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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

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    $\begingroup$ What's your question? $\endgroup$
    – noah
    Commented Feb 20, 2023 at 20:09
  • $\begingroup$ @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 $\endgroup$
    – Jack Jack
    Commented Feb 20, 2023 at 20:13
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    $\begingroup$ Fermi sphere is in k-space - what you have here is a constant rate of change for k (that rate of change of momentum is acceleration in terms of coordinates is not relevant here... especially viewed from QM point of view.) $\endgroup$
    – Roger V.
    Commented Feb 20, 2023 at 20:17
  • $\begingroup$ What is the question? If there is no resistance, as in this simple model, then yes, they accelerate "forever". $\endgroup$
    – Quillo
    Commented Feb 20, 2023 at 20:49

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The key part is

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.

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