Space-time properties of Gram-Schmidt vectors in classical Hamiltonian evolution

Not all tangent space directions play equivalent roles in the local chaotic motions of classical Hamiltonian many-body systems. These directions are numerically represented by basis sets of mutually orthogonal Gram-Schmidt vectors, whose statistical properties may depend on the chosen phase space-time domain of a trajectory. We examine the degree of stability and localization of Gram-Schmidt vector sets simulated with trajectories of a model three-atom Lennard-Jones cluster. Distributions of finite-time Lyapunov exponent and inverse participation ratio spectra formed from short-time histories reveal that ergodicity begins to emerge on different time scales for trajectories spanning different phase-space regions, in a narrow range of total energy and history length. Over a range of history lengths, the most localized directions were typically the most unstable and corresponded to atomic configurations near potential landscape saddles.

Figure 1

Three normal vibrational modes of the global potential minimum equilateral triangle $D_{3h}$ of the Lennard-Jones trimer. At the potential minimum, bond angles are $60°$ and bond lengths are $\sqrt[6]{2\sigma }$. Vectors indicate atomic displacements of the (a) $A_1^{\prime }$, (b) symmetric $E^{\prime }$, and (c) asymmetric $E^{\prime }$ modes. We call these the symmetric stretch, symmetric $E^{\prime }$, and asymmetric $E^{\prime }$. The $E^{\prime }$ modes are symmetric or asymmetric with respect to the vertical mirror plane.

Figure 2

The two normal vibrational modes of the linear configuration $D_{\infty h}$ of the Lennard-Jones trimer. Bond lengths are $\sqrt[6]{2\sigma }$. Vectors indicate atomic displacements of the (a) the asymmetric and (b) the symmetric stretches.

Figure 3

(Color online) Potential-energy level curves in the interval $-3.0ϵ\le V\le 0.0ϵ$ for linear combinations of the three vibrational modes of the trimer. Axes mark the integer number of steps in scaled Hessian eigenvector directions of the asymmetric $E^{\prime }$ mode (horizontal axes) and the symmetric $E^{\prime }$ mode (vertical axes). At the origin, in the center of each plot, are equilateral triangle configurations of the symmetric stretch. Their potential energies are (a) $V=-3.0ϵ$, (b) $-2.431ϵ$, (c) $-1.673ϵ$, and (d) $-1.114ϵ$. Curves in a given panel are separated by $\Delta V=0.25ϵ$. White areas are where $V>0.0$ and are energetically inaccessible.

Figure 4

(Color online) The four nonzero Lyapunov exponents ${\lambda }_1=\left|{\lambda }_{18}\right|$, ${\lambda }_2=\left|{\lambda }_{17}\right|$ and four inverse participation ratios $Y_1=Y_{18}$, $Y_2=Y_{17}$ as the total energy $E$, initially in the (a) symmetric stretch, (b) asymmetric $E^{\prime }$ mode, and (c) symmetric $E^{\prime }$ mode, of the equilateral triangle configuration is increased. The threshold energy required for chaos is greatest when the total energy is initially deposited in the symmetric stretch. $E$ is in $ϵ$ units and $\lambda$ is in natural units per reduced time.

Figure 5

Instantaneous atomic configuration and the GSV position 3 vectors of (a) ${\mathbf{g}}_1$ and (b) ${\mathbf{g}}_2$ in a trajectory initiated from a symmetric stretch geometry. The total energy was $E=1.367ϵ$. The configuration shown was approaching and about to traverse the linear saddle.

Figure 6

Instantaneous atomic configuration and the GSV position 3 vectors of (a) ${\mathbf{g}}_1$ and (b) ${\mathbf{g}}_2$ in a trajectory initiated from an asymmetric $E^{\prime }$ mode geometry. The total energy was $E=1.375ϵ$. The configuration shown had just traversed the linear saddle.

Figure 7

(Color online) One Lyapunov exponent pair was nonzero when the total energy $E$ was initially in the symmetric stretch of the linear configuration. Two exponent pairs were nonzero and identical when the asymmetric stretch was initially excited. The absolute values of these Lyapunov exponents and their inverse participation ratios with $E$ are shown as a function of total energy.

Figure 8

(Color online) Panels (a) and (b) show ${\lambda }_1$ distributions $f\left({\lambda }_1,l\right)$ for a range of time segment lengths $l$. Panels (c) and (d) show $Y_1$ distributions $f\left(Y_1,l\right)$ for a range of time segment lengths $l$. For (a) and (c), the total energy was initially deposited in the symmetric stretch $E=-1.582ϵ$. For (b) and (d), it was deposited in the asymmetric $E^{\prime }$ mode $E=-1.586ϵ$.

Figure 9

(Color online) Panels (a) and (b) show ${\lambda }_2$ distributions $f\left({\lambda }_2,l\right)$ for a range of time segment lengths $l$. Panels (c) and (d) show $Y_2$ distributions $f\left(Y_2,l\right)$ for a range of time segment lengths $l$. For (a) and (c), the total energy was initially deposited in the symmetric stretch $E=-1.582ϵ$. For (b) and (d), it was deposited in the asymmetric $E^{\prime }$ mode $E=-1.586ϵ$.

Figure 10

(Color online) Panels (a) and (b) show ${\lambda }_1$ distributions $f\left({\lambda }_1,200\right)$ over a range of total energy. Panels (c) and (d) show $Y_1$ distributions $f\left(Y_1,200\right)$ over a range of total energy. For (a) and (c), the total energy was initially deposited in the symmetric stretch, and for (b) and (d) it was deposited in the asymmetric $E^{\prime }$ mode.

Figure 11

(Color online) Panels (a) and (b) show ${\lambda }_2$ distributions $f\left({\lambda }_2,200\right)$ over a range of total energy. Panels (c) and (d) show $Y_2$ distributions $f\left(Y_2,200\right)$ over a range of total energy. For (a) and (c), the total energy was initially deposited in the symmetric stretch, and for (b) and (d) it was deposited in the asymmetric $E^{\prime }$ mode.