Numerical comparison of a constrained path ensemble and a driven quasisteady state

We investigate the correspondence between a nonequilibrium ensemble defined via the distribution of phase-space paths of a Hamiltonian system and a system driven into a steady state by nonequilibrium boundary conditions. To discover whether the nonequilibrium path ensemble adequately describes the physics of a driven system, we measure transition rates in a simple one-dimensional model of rotors with Newtonian dynamics and purely conservative interactions. We compare those rates with known properties of the nonequilibrium path ensemble. In doing so, we establish effective protocols for the analysis of transition rates in nonequilibrium quasisteady states. Transition rates between potential wells and also between phase-space elements are studied and found to exhibit distinct properties, the more coarse-grained potential wells being effectively further from equilibrium. In all cases the results from the boundary-driven system are close to the path-ensemble predictions, but the question of equivalence of the two remains open.

Figure 1

The model system under investigation: a one-dimensional chain of simple rotors with conservative nearest-neighbor interactions. The chain may be allowed to equilibrate, or can be driven by continuously twisting the boundary rotors. This many-body system is a simple model for a complex fluid in shear flow.

Figure 2

The interaction potential between neighboring rotors, $U\left(\Delta \theta \right)=-cos\left(\Delta \theta \right)-cos\left(4\Delta \theta \right)$, in terms of the angular difference $\Delta \theta$ between the neighbors.

Figure 3

The steady-state distribution of relative angles $\Delta \theta$ between neighbors in the chain without driving. The vertical axis shows $ln\left(f\right)$, where $f$ is the probability per unit angle of finding a given relative angle between neighbors. Also plotted as a continuous curve is $-\beta U\left(\Delta \theta \right)+c$ for the function $U$ defined in the legend of Fig. 2, where $\beta$ and $c$ are fitting parameters. Agreement between the curve and the data confirms that the deterministic system is at thermodynamic equilibrium, respecting Boltzmann's law: $f\propto exp(-\beta U)$.

Figure 4

Mean potential energy per rotor as a function of time, for one of the many simulations performed at a shear rate $\dot{\gamma }=2$. Vertical dashed lines delineate the interval during which data were obtained with the appropriate mean potential energy per rotor. Note the slow systematic energy increase, only apparent on the longest timescale. Note also the rapid noisy variation on a timescale much shorter than the measurement interval.

Figure 5

Graph of the standard deviation in mean potential energy per rotor (averaged over many simulations) as a function of the measurement interval $\Delta t$. The data shown are for simulations at a shear rate $\dot{\gamma }=1$.

Figure 6

The measured rates ${\omega }_{\mathrm{ba}}$, ${\omega }_{\mathrm{bc}}$, ${\omega }_{\mathrm{da}}$, and ${\omega }_{\mathrm{dc}}$ for a wide range of shear rates $\dot{\gamma }$, both in natural time units. Also plotted is the combination $({\omega }_{\mathrm{ba}}+{\omega }_{\mathrm{bc}})/({\omega }_{\mathrm{da}}+{\omega }_{\mathrm{dc}})$, which is the ratio of exit rates from wells b and d, predicted to remain constant at unity in a constrained ensemble, subject to the assumptions that momentum variables can be neglected, and that potential wells represent distinct microstates of a system weakly coupled to the nonequilibrium reservoir embodied by the other rotors. The red line is the theoretical prediction of this ratio (unity), while results of simulations are depicted by black squares. The mean potential energy per rotor is selected in the range $-0.1\pm 0.01$.

Figure 7

As described in the legend of Fig. 6, for the rates ${\omega }_{\mathrm{ab}}$, ${\omega }_{\mathrm{ba}}$, ${\omega }_{\mathrm{ad}}$, ${\omega }_{\mathrm{da}}$, and their combination $\left({\omega }_{\mathrm{ab}}{\omega }_{\mathrm{ba}}\right)/\left({\omega }_{\mathrm{ad}}{\omega }_{\mathrm{da}}\right)$.

Figure 8

As described in the legend of Fig. 6, for the rates ${\omega }_{\mathrm{cd}}$, ${\omega }_{\mathrm{dc}}$, ${\omega }_{\mathrm{cb}}$, ${\omega }_{\mathrm{bc}}$, and their combination $\left({\omega }_{\mathrm{cd}}{\omega }_{\mathrm{dc}}\right)/\left({\omega }_{\mathrm{cb}}{\omega }_{\mathrm{bc}}\right)$.

Figure 9

Discretization of the phase space occupied by the gaps between neighboring rotors, with coordinates $\Delta \theta$ (the relative angle between the pair) and $\Delta \dot{\theta }$ (their relative angular velocity). The states $\alpha$, $\beta$, and $\varepsilon$ are identified for analysis, as well as states reversed in $\Delta \theta$ only, denoted by $\widehat{}$, and time-reversed states denoted by *.

Figure 10

Histograms and density plots of occupancies of the phase-space bins defined in Fig. 9, measured in quasisteady states with mean potential energy density $-0.1\pm -0.01$ and shear rate (a) $\dot{\gamma }=0.2$, (b) $\dot{\gamma }=2$.

Figure 11

Test of the relationship ${\sum }_i{\omega }_{\varepsilon i}={\sum }_i{\omega }_{\widehat{\varepsilon }i}$ between exit rates from states $\varepsilon$ and $\widehat{\varepsilon }$, defined in Fig. 9, in quasisteady states with mean potential energy per rotor $-0.1\pm 0.01$ as for Fig. 6. The key gives symbols representing the individual rates ${\omega }_{\varepsilon \leftarrow }$ etc. where the arrow indicates the direction (relating to Fig. 9) of the transition out of the initial state. Black symbols: ${\sum }_i{\omega }_{\varepsilon i}/{\sum }_i{\omega }_{\widehat{\varepsilon }i}$ are in very close agreement with the theoretical value (red line) of unity.

Figure 12

Test of the relationship ${\omega }_{\alpha \beta }{\omega }_{\beta \alpha }={\omega }_{\widehat{\alpha }\widehat{\beta }}{\omega }_{\widehat{\beta }\widehat{\alpha }}$ for states $\alpha$ and $\beta$ defined in Fig. 9, in quasisteady states with mean potential energy per rotor $-0.1\pm 0.01$ as for Fig. 6. Black symbols: ${\omega }_{\alpha \beta }{\omega }_{\beta \alpha }/{\omega }_{\widehat{\alpha }\widehat{\beta }}{\omega }_{\widehat{\beta }\widehat{\alpha }}$ are in very close agreement with the theoretical value (red line) of unity.

Figure 13

As described in the legend of Fig. 12, for the relationship ${\omega }_{\alpha \beta }{\omega }_{\beta \alpha }={\omega }_{{\alpha }^{*}{\beta }^{*}}{\omega }_{{\beta }^{*}{\alpha }^{*}}$.

Figure 14

As described in the legend of Fig. 12, for the relationship ${\omega }_{\alpha \beta }{\omega }_{\beta \alpha }={\omega }_{{\widehat{\alpha }}^{*}{\widehat{\beta }}^{*}}{\omega }_{{\widehat{\beta }}^{*}{\widehat{\alpha }}^{*}}$.