We present the spectral and temporal radiative signatures expected within the ``supercritical pile'' model of gamma-ray bursts (GRBs). This model is motivated by the need for a process that provides the dissipation necessary in GRBs and presents a well-defined scheme for converting the energy stored in the relativistic protons of the relativistic blast waves (RBWs) associated with GRBs into radiation; at the same time, it leads to spectra that exhibit a peak in the burst νFν distribution at an energy Epeak~=1 MeV in the observer's frame, in agreement with observations and largely independent of the Lorentz factor Γ of the associated relativistic outflow. Furthermore, this scheme does not require (but does not preclude) acceleration of particles at the shock other than that provided by the isotropization of the flow bulk kinetic energy in the RBW frame. In the present paper we model in detail the evolution of protons, electrons, and photons from a RBW to produce detailed spectra of the prompt GRB phase as a function of time from across a very broad range in frequency, spanning roughly 4logΓ decades. The model spectra are in general agreement with observations and provide a means for delineating the model parameters through direct comparison with trends observed in GRB properties.

Aims.We analyse the influence of the stochastic particle acceleration for the evolution of the electron spectrum. We assume that all investigated spectra are generated inside a spherical, homogeneous source and also analyse the synchrotron and inverse Compton emission generated by such an object. Methods: .The stochastic acceleration is treated as the diffusion of the particle momentum and is described by the momentum-diffusion equation. We investigate the stationary and time dependent solutions of the equation for several different evolutionary scenarios. The scenarios are divided into two general classes. First, we analyse a few cases without injection or escape of the particles during the evolution. Then we investigate the scenarios where we assume continuous injection and simultaneous escape of the particles. Results: .In the case of no injection and escape the acceleration process, competing with the radiative cooling, only modifies the initial particle spectrum. The competition leads to a thermal or quasi-thermal distribution of the particle energy. In the case of the injection and simultaneous escape the resulting spectra depend mostly on the energy distribution of the injected particles. In the simplest case, where the particles are injected at the lowest possible energies, the competition between the acceleration and the escape forms a power-law energy distribution. We apply our modeling to the high energy activity of the blazar Mrk 501 observed in April 1997. Calculating the evolution of the electron spectrum self-consistently we can reproduce the observed spectra well with a number of free parameters that is comparable to or less than in the "classic stationary" one-zone synchrotron self-Compton scenario.