Debye length

Relevant youtube video
EdX EPFL course:
![[plasma_lecture1.mp4]]

Debye length

Initially neutral plasma :

n_{e} = n_{i} = n_{o}

What if a test charge is introduced in the initially neutral plasma?
What will the potential around the test charge be?
Using the Poisson Equation of the system:

\nabla^{2} V = \frac{1}{ϵ_{0}} [q_{t} δ (r) + e n_{i} (r) - e n_{e} (r)]

Assumptions:

Ions are very heavy : fixed positive charge background $n_{i} (r) = n_{0}$
Electrons move according to the Newton's 2nd Law under the influence of Electric Force and Pressure

m_{e} \frac{d \vec{u_{e}}}{d t} = - e \vec{E} - \frac{1}{n_{e}} \nabla p_{e} = - e \vec{E} - \frac{1}{n_{e}} \nabla (n_{e}, k_{B}, T_{e})

from the Equation of state

Hypothesis for electrons:

e are very light : $m_{e} \to 0$
Electric Field: $\vec{E} = - \nabla V$
$T_{e} = c o n s t$ - Closed system - Heat Bath - Canonical ensemble
Result:

0 = e \nabla V - T_{e} \frac{\nabla n_{e}}{n_{e}}

where $T_{e} = k_{B} * T$ since the electron temperature behaves according to the Maxwell - Boltzmann Distribution

0 = \nabla (e V - T_{e} l o g n_{e})

e V - T_{e} l o g n_{e} = c o n s t

n_{e} = n_{o} e^{e V / T_{e}}

The constant was calculated through the constrain that the initial density of the electrons is equal to $n_{e} = n_{0} = n_{i}$

Questions

how is the D.E of the potential solved ?
where did we use the statistical approach?
how did we find the condition (minimum radius) for one to one interactions to be weak