  We show that the standard theory of thermal production and chemical
decoupling of WIMPs is incomplete. The hypothesis that WIMPs are produced and
decouple from a thermal bath implies that the rate equation the bath particles
interacting with the WIMPs is an algebraic equation that constraints the actual
WIMPs abundance to have a precise analytical form down to the temperature
$x_\ast=m_\chi /T_\ast$. The point $x_\ast$, which coincides with the
stationary point of the equation for the quantity $\Delta= Y-Y_0$, is where the
maximum departure of the WIMPs abundance $Y$ from the thermal value $Y_0$ is
reached. For each mass $m_\chi$ and total annihilation cross section $\langle
\sigma_\text{ann}v_\text{r}\rangle$, the temperature $x_\ast$ and the actual
WIMPs abundance $Y(x_\ast)$ are exactly known. This value provides the true
initial condition for the usual differential equation that have to be
integrated in the interval $x\ge x_\ast$. The matching of the two abundances at
$x_\ast$ is continuous and differentiable. The dependence of the present relic
abundance on the abundance at an intermediate temperature is an exact result.
The exact theory suggests a new analytical approximation that furnishes the
relic abundance accurate at the level of $1\%-2\%$ in the case of $S$-wave and
$P$-wave scattering cross sections. We conclude the paper studying the
evolution of the WIMPs chemical potential and the entropy production using
methods of non equilibrium thermodynamics.
