  Both direct and indirect weak nonresonant interactions are shown to produce
entanglement between two initially disentangled systems prepared as a tensor
product of thermal states, provided the initial temperature is sufficiently
low. Entanglement is determined by the Peres-Horodeckii criterion, which
establishes that a composite state is entangled if its partial transpose is not
positive. If the initial temperature of the thermal states is higher than an
upper critical value $T_{uc}$ the minimal eigenvalue of the partially
transposed density matrix of the composite state remains positive in the course
of the evolution. If the initial temperature of the thermal states is lower
than a lower critical value $T_{lc}\leq T_{uc}$ the minimal eigenvalue of the
partially transposed density matrix of the composite state becomes negative
which means that entanglement develops. We calculate the lower bound $T_{lb}$
for $T_{lc}$ and show that the negativity of the composite state is negligibly
small in the interval $T_{lb}<T<T_{uc}$. Therefore the lower bound temperature
$T_{lb}$ can be considered as \textit{the} critical temperature for the
generation of entanglement.
