The telegraph equation in charged particle transport

We present a new derivation of the telegraph equation which modifies its coefficients. First, an infinite-order partial differential equation is obtained for the velocity-space solid angle-averaged phase-space distribution of particles which underwent at least a few collisions. It is shown that in the lowest order asymptotic expansion this equation simplifies to the well-known diffusion equation. The second-order asymptotic expansion for isotropic small-angle scattering results in a modified telegraph equation with a signal propagation speed of v(5/11)^1/2 instead of the usual v/3^1/2. Our derivation of a modified telegraph equation follows from an expansion of the Boltzmann equation in the relevant smallness parameters and not from a truncation of an eigen-function expansion. This equation is consistent with causality. It is shown that under steady state conditions in a convecting plasma the telegraph equation may be regarded as a diffusion equation with a modified transport coefficient, which describes a combination of diffusion and cosmic-ray inertia. This modified transport coefficient becomes negative for particles with random velocities less than the critical velocity, v_c. This negative value is a consequence of the second time derivative term in the telegraph equation and it is closely related to causality.