Introduction

Consider an unmagnetized, uniform, 1-dimensional plasma consisting of $N$ electrons and $N$ unit-charged ions. Now, ions are much more massive than electrons. Hence, on short time-scales, we can treat the ions as a static neutralizing background, and only consider the motion of the electrons. Let $r_i$ be the $x$-coordinate of the $i$th electron. The equations of motion of the $i$th electron are written:

$$\frac{dr_i}{dt} = v_i,$$ (289)

$$\frac{dv_i}{dt} = - \frac{e\,E(r_i)}{m_e},$$ (290)

where $e>0$ is the magnitude of the electron charge, $m_e$ the electron mass, and $E(x)$ the $x$-component of the electric field-strength at position $x$. Now, the electric field-strength can be expressed in terms of an electric potential:

$$E(x) = - \frac{d\phi(x)}{dx}.$$ (291)

Furthermore, from the Poisson-Maxwell equation, we have

$$\frac{d^2\phi(x)}{dx^2} = -\frac{e}{\epsilon_0}\,\{n_0- n(x)\},$$ (292)

where $\epsilon _0$ is the permittivity of free-space, $n(x)$ the electron number density (i.e., $n(x)\,dx$ is the number of electrons in the interval $x$ to $x+dx$), and $n_0$ the uniform ion number density. Of course, the average value of $n(x)$ is equal to $n_0$, since there are equal numbers of ions and electrons.

Let us consider an initial electron distribution function consisting of two counter-propagating Maxwellian beams of mean speed $v_b$ and thermal spread $v_{th}$: i.e.,

$$f(x,v) = \frac{n_0}{2}\left\{\frac{1}{\sqrt{2\,\pi}\,v_{th}}... ...,\pi}\,v_{th}}\,{\rm e}^{-(v+v_b)^2/2\,v_{th}^{\,2}} \right\}.$$ (293)

Here, $f(x,v)\,dx\,dv$ is the number of electrons between $x$ and $x+dx$ with velocities in the range $v$ to $v+dv$. Of course, $n(x) = \int_{-\infty}^{\infty} f(x,v)\,dv$. The beam temperature $T$ is related to the thermal velocity via $v_{th} = \sqrt{k_B\,T/m_e}$, where $k_B$ is the Boltzmann constant. It is well-known that if $v_b$ is significantly larger than $v_{th}$ then the above distribution is unstable to a plasma instability called the two-stream instability.³⁸ Let us investigate this instability numerically.