# Free electron transport theory derivation

I am going through the Introduction to solid state physics Book by Charles Kittel. In the section regrading Free electron transport theory, i am confused by a statement "In the absence of collisions the Fermi sphere (Fig. 10) moves in k space at a uniform rate by a constant applied electric field. " given in the book.

Since $$d𝐤/dt = -eE$$ this would imply a constant acceleration, and 𝐤 increasing without limit

• @noah, in the textbook it says '"In the absence of collisions the Fermi sphere (Fig. 10) moves in k space at a uniform rate by a constant applied electric field. " given in the book.' and since $dk/dt=−eE$ this would imply a constant acceleration, and 𝐤 increasing without limt. This seems contradictory since a unifrom rate means no acceleration Commented Feb 20, 2023 at 20:13
the Fermi sphere moves in $$k$$-space at a uniform rate.
The $$k$$-space is where the momentum and thus also the energies live. If the Fermi sphere (and thus also the occupied $$k$$-states) moves uniformly, that means the momentum (and therefore also the energy) grows at a constant rate. In the case of no collisions, even without limit, as you say. So constant change in momentum space means accelerated motion in real space.