Backward and covariant Lyapunov vectors and exponents for hard-disk systems with a steady heat current

The covariant Lyapunov analysis is generalized to systems attached to deterministic thermal reservoirs that create a heat current across the system and perturb it away from equilibrium. The change in the Lyapunov exponents as a function of heat current is described and explained. Both the nonequilibrium backward and covariant hydrodynamic Lyapunov modes are analyzed and compared. The movement of the converged angle between the hydrodynamic stable and unstable conjugate manifolds with the free flight time of the dynamics is accurately predicted for any nonequilibrium system simply as a function of their exponent. The nonequilibrium positive and negative $\mathbf{LP}$ mode frequencies are found to be asymmetrical, causing the negative mode to oscillate between the two functional forms of each mode in the positive conjugate mode pair. This in turn leads to the angular distributions between the conjugate modes to oscillate symmetrically about $\pi /2$ at a rate given by the difference between the positive and negative mode frequencies.

Figure 1

A visual interpretation of the QOD hard disk system, showing here the (H,P) boundary conditions. The shaded disks represent the particles while the unshaded disks represent their periodic images above and below the main channel.

Figure 2

The center of the $C$ matrix; brighter darker entries indicate magnitudes closer to unity. The red square indicates the relevant entries to the $v\mathbf{Z}$ modes, green squares indicate the relevant $v\mathbf{T}$ entries, blue squares the relevant $v\mathbf{LP}$ entries. The labels will be useful in the following analysis.

Figure 3

Normalized steady state nonequilibrium temperature profiles in $x$, $(T_{xi}-T_{xN})/(T_{x1}-T_{xN})$, as a function of the average particle position (a) for $\rho =0.003$, $N=40$ and (b) for $\rho =0.8$, $N=40$. As the temperature profiles are nonlinear, a cubic fitting function is overlaid for $\rho =0.003$ with fitting parameters given in Table 2.

Figure 4

Pointwise ratio of the Lyapunov exponents for each of the nonequilibrium systems compared to the equilibrium system $R_{J_Q}^{\left(j\right)}={\lambda }_{J_Q}^{\left(j\right)}/{\lambda }_{Eq}^{\left(j\right)}$ (a) for $\rho =0.003$, $N=40$ and (b) for $\rho =0.8$, $N=40$. Almost uniform pointwise scaling with heat current $R_{J_Q}^{\left(j\right)}=R_{J_Q}$ is seen for $\rho =0.8$.

Figure 5

Normalized pointwise ratios between the equilibrium and nonequilibrium exponent spectrums $({\lambda }_{J_Q}^{\left(j\right)}/{\lambda }_{J_Q}^{\left(1\right)})/({\lambda }_{Eq}^{\left(j\right)}/{\lambda }_{Eq}^{\left(1\right)})$ (a) for $\rho =0.003$, $N=40$ and panel (b) for $\rho =0.8$, $N=40$. Each full spectrum has been normalized with respect to their largest exponent ${\lambda }^{\left(j\right)}\to {\lambda }^{\left(j\right)}/{\lambda }^{\left(1\right)}$.

Figure 6

Normalized differences between equilibrium and nonequilibrium exponent spectrums $({\lambda }_{J_Q}^{\left(j\right)}/{\lambda }_{J_Q}^{\left(1\right)})-({\lambda }_{Eq}^{\left(j\right)}/{\lambda }_{Eq}^{\left(1\right)})=S_{J_Q}^{\left(j\right)}/{\lambda }^{\left(1\right)}$ for the high-density systems, $\rho =0.8$, $N=40$. The differences segment into regions displaying distinct scaling behavior. Black circles are for $J_Q=0$, red circles for $J_Q=1$, green circles for $J_Q=3$, and blue circles for $J_Q=5$.

Figure 7

Normalized symplectic differences between conjugate exponent pairs for nonequilibrium high-density states $({\lambda }_{J_Q}^{\left(j\right)}+{\lambda }_{J_Q}^{(4N+1-j)})/{\lambda }_{J_Q}^{\left(1\right)}$ indicating deviations from the conjugate pairing rule for $\rho =0.8$, $N=40$. The symplectic differences between conjugate pairs for the equilibrium system would be exactly zero. Black circles are for $J_Q=0$, red circles for $J_Q=1$, green circles for $J_Q=3$, and blue circles for $J_Q=5$.

Figure 8

The average zero modes of the high-density systems, $\rho =0.8$ and $N=40$ for both equilibrium and three nonequilibrium configurations (a) for the four Equ zero modes, (b) for the four ${\mathbf{J}}_{\mathbf{Q}}=\mathbf{0}$ zero modes, (c) for the four ${\mathbf{J}}_{\mathbf{Q}}=\mathbf{1}$ zero modes, and (d) for the four ${\mathbf{J}}_{\mathbf{Q}}=\mathbf{3}$ zero modes. As the total momenta is zero (the system does not move), the distribution of the momenta is removed when the modes are averaged. For each nonequilibrium system the exponent of the $g{\mathbf{Z}}^{-2}$ mode shifts downwards away from zero (shown in Fig. 6). The forms here verifies the nonequilibrium $\delta x$ components witnessed in Ref. [47].

