  We propose a criterion for the singularity formation of the pressureless
Euler-Poisson system equipped with the Boltzmann relation, which describes the
dynamics of cold ions in an electrostatic plasma. Under the proposed criterion,
we prove that the smooth solutions develop a $C^1$ blow-up in a finite time and
obtain their temporal blow-up rates. In general, it is known that smooth
solutions to nonlinear hyperbolic equations fail to exist globally in time when
the gradient of initial velocity is negatively large. In contrast, our blow-up
condition does not require the largeness of gradient of velocity, and our
result particularly implies that the smooth solutions can break down even if
the gradient of initial velocity is trivial.
