There is much interest within the mathematical biology and statistical physics community in converting stochastic agent-based models for random walkers into a partial differential equation description for the average agent density. Here a collection of noninteracting biased random walkers on a one-dimensional lattice is considered. The usual master equation approach requires that two continuum limits, involving three parameters, namely step length, time step and the random walk bias, approach zero in a specific way. We are interested in the case where the two limits are not consistent. New results are obtained using a Fokker–Planck equation and the results are highly dependent on the simulation update schemes. The theoretical results are confirmed with examples. These findings provide insight into the importance of updating schemes to an accurate macroscopic description of stochastic local movement rules in agent-based models when the lattice spacing represents a physical object such as cell diameter.

We study the Fokker–Planck equation as the many-particle limit of a stochastic particlesystem on one hand and as a Wasserstein gradient flow on the other. We write thepath-space rate functional, which characterises the large deviations from the expectedtrajectories, in such a way that the free energy appears explicitly. Next we use thisformulation via the contraction principle to prove that the discrete time rate functionalis asymptotically equivalent in the Gamma-convergence sense to the functional derived fromthe Wasserstein gradient discretization scheme.

We study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.

Nonlocal electron heat transport in plasmas heated by a high-intensity laser field is investigated. We show how the heat transport is strongly modified by the super-Gaussian character of the energy distribution caused by the strong collisional heating. The change in the collisional electron–ion energy exchange due to this modification of the shape of the electron distribution function is also studied.

The transport of hot electrons in solid density plasma created by a high-intensity subpicosecond laser pulse and the resulting heating and ionization of the bulk of plasma are simulated with the electron kinetic code “FPI.” Both the hot and the thermal electrons are treated kinetically. An analysis of the results in terms of a two Maxwellians fit to the numerically obtained distribution function is made.