the escape path.

Abstract. A machine-generated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer's 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the Coxeter group D n . Each of the 192 expressions is labeled by an element of D 4 . Of the 192, 24 are equivalent expressions for the local Heun function Hl, and it is shown that the resulting order-24 group of transformations of Hl is isomorphic to the symmetric group S 4 . The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.

When the Lattice Boltzmann Method (LBM) is used for simulating continuum fluid flow, the discrete mass distribution must satisfy imposed constraints for density and momentum along the boundaries of the lattice. These constraints uniquely determine the three-dimensional (3-D) mass distribution for boundary nodes only when the number of external (inward-pointing) lattice links does not exceed four. We propose supplementary rules for computing the boundary distribution where the number of external links does exceed four, which is the case for all except simple rectangular lattices. Results obtained with 3-D body-centered-cubic lattices are presented for Poiseuille flow, porous-plate Couette flow, pipe flow, and rectangular duct flow. The accuracy of the two-dimensional (2-D) Poiseuille and Couette flows persists even when the mean free path between collisions is large, but that of the 3-D duct flow deteriorates markedly when the mean free path exceeds the lattice spacing. Accuracy in general decreases with Knudsen number and Mach number, and the product of these two quantities is a useful index for the applicability of LBM to 3-D low-Reynolds-number flow.