On the Transport Properties of Charged Particles in One Dimension in Random Electric Fields

Abstract

view Abstract Citations (6) References (7) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On the Transport Properties of Charged Particles in One Dimension in Random Electric Fields Lerche, I. Abstract First-order smoothing theory is a commonly used approximation in obtaining estimates of the transport properties of charged particles in turbulent electric and magnetic fields. In order to demonstrate as succinctly as possible some of the changes introduced into the transport equations when the first-order smoothing approximation replaces a statistically exact description of the turbulent processes at play, we discuss the problem of one-dimensional charged-particle evolution under a turbulent electric field. We set up and discuss the behavior of charged particles in a one- dimensional turbulent electric field using a statistical pair of equations obtained by invoking the quasi-normality hypothesis. We demonstrate how the first-order smoothing approximation is obtained from this pair of equations. We also set up and discuss the behavior of charged particles in a one-dimensional turbulent electric field using a Gaussian distribution for the electric field. Comparison of the behavior of the three problems indicates several similarities and several differences. First, the evolution of the mean number density for Gaussian statistics of the turbulent electric field is governed by a diffusion equation for all time. Second, by consideration of a simple illustrative example we show that for times long compared to a correlation time of the turbulent electric field both the pair of equations obtained under the quasi-normality hypothesis, and the first-order smoothing approximation to the pair, reduce to diffusion equations but with numerically different diffusion coefficients. This implies that the first- order smoothing approximation gives a qualitatively correct picture of the evolution of the mean number density for long times-at least for our simple problem. Third, by considering the case of times short compared to a correlation time of the turbulent electric field we show that both the pair of equations obtained under the quasi-normality hypothesis, and the first-order smoothing approximation to the pair, reduce to wave equations for the mean number density, but with numerically different momentum propagation speeds. This implies that the first-order smoothing approximation also gives a qualitatively correct picture of the evolution of the mean number density for short times-at least for our simple problem. We also point out that the evolution of the mean number density under Gaussian statistics for the turbulent electric field does not give a wave equation for times short compared to a correlation time. We are of the opinion that the difference between these results for short times is due to the different statistical assumptions in force in both cases. Finally, we point out what remains to be done if we are to better understand both the regime of applicability of the commonly used first-order smoothing approximation and the transport properties of charged particles in turbulent fields. Publication: The Astrophysical Journal Pub Date: December 1972 DOI: 10.1086/151837 Bibcode: 1972ApJ...178..819L full text sources ADS |