  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.
