The telegraph equation in charged particle transport

T. I. Gombosi, J. R. Jokipii, J. Kota, K. Lorencz, L. L. Williams

Research output: Contribution to journalArticlepeer-review

46 Scopus citations


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/31/2. 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, vc. This negative value is a consequence of the second time derivative term in the telegraph equation and it is closely related to causality.

Original languageEnglish (US)
Pages (from-to)377-384
Number of pages8
JournalAstrophysical Journal
Issue number1
StatePublished - 1993


  • Acceleration of particles
  • Cosmic rays
  • Diffusion

ASJC Scopus subject areas

  • Astronomy and Astrophysics
  • Space and Planetary Science


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