Communications in Mathematical Sciences

Volume 9 (2011)

Number 3

A transport equation for confined structures derived from the Boltzmann equation

Pages: 829 – 857



Clemens Heitzinger (Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom)

Christian Ringhofer (Department of Mathematics, Arizona State University, Tempe, Arizona)


A system of diffusion-type equations for transport in 3d confined structures is derived from the Boltzmann transport equation for charged particles. Transport takes places in confined structures and the scaling in the derivation of the diffusion equation is chosen so that transport and scattering occur in the longitudinal direction and the particles are confined in the two transversal directions. The result are two diffusion-type equations for the concentration and fluxes as functions of position in the longitudinal direction and energy. Entropy estimates are given. The transport coefficients depend on the geometry of the problem that is given by arbitrary harmonic confinement potentials. An important feature of this approach is that the coefficients in the resulting diffusion-type equations are calculated explicitly so that the six position and momentum dimensions of the original 3d Boltzmann equation are reduced to a 2d problem. Finally, numerical results are given and discussed. Applications of this work include the simulation of charge transport in nanowires, nanopores, ion channels, and similar structures.


Boltzmann transport equation, many-body system, transport equation, diffusion equation, confined structure, entropy estimate, nanowire, nanopore, nano-structure, ion channel

2010 Mathematics Subject Classification

35Q20, 76R50, 82D80

Published 11 March 2011