Approach to the thermodynamic limit in lattice QCD at μ≠0

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 μ<mπ/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 Li→∞ at fixed temperature 1/L0, the average phase factor vanishes. In the low temperature limit where Li/L0 is fixed as Li becomes large, the average phase factor approaches 1 for μ<mπ/2. The results for a finite volume compare well with lattice results obtained by Allton et al. After taking appropriate limits, we reproduce previously derived results for the regime and for one-dimensional QCD. The distribution of the phase itself is also computed.