Low-frequency fluctuations in plasma magnetic fields

Abstract

It is shown that even a nonmagnetized plasma with temperature T sustains zero-frequency magnetic fluctuations in thermal equilibrium. Fluctuations in electric and magnetic fields, as well as in densities, are computed. Four cases are studied: a cold, gaseous, isotropic, nonmagnetized plasma; a cold, gaseous plasma in a uniform magnetic field; a warm, gaseous plasma described by kinetic theory; and a degenerate electron plasma. For the simple gaseous plasma, the fluctuation strength of the magnetic field as a function of frequency and wave number is calculated with the aid of the fluctuation-dissipation theorem. This calculation is done for both collisional and collisionless plasmas. The magnetic-field fluctuation spectrum of each plasma has a large zero-frequency peak. The peak is a Dirac \ensuremath{\delta} function in the collisionless plasma; it is broadened into a Lorentzian curve in the collisional plasma. The plasma causes a low-frequency cutoff in the typical blackbody radiation spectrum, and the energy under the discovered peak approximates the energy lost in this cutoff. When the imposed magnetic field is weak, the magnetic-field wave-vector fluctuation spectra of the two lowest modes are independent of the strength of the imposed field. Further, these modes contain finite energy even when the imposed field is zero. It is the energy of these modes that forms the zero-frequency peak of the nonmagnetized plasma. In deriving these results, a simple relationship between the dispersion relation and the fluctuation power spectrum of electromagnetic waves is found. The warm plasma is shown, by kinetic theory, to exhibit a zero-frequency peak in its magnetic-field fluctuation spectrum as well. For the degenerate plasma, we find that electric-field fluctuations and number-density fluctuations vanish at zero frequency; however, the magnetic-field power spectrum diverges at zero frequency.