  Charged plasma and Fermi liquid are two distinct states of electronic matter
intrinsic to dilute two-dimensional electron systems at elevated and low
temperatures, respectively. Probing their thermodynamics represents challenge
because of lacking an adequate technique. Here we report thermodynamic method
to measure the entropy per electron in gated structures. Our technique appears
to be three orders of magnitude superior in sensitivity to the ac calorimetry,
allowing entropy measurements with only $10^8$ electrons. This enables us to
investigate the correlated plasma regime, previously inaccessible
experimentally in two-dimensional electron systems in semiconductors. In
experiments with clean two-dimensional electron system in Si-based structures
we traced entropy evolution from the plasma to Fermi-liquid regime by varying
electron density. We reveal that the correlated plasma regime can be mapped
onto the ordinary non-degenerate Fermi gas with an interaction-enhanced
temperature dependent effective mass. Our method opens up new horizons in
studies of low-dimensional electron systems.
