### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 377-384 |

Number of pages | 8 |

Journal | Astrophysical Journal |

Volume | 403 |

Issue number | 1 |

State | Published - 1993 |

### Fingerprint

### Keywords

- Acceleration of particles
- Cosmic rays
- Diffusion

### ASJC Scopus subject areas

- Space and Planetary Science

### Cite this

*Astrophysical Journal*,

*403*(1), 377-384.

**The telegraph equation in charged particle transport.** / Gombosi, T. I.; Jokipii, J. Randy; Kota, J.; Lorencz, K.; Williams, L. L.

Research output: Contribution to journal › Article

*Astrophysical Journal*, vol. 403, no. 1, pp. 377-384.

}

TY - JOUR

T1 - The telegraph equation in charged particle transport

AU - Gombosi, T. I.

AU - Jokipii, J. Randy

AU - Kota, J.

AU - Lorencz, K.

AU - Williams, L. L.

PY - 1993

Y1 - 1993

N2 - 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.

AB - 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.

KW - Acceleration of particles

KW - Cosmic rays

KW - Diffusion

UR - http://www.scopus.com/inward/record.url?scp=12044251484&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=12044251484&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:12044251484

VL - 403

SP - 377

EP - 384

JO - Astrophysical Journal

JF - Astrophysical Journal

SN - 0004-637X

IS - 1

ER -