We probe the physical conditions in high redshift galaxies, specifically, the Damped Lyman-alpha Systems (DLAs) using neutral carbon (CI) fine structure lines and molecular hydrogen (H2). We report five new detections of CI and analyze the CI in an additional 2 DLAs with previously published data. We also present one new detection of H2 in a DLA. We present a new method of analysis that simultaneously constrains emph{both} the volume density and the temperature of the gas, as opposed to previous studies that a priori assumed a gas temperature. We use only the column density of CI measured in the fine structure states and the assumption of ionization equilibrium in order to constrain the physical conditions in the gas. We present a sample of 11 CI velocity components in 6 DLAs and compare their properties to those derived by the global CII* technique. The resulting median values for this sample are: <n(HI)> = 69 cm^{-3}, <T> = 50 K, and <log(P/k)> = 3.86 cm^{-3} K, with standard deviations, sigma_{n(HI)} = 134 cm^{-3}, sigma_T = 52 K, and sigma_{log(P/k)} = 3.68 cm^{-3} K. This can be compared with the integrated median values for the same DLAs : <n(HI)> = 2.8 cm^{-3}, <T> = 139 K, and <log(P/k)> = 2.57 cm^{-3} K, with standard deviations sigma_{n(HI)} = 3.0 cm^{-3}, sigma_T = 43 K, and sigma_{log(P/k)} = 0.22 cm^{-3} K. Interestingly, the pressures measured in these high redshift CI clouds are similar to those found in the Milky Way. We conclude that the CI gas is tracing a higher-density, higher-pressure region, possibly indicative of post-shock gas or a photodissociation region on the edge of a molecular cloud. We speculate that these clouds may be direct probes of the precursor sites of star formation in normal galaxies at high redshift.
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