Maxwell-Jüttner distribution for T = mc² and . Upper plot: contours are drawn from the exact expression 2πrg(r,0,p_y), with g from Eq. (13) and (r,0,p_y) defined in Appendix D. The background color map is the 2D histogram of for 10⁶ particles generated according to our method. Middle plot: normalized histogram of p_y for the particles (red dotted), to be compared to the exact expression in Eq. (D.2) (blue line), and for comparison (black dots) the histogram of p_y for particles generated the wrong way (initialization in the rest frame and boost of Γ₀). Bottom plot: difference between the red points and the blue curve.