Possibly the greatest challenge of disruptions to ITER and future tokamaks is the transfer of the plasma current from thermal to runaway-electron carriers of relativistic energy. When relativistic electrons carry the full plasma current in ITER, the fraction of the electrons that are relativistic is approximately $2\times10^{-4}$ and those electrons have a typical energy of 10~MeV, a thousand times the typical 10 keV thermal energy. Consequently, the total energy deposition when relativistic electrons strike the chamber walls is approximately 10\% of the energy in thermal quench associated with a disruption. Nevertheless, the damage can be far greater because the deposition can be highly localized in space and time. Although this localization is often seen in experiments, it does not always occur. When runaway-containing plasmas in JET [1] and DIII-D [2] were made highly unstable, relativistic electrons were deposited sufficiently uniformly on the walls to avoid the energy concentration problems. The success of ITER and the fusion relevance of tokamaks requires the extreme damage of runaways be avoided, and thus defines the importance of determining why runaway loss is sometimes concentrated and sometimes not. Runaways are small-gyroradius passing electrons and follow the lines of an effective magnetic field. These field lines and are given by a 1.5 degree of freedom Hamiltonian, which is the poloidal flux $\psi_p(\psi,\theta,\varphi)$, where $\psi$ is the toroidal flux, $\theta$ is a poloidal and $\varphi$ is a toroidal angle. The properties of such Hamiltonians can be studied by iterating an area-preserving map. The map we chose [3] was $\varphi_{j+1}=\varphi_j +\delta\varphi$, $\psi_{j+1} = \psi_j +\Delta_j(\theta_j)\delta\varphi$ and $\theta_{j+1}=\theta_j +\iota(\psi_{j+1})\delta\varphi$. Time evolves as $t _{j+1}= t_j+ \tau_t \delta\varphi/2\pi$, where the toroidal transit time of a relativistic electron is $t_t\equiv 2\pi R/c\approx 10^{-7}~$s in ITER . After each iteration, $\Delta_j(\theta)$ is changed to represent an evolving perturbation. The choices were $\iota=1-0.8\psi/\psi_a$, with $\psi_a$ a constant, $a$ is the minor radius of the annulus, and $\Delta_j(\theta)=\delta_j(\theta) + 5\psi_a\sin(5\theta-2\varphi_j)$ with $\delta_j =3(\delta_0+t_j/\tau_{ev})\sin(3\theta-\varphi_j)$ and $\delta_0=0.7\times 10^{-7}$. Initially, a large chaotic field line region is separated from the wall at $\psi_w=1.21\psi_a$ by an annulus of magnetic surfaces. The surfaces in the annulus break as the magnetic field evolves on the timescale $\tau_{ev}$. Once the last magnetic surface is broken, a turnstile opens [4], and the magnetic flux leaks in and out through the turnstile, which collimates the flux tube along which the electrons escape confinement into a tube that is a fraction $f\approx7\sqrt{\tau_t/\tau_{ev}}$ of the toroidal flux in the original chaotic region. The factor of $\approx7$ is model dependent. The $\tau_{ev}$ range of 100~ms to 0.1~ms gives $f\approx 7\times 10^{-3}$ to $f\approx 22\%$. A second possibility is that the perturbation does not evolve, but the plasma approaches the wall at some location at a speed $v=a/\tau_a$, maybe due to an ideal instability. Again a turnstile opens and the flux in the tube along which runaways strike the wall is $f_a\approx 2.7 (\tau_t/\tau_a)^{4/3}$ times the flux in the original chaotic region. An ideal instability can move at the poloidal Alfv\'en speed, which divided by the speed of light is $\sim3\times10^{-3}$. But, $2\pi R/a \sim20$, giving $f_a\sim 6\times10^{-2}$. When the instability grows ten times slower, $f_a\approx3\times10^{-3}$. On ITER a number of such relativistic electron dumps would be required since the transfer of the plasma current between the thermal and relativistic electrons can occur a number of times before the current is quenched [5]. Support is acknowledged from the U.S. Department of Energy, DE-SC0020107 to Hampton University and DE-FG02-95ER54333, DE-FG02-03ER54696, DE-SC0018424, and DE-SC0019479 to Columbia University. 1. C. Reux, C. Paz-Soldan, P. Aleynikov, et al., \emph{Demonstration of Safe Termination of Megaampere Relativistic Electron Beams in Tokamaks}, Phys. Rev. Lett. \textbf{126}, 175001 (2021). 2. C. Paz-Soldan, C. Reux, K. Aleynikova, et al., \emph{A novel path to runaway electron mitigation via deuterium injection and current-driven MHD instability}, Nucl. Fusion \textbf{61}, 116058 (2021). 3. A. H. Boozer and A. Pujabi, \emph{Loss of relativistic electrons when magnetic surfaces are broken}, Phys. Plasmas \textbf{23}, 102513 (2016); doi:10.1063/1.4966046. 4. R. S. MacKay, J. D. Meiss, and I. C. Percival \emph{Stochasticity and transport in Hamiltonian-systems}, Phys. Rev. Lett. \textbf{52},697 (1984); DOI10.1103/PhysRevLett.52.697. 5. J. McDevitt and X. Tang, \emph{Runaway electron current reconstitution after a non-axisymmetric magnetohydrodynamic flush}, $<$https://arxiv.org/pdf/2211.02160.pdf$>$ November 2022; DOI: 10.48550/arXiv.2211.02160.