  We recently found that streaming cosmic rays (CRs) induce a resistive
electric field that can accelerate secondary electrons produced by CR
ionization. In this work, we study the evolution of the energy spectrum of
secondary electrons by numerically solving the one-dimensional Boltzmann
equation and Ohm's law. We show that the accelerated secondary electrons
further ionize a gas, that is, the electron avalanche occurs, resulting in
increased ionization and excitation of the gas. Although the resistive electric
field becomes weaker than one before the CR discharge, the weak resistive
electric field weakly accelerates the secondary electrons. The quasi-steady
state is almost independent of the initial resistive electric field, but
depends on the electron fraction in the gas. The resistive electric field in
the quasi-steady state is larger for the higher electron fraction, which makes
the number of secondary electrons that can ionize the gas larger, resulting in
a higher ionization rate. The CR discharge could explain the high ionization
rate that are observed in some molecular clouds.
