Unexpected Behavior of the Local Compressibility near theB=0Metal-Insulator Transition

We have measured the local electronic compressibility of a two-dimensional hole gas as it crosses the $B\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ metal-insulator transition. In the metallic phase, the compressibility follows the mean-field Hartree-Fock (HF) theory and is found to be spatially homogeneous. In the insulating phase it deviates by more than an order of magnitude from the HF predictions and is spatially inhomogeneous. The crossover density between the two types of behavior agrees quantitatively with the transport critical density, suggesting that the system undergoes a thermodynamic change at the transition.