  On a fine grained scale the Gibbs entropy of an isolated system remains
constant throughout its dynamical evolution. This is a consequence of
Liouville's theorem for Hamiltonian systems and appears to contradict the
second law of thermodynamics. In reality, however, there is no problem since
the thermodynamic entropy should be associated with the Boltzmann entropy,
which for non-equilibrium systems is different from Gibbs entropy. The
Boltzmann entropy accounts for the microstates which are not accessible from a
given initial condition, but are compatible with a given macrostate. In a sense
the Boltzmann entropy is a coarse grained version of the Gibbs entropy and will
not decrease during the dynamical evolution of a macroscopic system. In this
paper we will explore the entropy production for systems with long range
interactions. Unlike for short range systems, in the thermodynamic limit, the
probability density function for these systems decouples into a product of one
particle distribution functions and the coarse grained entropy can be
calculated explicitly. We find that the characteristic time for the entropy
production scales with the number of particles as $N^\alpha$, with $\alpha >
0$, so that in the thermodynamic limit entropy production takes an infinite
amount of time.
