Mathematical and Numerical Methods for Complex Quantum Systems

Transport equations for confined structures derived from the Boltzmann equation

Clemens Heitzinger

TU Vienna


It is well-known that the drift-diffusion equations can be derived from the Boltzmann equation. Furthermore, it is clear that bulk diffusion coefficients are not a good model for 1D confined structures such as ion channels and nanowires and for 2D confined structures. Therefore the question arises whether it is possible at all to derive a transport equation from the 3D Boltzmann transport equation such that the transport equation takes into account the confinement of the particles. We have shown that the answer is positive, and---quite surprisingly---we even found explicit expressions for the geometry dependent transport coefficients. There is also an important computational component: The $(6+1)$-dimensional problem (three space dimensions, three momentum dimensions, and time) is reduced to a $(2+1)$-dimensional problem (one space dimension, one energy dimension, and time). The significance for applications such as ion channels and nanopores is that currents can be calculated immediately. The diffusion-type equation is a confined-transport model using relaxation against a phenomenologically squeezed Maxwellian distribution. The basic assumption here is that the collisions thermalize in the transport direction and conserve energy in the confinement direction. For part of the argument, the confinement potential is assumed to be harmonic. Finally, we have applied the transport equation to two ion channels, namely Gramicidin~A (an antibiotic) and the KcsA channel (a potassium channel). Excellent agreement between simulation and measurement was found.