On particle number conservation

In the above, we see that the integration $\vert \int_{- \infty}^{\infty} \hat{F}_1 d v_z \vert$ decreases with time, which seems to be inconsistent with the conservation of particle number. Note the spatial dependence of the perturbed distribution function is $e^{i k z}$, i.e., the perturbed distribution function is given by $\hat{F}_1 e^{i k z}$, the real part of which corresponds to the physical distribution function, i.e., $F_1 (t, v_z, z) = A (t, v) \cos (k z + \alpha)$, where $A = \vert \hat{F}_1 \vert$ and $\alpha$ is the angle of $\hat{F}_1$ on the complex plane. The particle number for the distribution function $F_1$ in a region of the wave length $2 \pi / k$ is given by

$$N = \int_0^{2 \pi / k} \int_{- \infty}^{\infty} F_1 d v_z d z$$

$$= \int_0^{2 \pi / k} d z \int_{- \infty}^{\infty} A (t, v) \cos (k z + a) d v_z$$

$$= \int_{- \infty}^{\infty} A (t, v) \left( \int_0^{2 \pi / k} d z \cos (k z + \alpha) \right) d v_z$$ (90)

Note that the term in the parenthesises is zero. Therefore the above expression is always zero no matter what value the integration $\int_{- \infty}^{\infty} \hat{F}_1 d v_z$ is . Thus the number of particles is conserved in this case.