Small-signal power flow and energy density for streaming carriers in the presence of collisions

The Poynting theorem, or equation of continuity relating energy density and power flow, is extended to include the effects of diffusion and collisions. It is shown that the presence of the thermal power due to diffusion is accompanied by an increase in electrokinetic power. The effects of collisions on the electrokinetic power and energy density are examined in detail in the absence of diffusion. It is found that the kinetic power is zero in isolated streams when collisions are frequent (
\nu > 2\omega_{p}
). However, when such streams couple to "circuit-like" positive-energy waves (as in the acoustic amplifier) the stream's kinetic power becomes finite; in particular, it becomes negative if the stream drifts faster than the wave. Thus the usual picture, used in collision-free theory, in which the active wave must carry negative power, is preserved. It is also shown that, on the other hand, if the stream is lossless but interacts with a lossy and nonpropagating medium (as in a stationary collisional plasma), then the stream's finite-energy waves are coupled by the lossy medium. In contrast to the previous case where both systems propagate, it is now the negative-energy wave that grows. It is found that the electrokinetic energy density in a collision-dominant stream is negative for both modes due to collisional losses.