Stationary states of activity-driven harmonic chains

We study the stationary state of a chain of harmonic oscillators driven by two active reservoirs at the two ends. These reservoirs exert correlated stochastic forces on the boundary oscillators which eventually leads to a nonequilibrium stationary state of the system. We consider three most well-known dynamics for the active force, namely, the active Ornstein-Uhlenbeck process, run-and-tumble process, and active Brownian process, all of which have exponentially decaying two-point temporal correlations but very different higher-order fluctuations. We show that, irrespective of the specific dynamics of the drive, the stationary velocity fluctuations are Gaussian in nature with a kinetic temperature which remains uniform in the bulk. Moreover, we find the emergence of an “equipartition of energy” in the bulk of the system—the bulk kinetic temperature equals the bulk potential temperature in the thermodynamic limit. We also calculate the stationary distribution of the instantaneous energy current in the bulk which always shows a logarithmic divergence near the origin and asymmetric exponential tails. The signatures of specific active driving become visible in the behavior of the oscillators near the boundary. This is most prominent for the RTP- and ABP-driven chains where the boundary velocity distributions become non-Gaussian and the current distribution has a finite cutoff.

Figure 1

Schematic representation of a harmonic chain of oscillators connected to two nonequilibrium reservoirs at the two ends. Apart from the usual thermal noise the boundary oscillators are driven by autocorrelated active forces $f_1\left(t\right)$ and $f_N\left(t\right)$.

Figure 2

Plot of kinetic temperature profile ${\widehat{T}}_l$ (open symbols) and potential temperature profile ${\widehat{\mathcal{T}}}_l$ (filled symbols) for the (a) AOUP-, (b) RTP-, and (c) ABP-driven chains, for fixed ${\tau }_1=2.0$ and two different values of ${\tau }_N$. The symbols correspond to data obtained from numerical simulations with a chain of $N=512$ oscillators and, $D_1=D_N=1$ in panel (a), $A_1=A_N=1$ in panel (b) and $D_1^R=D_N^R=1$ in panel (c). The other parameters are $\gamma =1=k=m$. The black dashed lines correspond to the value of the bulk temperature according to Eq. (18). The inset in panel (a) shows zoomed in temperature profiles near the left boundary. The solid black lines in the inset corresponds to the analytical predictions of ${\widehat{\mathcal{T}}}_l$ (see Appendix pp3) and ${\widehat{T}}_l$ [30].

Figure 3

(a) Plot of the scaled velocity distribution of the middle oscillator $N/2$ for ${\tau }_1=5.0$ and ${\tau }_N=1.50$. The data corresponding to AOUP-, RTP-, and ABP-driven chains show a perfect collapse with the standard normal distribution indicated by the solid black line. (b) Plot of the velocity distributions of the oscillators at the left boundary for the three models AOUP, RTP, and ABP, with ${\tau }_1=5.0$ and ${\tau }_N=1.50$. The black solid line corresponds to the Gaussian distribution for AOUP, and the dashed line corresponds to a Gaussian with the variance ${\widehat{T}}_1$ for ABP. For both panels (a) and (b), $N=512$ and $m=\gamma =k=1$. The other parameters are $D_1=D_N=1$ for AOUP, $A_1=A_N=1$ for RTP, and $D_1^R=D_N^R=1$ for ABP.

Figure 4

Plot of the average active current $J_{\text{act}}$ as functions of activity ${\tau }_1$, for (a) AOUP, (b) RTP- and (c) ABP-driven chains for different values of ${\tau }_N$. Symbols corresponds to the data obtained from numerical simulations with $N=64$ oscillators and the other parameters are $D_1=D_N=1$ for AOUP, $A_1=A_N=1$ for RTP, $D_1^R=D_N^R=1$ for ABP, and $\gamma =1=k=m$. Black solid lines correspond to Eq. (34).

Figure 5

(a) Plot of the instantaneous bulk current distribution $P\left({\mathcal{J}}_l\right)$ for the AOUP-driven chain with ${\tau }_N=1.50$ and different values of ${\tau }_1$. Black solid lines correspond to the analytical prediction Eq. (40) and the red dashed line corresponds to the exponential decay predicted in Eq. (43). (b) Plot of $P\left({\mathcal{J}}_l\right)$ near ${\mathcal{J}}_l=0$. Black solid lines correspond to Eq. (42). For both panels (a) and (b), the symbols correspond to numerical simulations performed on a chain of $N=512$ with $m=D_1=D_2=\gamma =k=1$.

Figure 6

(a) Plot of the distribution of the scaled bulk instantaneous current ${\mathcal{J}}_l/a_1^2$ for RTP-driven (open symbols), and ABP-driven (filled symbols) chains for ${\tau }_N=1.50$ and different values of ${\tau }_1$. Black solid lines correspond to Eq. (40). (b) Plot of the same data as in panel (a) zoomed near the origin. The black solid lines here correspond to the logarithmic behavior predicted by Eq. (42). The simulations are performed on a chain of $N=512$ oscillators with $A_1=A_N=1$ and $D_1^R=D_N^R=1$ for ABP; $m=\gamma =1=k$ here.

Figure 7

Plot of the instantaneous boundary current distribution $P\left({\mathcal{J}}_1\right)$ for ${\tau }_N=1.50$ and different values of ${\tau }_1$ for (a) AOUP-, (b) RTP-, and (c) ABP-driven chains. The symbols correspond to the data obtained from numerical simulations performed on a chain of $N=512$ oscillators with $D_1=D_N=1$ for AOUP, $A_1=A_N=1$ for RTP, and $D_1^R=D_N^R$ for ABP. The other parameters are $\gamma =k=m=1$. In panel (a), the black solid line corresponds Eq. (50). In panels (b) and (c) the dashed lines indicate the upper bound ${\mathcal{J}}_1^{\text{max}}$, see Eq. (51).

Figure 8

A schematic representation of the contours used to evaluate the complex integral in Eq. (D8).

Figure 9

Plot of the second moment of the bulk energy current ${\mathcal{J}}_l^2$ as functions of the activity ${\tau }_1$, for (a) AOUP-, (b) RTP-, and (c) ABP-driven chains, for different values of ${\tau }_N$. The symbols correspond to the data obtained from numerical simulations performed on a chain of $N=512$ oscillators with $D_1=D_N=1$ for AOUP, $A_1=A_N=1$ for RTP, $D_1^R=D_N^R=1$ for ABP and $\gamma =k=m=1$. Black solid lines corresponds to the analytical prediction Eq. (D21).

Figure 10

Plot of the second moment of the boundary current ${\mathcal{J}}_1^2$ as functions of the activity ${\tau }_1$ for (a) AOUP-, (b) RTP-, and (c) ABP-driven chains, for different values of ${\tau }_N$. The symbols correspond to the data obtained from numerical simulations performed on a chain of $N=512$ oscillators with $D_1=D_N=1$ for AOUP, $A_1=A_N=1$ for RTP, $D_1^R=D_N^R=1$ for ABP, and $\gamma =k=m=1$. Black solid lines in panel (a) correspond to the analytical prediction (D22).