In this paper, we prove that the kappa distribution is the stationary solution of the Vlasov-Poisson system in an inhomogeneous plasma under the polytropic equation of state and an assumption restricting the local velocity distribution to a specific mathematical form. The profiles of density, temperature, and electric potential are obtained theoretically. The kappa index can be determined if the initial state is known. In order to verify the theory, particle-in-cell simulations are made and the results show excellent agreement with the theoretical predictions for density, temperature, and velocity distributions of electrons. It is shown that the electron velocity distribution of spatially inhomogeneous plasma evolves from an initial Maxwellian to the final kappa distribution. It is also found that the value of kappa index in the final stationary state depends on the initial state of plasma.

中文翻译：

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多方等离子体的静止状态

在本文中，我们证明了 kappa 分布是非均匀等离子体中 Vlasov-Poisson 系统在多方状态方程和将局部速度分布限制为特定数学形式的假设下的平稳解。密度、温度和电势的分布是从理论上获得的。如果初始状态已知，则可以确定 kappa 指数。为了验证该理论，进行了细胞内粒子模拟，结果显示与电子密度、温度和速度分布的理论预测非常吻合。结果表明，空间非均匀等离子体的电子速度分布从最初的麦克斯韦分布演变为最终的 kappa 分布。