  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.
