Fig. 6

Upper left: the equilibrium distribution \(\Pi (r,u)\) at the right boundary R/2. Its peak height scales with the cutoff \(\Lambda\) and its width scales with \(\Lambda ^{-1/3}\). Here we used \(\Lambda =20^3\) for illustration. Upper right: the function B(u) that controls the contribution of arena boundaries to the velocity distribution. Its peak height and width scale as \(2\ln \Lambda\) and \(1/\ln \Lambda\), respectively. We used \(\Lambda =10^6\) in this plot. Bottom left: the velocity distribution P(u), with the parameter \(\alpha =1.5\). Bottom right: the density \(n(r)-n(0)\) is plotted versus the distance d from the right boundary \(d=R/2-r\) (or left boundary \(d=r+R/2\)) using \(u_0=1.5\). The dashed line is the analytic approximation, for \(d<<1\), in Eq. 39