  Nonthermal radiation observed from astrophysical systems containing
relativistic jets and shocks, e.g., gamma-ray bursts (GRBs), active galactic
nuclei (AGNs), and Galactic microquasar systems usually have power-law emission
spectra. Recent PIC simulations of relativistic electron-ion
(electron-positron) jets injected into a stationary medium show that particle
acceleration occurs within the downstream jet. In the presence of relativistic
jets, instabilities such as the Buneman instability, other two-streaming
instability, and the Weibel (filamentation) instability create collisionless
shocks, which are responsible for particle (electron, positron, and ion)
acceleration. The simulation results show that the Weibel instability is
responsible for generating and amplifying highly nonuniform, small-scale
magnetic fields. These magnetic fields contribute to the electron's transverse
deflection behind the jet head. The ``jitter'' radiation from deflected
electrons in small-scale magnetic fields has different properties than
synchrotron radiation which is calculated in a uniform magnetic field. This
jitter radiation, a case of diffusive synchrotron radiation, may be important
to understand the complex time evolution and/or spectral structure in gamma-ray
bursts, relativistic jets, and supernova remnants.
