  We investigate nonequilibrium underdamped Langevin dynamics of Brownian
particles that interact through a harmonic potential with coupling constant $K$
and are in thermal contact with two heat baths at different temperatures. The
system is characterized by a net heat flow and an entropy production in the
steady state. We compare the entropy production of the harmonic system with
that of Brownian particles linked with a rigid rod. The harmonic system may be
expected to reduce to the rigid rod system in the infinite $K$ limit. However,
we find that the harmonic system in the $K\to\infty$ limit produces more
entropy than the rigid rod system. The harmonic system has the center of mass
coordinate as a slow variable and the relative coordinate as a fast variable.
By identifying the contributions of the degrees of freedom to the total entropy
production, we show that the hidden entropy production by the fast variable is
responsible for the extra entropy production. We discuss the $K$ dependence of
each contribution.
