The effect of density-gradient terms in the free energy on density fluctuations near the critical point

Density-gradient terms may play a role in controlling density fluctuations in a fluid. It is shown that if these terms depend on the square of the density gradient they diminish in importance as the critical point is approached, in ordinary fluids and in the two-dimensional Ising lattice, but not in the three-dimensional Ising lattice or in four-dimensional systems. However, with a slight deviation from the square-gradient law, the gradient term may parallel the compressibility term, and it seems probable that this does occur, in conformity with Widom's hypothesis that the correlation length parallels the thickness of an interface between two phases. Critical-exponent relationships involved are discussed, and possible effects on the correlation length are considered. The introduction of a new critical singularity is discussed.