Context. Mrk 421 displayed its highest flux state ever observed in February 2010 with very high tera-electronvolt fluxes and interesting cross-band correlations and a spectral energy distribution (SED) evolution not entirely consistent with the standard single zone leptonic synchrotron self-Compton model. The source was already in a high state in January 2010 and displayed strong variability in the days preceding the highest state. We study the temporal evolution of the spectra in January to extract information about the particle dynamics and the physical properties of the emission region.

Aims. We build on the temporal variability and correlations studied in our previous work and attempt to improve the SED model fits with a physics-oriented approach.

Methods. The The multi-wavelength data were processed and the SEDs were fit using JetSeT. The SED evolution and cross-band correlations were modelled using leptonic log-parabola with a low energy power-law branch (LPPL) and pile-up distributions that are predicted in a stochastic acceleration scenario. A simplified temporal evolution model was developed and fitted to the SEDs and the resulting trends and phenomenology were characterised in the context of the theoretical literature. An expanding emission region model was also tested.

Results. We find the spectral variability to be in good agreement with stochastic acceleration. Our analysis suggests that the standard LPPL distribution develops a Maxwellian pile-up component at the transition from the acceleration-dominated to the cooling-dominated phase on 3 nights in the dataset, as is also hinted at by the very-high energy and X-ray light curves. The resulting phenomenology of our sequential snapshot evolution SED model agrees well with theoretical and numerical simulation studies on temporal evolution using the diffusion equation approach. Curvature in the electron energy distribution (EED) anti-correlates with the synchrotron peak frequency, as was expected from stochastic acceleration. We tested an alternate model with expanding emission region and the resulting EED spectral index, expansion velocity, and magnetic field index agree with prior works using datasets collected in different flaring states of the source.