We consider a homogeneous plasma composed of N particles of the same electric charge which interact through a Coulomb potential. In the large plasma parameter limit, classical kinetic theories justify that the empirical density is the solution of the Balescu–Guernsey–Lenard equation, at leading order. This is a law of large numbers. The Balescu–Guernsey–Lenard equation is approximated by the Landau equation for scales much smaller than the Debye length. In order to describe typical and rare fluctuations, we compute for the first time a large deviation principle for dynamical paths of the empirical density, within the Landau approximation. We obtain a large deviation Hamiltonian that describes fluctuations and rare excursions of the empirical density, in the large plasma parameter limit. We obtain this large deviation Hamiltonian either from the Boltzmann large deviation Hamiltonian in the grazing collision limit, or directly from the dynamics, extending the classical kinetic theory for plasmas within the Landau approximation. We also derive the large deviation Hamiltonian for the empirical density of N particles, each of which is governed by a Markov process, and coupled in a mean field way. We explain that the plasma large deviation Hamiltonian is not the one of N particles coupled in a mean-field way.

我们考虑由N组成的均匀等离子体具有相同电荷的粒子通过库仑势相互作用。在大等离子体参数限制中，经典动力学理论证明经验密度是 Balescu-Guernsey-Lenard 方程的解，处于领先阶。这是大数定律。对于远小于德拜长度的尺度，Balescu-Guernsey-Lenard 方程近似于 Landau 方程。为了描述典型和罕见的波动，我们首次在朗道近似内计算了经验密度动力学路径的大偏差原理。我们获得了一个大偏差哈密顿量，它描述了在大等离子体参数限制下的经验密度的波动和罕见的偏移。我们从掠碰撞极限中的玻尔兹曼大偏差哈密顿量或直接从动力学中获得这个大偏差哈密顿量，扩展了朗道近似内等离子体的经典动力学理论。我们还推导出了经验密度的大偏差哈密顿量N 个粒子，每个粒子都受马尔可夫过程控制，并以平均场方式耦合。我们解释了等离子体大偏差哈密顿量不是以平均场方式耦合的N 个粒子之一。