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We report the results of a new analysis of 13 wide band BeppoSAX observations of the BL Lac object Mkn 421. The data from LECS, MECS and PDS, covering an energy interval from 0.1 to over 100keV, have been used to study the spectral variability of this source. We show that the energy distributions in different luminosity states can be fitted very well by a log-parabolic law F(E)=K(E/E₁)^-(a+b Log(E/E₁))^, which provides good estimates of the energy and flux of the synchrotron peak in the SED. In the first four short observations of 1997 Mkn 421 was characterized by a very stable spectral shape, with average values a=2.25 and b=0.45, independently of the source brightness and of the fact that the source luminosity was increasing or decreasing. In the observations of 1998 smaller values for both parameters, a≃2.07 and b≃0.34, were found and the peak energy in the SED was in the range 0.5-0.8keV. The observations of May 1999 and April-May 2000 covered runs of a duration of several days and provided a very high number of events for all the instruments. The resulting spectral fits were then limited by some instrumental systematics. Also in these cases, the log-parabolic model gave a satisfactory description of the overall SED of Mkn 421. In particular, in the observations of spring 2000 the source was brighter than the other observations and showed a large change of the spectral curvature. Spectral parameters estimates gave a≃1.8 and b≃0.19 and the energy of the maximum in the SED moved to the range 1-5.5keV. We give a possible interpretation of the log-parabolic spectral model in terms of particle acceleration mechanisms. An energy distribution of emitting particles with curvature close to the one observed can be explained by a simple model for statistical acceleration with the assumption that the probability for a particle to increase its energy is a decreasing function of the energy itself. A consequence of this mechanism is the existence of a linear relation between the spectral parameters a and b, well confirmed by the estimated values of these two parameters for Mkn 421.