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Diffusive Shock Acceleration, resulting from first-order Fermi accelerationoccurring near Magnetohydrodynamic shock waves, is essential in explaining thepower law spectrum in various astrophysical radiation and cosmic rays. Weperform Monte Carlo simulations to model the stochastic scattering process inFermi acceleration, capturing the confinement of particles around the shockwithin the ambient fluid. The model is tested and validated by comparing itwith the spectral index obtained with analytical calculation. Assuming arelativistic EoS, we calculate the power-law spectral index for differentdiffusion coefficients. With constant diffusion co-efficient and stiffer EoS,the observed range of the spectral index is very narrow; however, as the EoSbecomes softer, the range increases. With varying diffusion co-efficientstiffer EoS fails to give a well-defined spectral index (no linear spectrum);however, as the EoS becomes softer, the spectral index lies between $2-4$. Forultra-relativistic shocks, we consistently obtained a linear spectrum; however,the spectral index range varied considerably with the diffusion coefficient.

Effect of relativistic equation of state and diffusion coefficient on diffusive shock acceleration