Figure 9

A comparison between the numerical $\delta x$ components and the predicted $\delta x$ components of the average $g{\mathbf{Z}}^2$ and $g{\mathbf{Z}}^1$ zero modes as given by Eqs. (8) and (9) for the $J_Q=0$, $\rho =0.8$, $N=40$ system of Fig. 8 (b). (a) For $\delta x$ for the $g{\mathbf{Z}}^2$ mode and (b) for $\delta x$ for the $g{\mathbf{Z}}^1$ mode.

Figure 10

(a) The $\delta x$ components for all nonequilibrium $g{\mathbf{Z}}^2$ and $g{\mathbf{Z}}^1$ modes, showing the exact average functional form as given by Eqs. (8) and (9) when the $a$ and $b$ factors are removed. (b) The differences between the $J_Q\ne 0$ and $J_Q=0$ $\delta x$ components, giving the $f(J_Q,x_i)$ function.

Figure 11

The $\delta y$ components of the nonequilibrium backward transverse modes in comparison to the equilibrium transverse mode, for $\rho =0.8$ and $N=40$ (a) for the $g{\mathbf{T}}^1$ modes, (b) for the $g{\mathbf{T}}^2$ modes. The top of each figure shows the transverse modes themselves (magnitude given by the left-hand axis), the bottom gives the (enlarged) differences between the nonequilibrium and equilibrium transverse modes (magnitude given by the right-hand axis).

Figure 12

The center of an evolved nonequilibrium $C$ matrix corresponding to the $N=40$, $\rho =0.8$, $J_Q=3$ system. Shown over the same region as the equilibrium $C$ matrix of Fig. 2 and utilizing the same mode identification color boxes, there are clear differences between the equilibrium and nonequilibrium matrices.

Figure 13

The average four covariant zero modes of equilibrium and nonequilibrium $\rho =0.8$, $N=40$ systems (a) for the four Equ zero modes, (b) for the four ${\mathbf{J}}_{\mathbf{Q}}=\mathbf{3}$ zero modes. Compare directly with the BLVs of Figs. 8 and 8. The center $C$ matrix of Fig. 12 gives the covariant zero modes of panel (b).

Figure 14

The $\delta y$ components of the nonequilibrium covariant transverse modes in comparison to the equilibrium covariant transverse mode, for $\rho =0.8$ and $N=40$ (a) for the $v{\mathbf{T}}^1$ modes, (b) for the $v{\mathbf{T}}^2$ modes. The top of each panel shows the transverse modes themselves (magnitude given by the left axis), the bottom gives the (enlarged) differences between the nonequilibrium and equilibrium transverse modes (magnitude given by the right axis).

Figure 15

The difference in the average BLV and CLV localizations between equilibrium and nonequilibrium systems ${\langle L^{\left(j\right)}\rangle }_{J_Q}-{\langle L^{\left(j\right)}\rangle }_{Eq}$ for $\rho =0.8$ and $N=40$ (a) for the BLVs and (b) for the CLVs. Both figures are shown over the same range.

Figure 16

The difference in the average BLV and CLV localizations between equilibrium and nonequilibrium systems ${\langle L^{\left(j\right)}\rangle }_{J_Q}-{\langle L^{\left(j\right)}\rangle }_{Eq}$ for $\rho =0.003$ and $N=40$ (a) for the BLVs and (b) for the CLVs. Both panels are shown over the same range.

Figure 17

The angles distributions between the $v{\mathbf{T}}^1$ and $v{\mathbf{T}}^2$ conjugate pairs for equilibrium and nonequilibrium $N=40$ and $\rho =0.8$ systems (a) for the $v{\mathbf{T}}^1$ modes and (b) for the $v{\mathbf{T}}^2$ modes. The increase in the peak angle towards $\pi /2$ is due to the increase in the ${\lambda }^T$ of the modes.

Figure 18

The angles distributions between the $v{\mathbf{LP}}^{1,i}.v{\mathbf{LP}}^{-1,i}$ (circles) and $v{\mathbf{LP}}^{2,i}.v{\mathbf{LP}}^{-2,i}$ (crosses) conjugate pairs for equilibrium and nonequilibrium $N=40$ and $\rho =0.8$ systems, corresponding to the diagonal elements of the $C$ matrix (a) for the $v{\mathbf{LP}}^1$ modes and (b) for the $v{\mathbf{LP}}^2$ modes. The modes move towards $\pi /2$ with increasing heat current.

Figure 19

In equilibrium systems the positive and negative $\mathbf{LP}$ mode frequencies are equal; therefore the $C_{LP}^{*}$ matrix entries of the $v{\mathbf{LP}}^{-1,1}$ mode maintain steady values [the plotted entries corresponding to the left-hand side of Eq. (11)]. In nonequilibrium systems, the negative $v{\mathbf{LP}}^{-1,1}$ (and therefore $g{\mathbf{LP}}^{-1,1}$) mode oscillates slower than the positive modes ${\omega }_{-1}<{\omega }_1$ (from Table 4); therefore the negative mode oscillates within the subspace of the positive $v{\mathbf{LP}}^{1,1}v{\mathbf{LP}}^{1,2}$ mode pair at a rate determined by the difference in the positive and negative frequencies ${\omega }_{J_Q}={\omega }_1-{\omega }_{-1}$. This causes the $C_{LP}^{*}$ matrix entries of the $v{\mathbf{LP}}^{-1,1}$ mode to oscillate at this ${\omega }_{J_Q}$ frequency [the plotted entries corresponding to the right-hand side of Eq. (11)]. The periods here are indicative only and serve as illustrative approximations.