Relaxation of weights in a $\delta f$ code

The weight of particle $i$ in a $\delta f$ particle following code obeys $dw_i/dt=\dot{W}(\vec{x}_i,\vec{p}_i)$. The phase space location of the particle is $(\vec{x}_i,\vec{p}_i)$, and $\dot{W}$ is given by derivatives of a background Maxwellian. The weights $w_i$ tend to increase without limit. When the $w_i$ become sufficiently large the approximations used in $\delta f$ codes become invalid. The long-term increase in $w_i$ is unphysical since a particle should loose its history within a collision time. The addition of a term to the weight evolution equation solves the problem, $dw_i/dt=\dot{W}-\nu_ww_i$. If the constant $\nu_w$ is chosen to be comparable to, or smaller than, the collision frequency it should have no effect on physically correct outputs of the code, but should keeps the weights at a low amplitude forever. In order to conserve particles, the background Maxwellian must be modified so its density obeys $dn(\psi)/dt=\nu_wn \sum_iw_if_i/\sum_if_i$, where the sum is over an annular radial region. Similar changes in flow and temperature of the background Maxwellian are required to conserve momentum, and energy.