Nonlinear Drift-diffusion Equations Research Articles

Convergence of a finite-volume scheme for the drift-diffusion equations in 1D

We construct an approximate solution to one‐dimensional nonlinear drift–diffusion equations by a finite‐volume scheme. The convergence of the scheme is proved, which yields the global existence of solutions to the equations. This result is valid even in the presence of vacuum state: the densities of free carriers of charge vanish for some times and positions, which implies that the continuity equations are degenerate parabolic equations. Finally, numerical simulations are performed by the finite‐volume scheme.

Entropy methods for kinetic models of traffic flow

In these notes we first introduce logarithmic entropy methods for time-dependent drift-diffusion equations and then consider a kinetic model of Vlasov-Fokker-Planck type for traffic flows. In the spatially homogeneous case the model reduces to a special type of nonlinear driftdiffusion equation which may permit the existence of several stationary states corresponding to the same density. Then we define general convex entropies and prove a convergence result for large times to steady states, even if more than one exists in the considered range of parameters, provided that some entropy estimates are uniformly bounded.

LARGE TIME ASYMPTOTICS OF NONLINEAR DRIFT-DIFFUSION SYSTEMS WITH POISSON COUPLING

We study the asymptotic behavior as t→ +∞ of a system of densities of charged particles satisfying nonlinear drift-diffusion equations coupled by a damped Poisson equation for the drift-potential. In plasma physics applications the damping is caused by a spatio-temporal rescaling of an “unconfined” problem, which introduces a harmonic external potential of confinement. We present formal calculations (valid for smooth solutions) which extend the results known in the linear diffusion case to nonlinear diffusion of e.g. Fermi–Dirac or fast diffusion/porous media type.

Hydrodynamics and fluctuations outside of local equilibrium: Driven diffusive systems

We derive hydrodynamic equations for systems not in local thermodynamic equilibrium, that is, where the local stationary measures are “non-Gibbsian” and do not satisfy detailed balance with respect to the microscopic dynamics. As a main example we consider thedriven diffusive systems (DDS), such as electrical conductors in an applied field with diffusion of charge carriers. In such systems, the hydrodynamic description is provided by a nonlinear drift-diffusion equation, which we derive by a microscopic method ofnonequilibrium distributions. The formal derivation yields a Green-Kubo formula for the bulk diffusion matrix and microscopic prescriptions for the drift velocity and “nonequilibrium entropy” as functions of charge density. Properties of the hydrodynamic equations are established, including an “H-theorem” on increase of the thermodynamic potential, or “entropy”, describing approach to the homogeneous steady state. The results are shown to be consistent with the derivation of the linearized hydrodynamics for DDS by the Kadanoff-Martin correlation-function method and with rigorous results for particular models. We discuss also the internal noise in such systems, which we show to be governed by a generalizedfluctuation-dissipation relation (FDR), whose validity is not restricted to thermal equilibrium or to time-reversible systems. In the case of DDS, the FDR yields a version of a relation proposed some time ago by Price between the covariance matrix of electrical current noise and the bulk diffusion matrix of charge density. Our derivation of the hydrodynamic laws is in a form—the so-called “Onsager force-flux form” which allows us to exploit the FDR to construct the Langevin description of the fluctuations. In particular, we show that the probability of large fluctuations in the hydrodynamic histories is governed by a version of the Onsager “principle of least dissipation,” which estimates the probability of fluctuations in terms of the Ohmic dissipation required to produce them and provides a variational characterization of the most probable behavior as that associated to least (excess) dissipation. Finally, we consider the relation of longrange spatial correlations in the steady state of the DDS and the validity of ordinary hydrodynamic laws. We also discuss briefly the application of the general methods of this paper to other cases, such as reaction-diffusion systems or magnetohydrodynamics of plasmas.