Hidden entropy production by fast variables

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.

Figure 1

Comparison of a rigid rod system in (a) and a harmonic system in (b).

Figure 2

Heat-flow rates as a function of $K$. Parameter values are ${\gamma }_1=2,{\gamma }_2=1,m=1$, and $(T_1-T_2)=1$.

Figure 3

Integration contour for the evaluation of (17) in complex $\omega$ plane. Simple poles are marked with $\times$ symbols. The nonzero roots ${\omega }_{4,5,6}$ of $D(-\omega )$ are in the upper half-plane, while the nonzero roots ${\omega }_{1,2,3}$ of $D\left(\omega \right)$ are in the lower half-plane.