asymptotic solution details for Boltzmann equation for a hole has a uniform electric field
n + - n = 0
Steps to solve Boltzmann equation
for the previous equation let consider the asymptotic solution has the form:
In spherical coordinates:
which is symmetric in direction.
Assumingthe solution of the zenith angle direction is the Legendre polynomial, and can be written as:
The modified Bessel functions, first I_k and second kind K_k, are the solution for the previous equation but the boundary conditions determines which one to use, in this case , , and as . so