  We present a detailed analysis of the dayside atmosphere of the hot-Neptune
GJ~436b, based on recent Spitzer observations. We report statistical
constraints on the thermal and chemical properties of the planetary atmosphere,
study correlations between the various molecular species, and discuss scenarios
of equilibrium and non-equilibrium chemistry in GJ 436b. We model the
atmosphere with a one-dimensional line-by-line radiative transfer code with
parameterized molecular abundances and temperature structure. We explore the
model parameter space with 10^6 models, using a Markov chain Monte Carlo
scheme. Our results encompass previous findings, indicating a paucity of
methane, an overabundance of CO and CO2, and a slight underabundance of H2O, as
compared to equilibrium chemistry with solar metallicity. The concentrations of
the species are highly correlated. Our best-fit, and most plausible,
constraints require a CH4 mixing ratio of 1.0E-7 to 1.0E-6, with CO >= 1.0E-3,
CO2 ~ 1.0E-6 to 1.0E-4, and H2O <= 1.0E-4; higher CH4 would require much higher
CO and CO2. Based on calculations of equilibrium and non-equilibrium chemistry,
we find that the observed abundances can potentially be explained by a
combination of high metallicity (~ 10 x solar) and vertical mixing with Kzz ~
10^6 - 10^7 cm^2/s. The inferred metallicity is enhanced over that of the host
star which is known to be consistent with solar metallicity. Our constraints
rule out a dayside thermal inversion in GJ 436b. We emphasize that the
constraints reported in this work depend crucially on the observations in the
two Spitzer channels at 3.6 micron and 4.5 micron. Future observations with
warm Spitzer and with the James Webb Space Telescope will be extremely
important to improve upon the present constraints on the abundances of carbon
species in the dayside atmosphere of GJ 436b.
