We study near-equilibrium thermodynamics of bosonic atoms in a two-dimensional optical lattice by ramping up the lattice depth to convert a superfluid into an inhomogeneous mixture of superfluid and Mott insulator. Detailed study of in situ density profiles shows that, first, locally adiabatic ramps do not guarantee global thermal equilibrium. Indeed, full thermalization for typical parameters only occurs for experiment times which exceed one second. Secondly, ramping non-adiabatically to the Mott insulator regime can result in strong localized cooling at short times and global cooling once equilibrated. For an initial temperature estimated as 20 nK, we observe local temperatures as low as 1.5 nK, and a final global temperature of 9 nK. Possible cooling mechanisms include adiabatic decompression, modification of the density of states near the quantum critical regime, and the Joule-Thomson effect. **NOTE: Following submission of arXiv:0910.1382v1, a systematic correction was discovered in the density measurement, stemming from three-body losses during the imaging process. New measurements were performed, and the result is in support of the claim on the slow global dynamics. Due to the substantially altered methods and analysis, a new text has been posted as arXiv:1003.0855.
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