NICER Perspective on TeV Blazar Mrk~421: X-ray Variability and Particle Acceleration

Mrk~421 is one of the most fascinating blazars, widely studied across the electromagnetic spectrum using observations at various wavebands, from radio to the TeV gamma ray bands. We present the first detailed spectral and timing analysis of the TeV blazar Mrk~421 based on 45 X-ray observations from the \textit{NICER} X-ray telescope, collected over two years from 2022 to 2024. The source exhibits strong X-ray variability across intraday and long-term timescales. During this period, we observe a dramatic change in flux, from $\sim 50$ to $\sim 1380$~cts~s$^{-1}$, representing a $\sim 28$-fold increase. Spectral modeling with power-law, broken power-law, and log-parabolic functions shows that the log-parabola provides the most accurate description of the X-ray spectra. The hardness ratio analysis confirms a \textit{harder-when-brighter} trend, consistent with the anticorrelation between flux and photon index($\Gamma$). Correlation studies reveal a positive relation between the photon index ($\alpha$) and the curvature parameter ($\beta$) of the log-parabola model, a negative correlation between $\beta$ and synchrotron peak energy ($E_{\mathrm{p}}$), and a positive correlation between $E_{\mathrm{p}}$ and flux. In addition, the observed rapid variability indicates that the X-ray emission originates from a compact region located close to the central engine. Furthermore, using a log-parabolic electron energy distribution within the synchrotron jet scenario, we simulate the observed anti-correlation between the $E_{\rm p}$ and $\beta$. These features can be interpreted within the framework of energy-dependent particle acceleration in blazar jets, which are often associated with turbulence, strong magnetic fields, and relativistic outflows.