\neq k$, however, the two eigenfunctions are different and the time-dependence of the integrand is therefore more complicated and does affect the value of the integral. The average momentum therefore has a time-independent part $K_0$ consisting of the first two terms in Eq. and a time-dependent part $K_t(t)$ which consists of the last term.

The effect of the underlying many-body phase of the TG gas on the average transport in the system should be reflected in the average momentum and Eq.  allows us to explain the observed behavior in some mathematical detail. Furthermore, the same behaviour appears for a spin-polarised Fermi gas, even though its momentum distribution is entirely different. This is because the expectation value of the momentum operator is the same for the Fermi and TG gas, which means that the average momentum alone is insufficient for fully characterising the dynamics of the different many-body bosonic phases. We will therefore investigate and compare the average momentum, the coherence and the momentum distribution.

Preliminary consideration - rotation speed {#subsubsec:rotationspeed}
------------------------------------------

![$\langle K \rangle/N$ as a function of $q$ for different filling ratios and depths. In (a) the black line corresponds to $F=\frac{2}{5}$ and $V_0=0.02 E_r$, while the red line 