Recently, Gillespie introduced the τ-leap approximate, accelerated stochastic Monte Carlo method for well-mixed reacting systems [J. Chem. Phys. 115, 1716 (2001)]. In each time increment of that method, one executes a number of reaction events, selected randomly from a Poisson distribution, to enable simulation of long times. Here we introduce a binomial distribution τ-leap algorithm (abbreviated as BD-τ method). This method combines the bounded nature of the binomial distribution variable with the limiting reactant and constrained firing concepts to avoid negative populations encountered in the original τ-leap method of Gillespie for large time increments, and thus conserve mass. Simulations using prototype reaction networks show that the BD-τ method is more accurate than the original method for comparable coarse-graining in time.

A novel traffic flow model based on stochastic microscopic dynamics is introduced and analyzed. Vehicles advance based on the energy profile of their surrounding traffic implementing the "look-ahead" rule and following an underlying asymmetric exclusion process with Arrhenius spin-exchange dynamics. Monte Carlo simulations produce numerical solutions of the microscopic traffic model. Fluctuations play an important role in profiling observationally documented but, at the simulation level, elusive traffic phenomena. Furthermore, based on scaling and limit arguments we obtain a macroscopic description of this microscopic dynamics formulation which up to leading term of the expansions takes the form of integrodifferential Burgers or higher-order dispersive partial differential equations. We outline connections and comparisons of the hierarchical models presented here (microscopic, macroscopic) with other well-known traffic flow models.

Uncertainty quantification is a primary challenge for reliable modeling and simulation of complex stochastic dynamics. Such problems are typically plagued by incomplete information that may enter a...