Particle-conserving displacement of a Fermi circle ($\nu (\theta )=p_0\cos \theta +\sqrt{p_F^2-p_0^2{\sin }^2\theta }-p_F$)

Translating a Fermi circle of radius $p_F$ by $p_0\hat{\mathbf{x}}$ preserves its area. For $p_0<p_F$, its radial displacement from the original centre is the displayed expression. Its Taylor expansion is $\nu =p_0\cos \theta -p_0^2{\sin }^2\theta /(2p_F)+O(p_0^4/p_F^3)$. The second-order term cancels the apparent area increase from the first-order shape: ${\int }_0^{2\pi }(p_F\nu +{\nu }^2/2)d\theta =0$. This distinction matters when converting a grand-potential variation into a fixed-particle number energy change.