Collisionless Shocks

Collisionless fluids

Plasmas can be considered a neutral gas if their microscopic electric fields are screened out over a distance known as the Debye length \[\lambda_{\rm D} = \sqrt{\frac{\epsilon_0 k T}{n e^2}} = \frac{v_{\rm th}}{\omega} \ ,\] where

\[\begin{aligned} \epsilon_0 &= {\rm permittivity~of~free~space} \ , \\ k &= {\rm Boltzmann~constant} \ , \\ T &= {\rm plasma~temperature} \ , \\ n &= {\rm number~density} \ , \\ v_{\rm th} &= {\rm thermal~speed} \ , \\ \omega &= {\rm plasma~frequency} \ .\end{aligned}\]

The plasma frequency is defined as \[\omega \equiv \sqrt{\frac{n q^2}{m \epsilon_0}} \ ,\] where the condition for defining a gas as a plasma is \[n \lambda_{\rm D}^3 \ll 1 \ ,\] which states that the number of particles in a Debye sphere is large. Most plasmas are considered collisionless, which means that the Coulomb collision time, \(\tau_{\rm c}\) is much larger than any relevant plasma time scale. In other words, the average mean free path of a particle is often much larger than the gradient scale lengths of waves or density fluctuations.

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