Abstract:
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. |