  The scalar particle production through a scalar field non-minimally coupled
with geometry is investigated in the context of a spatially homogeneous and
isotropic universe. In this paper, in order to study the evolution of particle
production over time in the case of analytical solutions, we focus on a simple
Horndeski theory. We first suppose that the universe is dominated by a scalar
field and derive the energy conservation condition. Then from the thermodynamic
point of view, the macroscopic non-conservation of the scalar field
energy-momentum tensor can be explained as an irreversible production of the
scalar particles. Based on the explanation, we obtain a scalar particle
production rate and the corresponding entropy. Finally, since the universe, in
general, could be regarded as a closed system satisfying the laws of
thermodynamics, we naturally impose some thermodynamic constraints on it. The
thermodynamic properties of the universe can provide additional constraints on
the simple Horndeski theory.
