Nonequilibrium steady states in bead-spring models: Entropy production and probability distributions

Abstract

We study non-equilibrium models comprising of beads connected by springs. The system is coupled to two thermal baths kept at different temperatures. We derive the steady state probability distributions of positions of the bead for the one-bead system in the underdamped case. We employ the recently proposed technique of an effective temperature, along with numerical simulations to solve the Langevin equations and obtain their corresponding probability distributions. It is observed that the marginal probability distributions in the position are independent of mass. We also obtain theoretically and numerically the rate of entropy production for the one-bead system. The probability distribution of the positions in the two-beads system are obtained theoretically and numerically, both in the underdamped and overdamped case. Lastly, we discuss the notion of ergodicity and have tested the convergence of the time-averaging and the ensemble-averaging protocols.

Figures (12)

• Figure 1: A bead with mass $m$ simultaneously coupled to two baths at temperatures $T_{L}$ and $T_{R}$, with springs having spring constants $k_{1}$ and $k_{2}$. The frictional drag coefficients are given by $\gamma_{L}$ and $\gamma_{R}$.
• Figure 2: The marginal probability distribution ($P(x_{1})$) is derived from the three approaches. Here, $k=1$, $\gamma_{L} = \gamma_{R}=1$, $T_{L}=99$, and $T_{R}=1$. These results are independent of mass ($m$).
• Figure 3: (a) The numerically computed rate of entropy production and (b) its deviation from the theoretical prediction [eq. (\ref{['EP_Prod_thermo']})] versus bath temperatures. The small deviation (one order less) of theoretical prediction from the numerical results indicates a good match. The simulations assume $\gamma_{L}=\gamma_{R}=1$, $k=1$, and $m=1$.
• Figure 4: Two beads, each of mass $m$, simultaneously coupled to each other with a spring having spring constant $\kappa$ as well as with a bath at temperature $T_{L}$ ($T_{R}$) with spring constant $k_{1}$ ($k_{2}$). The frictional drag coefficient is $\gamma_{L}$ ($\gamma_{R}$) for the left (right) particle.
• Figure 5: The numerically computed marginal probability distribution ($P(x_{1})$) in comparison to the distribution obtained from effective temperature theory. The simulations assume $T_{L}=9$ (a) and $T_{L}=99$ (b), $T_{R}=1$, $\gamma_{L}=\gamma_{R}=1$, $k_{1}=k_{2}=1$, $\kappa = 2$, and $m=0.01$.