具有长程相互作用的线性化平均场Kramers方程的初值问题
Initial value problem for the linearized mean field Kramers equation with long-range interactions
摘要：
在$N\rightarrow +\infty$极限下，我们求解了描述长程相互作用布朗粒子的线性化平均场Kramers方程的初值问题.我们表明，介电函数可以表示在不完整的伽马函数。与线性化的Vlasov方程和线性化的平均场Smoluchowski方程相关联的介电函数被恢复为分别对应于无摩擦极限或强摩擦极限的特殊情况。虽然麦克斯韦-玻尔兹曼分布的稳定性与摩擦参数无关，但扰动的演化以非平凡的方式依赖于摩擦参数。为了说明，我们将我们的结果应用到自引力系统，等离子体，并吸引和排斥BMF模型。
英文摘要：
We solve the initial value problem for the linearized mean field Kramers equation describing Brownian particles with long-range interactions in the $N\rightarrow +\infty$ limit. We show that the dielectric function can be expressed in terms of incomplete Gamma functions. The dielectric functions associated with the linearized Vlasov equation and with the linearized mean field Smoluchowski equation are recovered as special cases corresponding to the no friction limit or to the strong friction limit respectively. Although the stability of the Maxwell-Boltzmann distribution is independent on the friction parameter, the evolution of the perturbation depends on it in a non-trivial manner. For illustration, we apply our results to self-gravitating systems, plasmas, and to the attractive and repulsive BMF models.
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