  The dynamics of two initially unmagnetized relativistic counter-streaming
homogeneous ion-electron plasma beams are simulated in two dimensions using the
particle-in-cell (PIC) method. It is shown that current filaments, which form
due to the Weibel instability, develop a large scale longitudinal electric
field in the direction opposite to the current carried by the filaments as
predicted by theory. Fast moving ions in the current filaments decelerate due
to this longitudinal electric field. The same longitudinal electric field,
which is partially inductive and partially electrostatic, is identified as the
main source of acceleration of electrons in the current filaments. The
transverse electric field, though larger than the longitudinal one, is shown to
play a smaller role in heating electrons, contrary to previous claims. It is
found that, in 1D, the electrons become strongly magnetized and are
\textit{not} accelerated beyond their initial kinetic energy. Rather, the
heating of the electrons is enhanced by the bending and break-up of the
filaments, which releases electrons that would otherwise be trapped within a
single filament and hence slow the development of the Weibel instability (i.e.
the magnetic field growth) via induction as per Lenz's law. In 2D simulations
electrons are heated to about one quarter of the initial kinetic energy of
ions. The magnetic energy at maximum is about 4 percent, decaying to less than
1 percent by the end of the simulation. Most of the heating of electrons takes
place while the longitudinal electric field is still growing while only a small
portion of the heating is a result of subsequent magnetic field decay. The ions
are found to gradually decelerate until the end of the simulation by which time
they retain a residual anisotropy less than 10 percent.
