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:

so

where

and

In spherical coordinates:

which is symmetric in direction.

Assuming the solution of the zenith angle direction is the Legendre polynomial, and can be written as:

and

so,

The modified Bessel functions, first and second kind, are the solutions for the previous equation but the boundary conditions determines which one to use, in this case , , and as . so only the modified Bessel of second kind K_k are the non-zaro terms. so the the general solution for the equation can be written as :