The Approach to the Thermodynamic Limit in Lattice QCD at μneq0

The expectation value of the complex phase factor of the fermion determinant is computed to leading order in the $p$-expansion of the chiral Lagrangian. The computation is valid for $\mu<m_\pi/2$ and determines the dependence of the sign problem on the volume and on the geometric shape of the volume. In the thermodynamic limit with $ L_i \to \infty $ at fixed temperature $1/L_0$, the average phase factor vanishes. In the low temperature limit where $L_i/L_0$ is fixed as $L_i$ becomes large the average phase factor approaches one. The results for a finite volume compare well with lattice results obtained by Allton {\it et al}.. After taking appropriate limits, we reproduce previously derived results for the $\epsilon$-regime and for 1-dimensional QCD. The distribution of the phase itself is also computed.