A transport equation for confined structures derived from the Boltzmann equation

Clemens Heitzinger, Christian Ringhofer

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)829-857
Number of pages29
JournalCommunications in Mathematical Sciences
Issue number3
StatePublished - Sep 2011


  • Boltzmann transport equation
  • Confined structure
  • Diffusion equation
  • Entropy estimate
  • Ion channel
  • Many-body system
  • Nano-structure
  • Nanopore
  • Nanowire
  • Transport equation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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