  We systematically investigate the thermodynamic properties of homogeneous
nuclear matter with light clusters at low densities and finite temperatures
using a generalized nonlinear relativistic mean-field (gNL-RMF) model, in which
light clusters up to $\alpha $ ($1 \le A \le 4$) are included as explicit
degrees of freedom and treated as point-like particles with their interactions
described by meson exchanges and the medium effects on the cluster binding
energies are described by density- and temperature-dependent energy shifts with
the parameters obtained by fitting the experimental cluster Mott densities. We
find that the composition of low density nuclear matter with light clusters is
essentially determined by the density- and temperature-dependence of the
cluster binding energy shifts. Compared with the values of the conventional
(second-order) symmetry energy, symmetry free energy and symmetry entropy,
their fourth-order values are found to be significant at low densities ($\rho
\sim 10^{-3}$ fm$^{-3}$) and low temperatures ($T \lesssim 3$ MeV), indicating
the invalidity of the empirical parabolic law for the isospin asymmetry
dependence of these nuclear matter properties. Our results indicate that in the
density region of $\rho \gtrsim 0.02$ fm$^{-3}$, the clustering effects become
insignificant and the nuclear matter is dominated by nucleon degree of freedom.
In addition, we compare the gNL-RMF model predictions with the corresponding
experimental data on the symmetry energy and symmetry free energy at low
densities and finite temperatures extracted from heavy-ion collisions, and
reasonable agreement is found.
