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Scientific Report 2007-2009<br />
Condensed matter physics and biophysics<br />
C15. Ordered and chaotic dynamics in molecules<br />
The very first computer study of a condensed matter<br />
system that showed the (unexpected) existence of<br />
an ordered dynamical regime is a well known simulation<br />
by Fermi, Pasta, and Ulam, back in 1953. Since then,<br />
many papers have been published in this field, dealing<br />
with the dynamical behaviour of model or realistic systems.<br />
Larger systems have been usually found to be<br />
chaotic, while ordered behaviour has been found in some<br />
systems with few degrees of freedom (DOFs). The theoretical<br />
explanation of the appearance of ordered motions<br />
in nonlinear systems begun at the same time - but<br />
independently - as the FPU experiment, and is known<br />
as the KAM theorem. This theorem explained why a<br />
nonlinear system may be endowed with regular motions,<br />
provided the nonlinearity is not too strong; this property<br />
was attributed to the system as a whole. A later theorem<br />
by Nekhoroshev foresaw the possibility that within a<br />
chaotic system different DOFs may exhibit their chaotic<br />
behaviour on very different time scales. A computer<br />
simulation that yielded evidence of the type of dynamics<br />
foreseen by Nekhoroshev has been done for 2D and<br />
3D lattices of particles interacting via a Lennard Jones<br />
potential; there, at low energy, the dynamics showed a<br />
mixed pattern, as different normal modes became chaotic<br />
over times that differed by several orders of magnitude<br />
for normal modes of different frequency.<br />
In this framework we investigate the ordered and<br />
chaotic dynamics of molecules. We have simulated the<br />
dynamics of a butane molecule, and computed the time<br />
evolution of collective internal variables (three stretchings,<br />
two bendings, and the dihedral angle).<br />
Figure 1: Model of the butane molecule.<br />
The system is strongly nonlinear at high temperature<br />
because of the dihedral potential, as shown in Fig. 2.<br />
45<br />
40<br />
35<br />
A chaotic system is usually characterized by a fast, exponential<br />
rate of divergence of trajectories beginning at<br />
near points in the phase space. This rate is measured by<br />
the maximum Lyapunov exponent λ 1 . Coherence is the<br />
opposite of chaoticity, namely a slow divergence of near<br />
trajectories, and each collective variable can be characterized<br />
by a coherence time, the time needed to develop<br />
its chaotic behaviour. Table 1 shows the Lyapunov time<br />
λ −1<br />
1 of the molecule and the coherence times ˜τ c<br />
(l) of the<br />
six internal coordinates: stretchings (b i ), bendings (θ i ),<br />
and dihedral angle (γ).<br />
T = 54 K T = 168 K T = 250 K<br />
λ −1<br />
1 = 222 λ −1<br />
1 = 0.38 λ −1<br />
1 = 0.26<br />
˜τ (l)<br />
c<br />
˜τ (l)<br />
c<br />
˜τ (l)<br />
c<br />
cos θ 1 3810 1.95 0<br />
cos θ 2 3312 1.89 0.04<br />
b 1 1596 6.79 0<br />
b 2 0 0.87 0.87<br />
b 3 1243 8.22 0<br />
γ 697 0 0.04<br />
Table 1: λ −1<br />
1 and all coherence times are in ps. ˜τ c is the<br />
coherence time relative to each DOF.<br />
The coherence times diminish significantly when the<br />
temperature is raised above T = 150 K, where conformational<br />
transitions of the dihedral angle set in. Below<br />
this temperature the coherence times of some variables<br />
reach nanoseconds; moreover, there are large differences<br />
among variables, as their coherence times can be much<br />
larger or much smaller than the Lyapunov time of the<br />
whole molecule. This hierarchy of coherence reflects the<br />
prediction of Nekhoroshev’ s theorem. Raising T above<br />
the transition region the coherence times drop to few picoseconds,<br />
and the differences among variables diminish,<br />
as the whole molecule becomes chaotic. At T = 250 K<br />
the central stretching b 2 , which is the most chaotic at<br />
low temperature, becomes the most coherent.<br />
We now aim at extending this analysis to larger<br />
molecules, where the coherence hierarchy may yield<br />
new insight into a variety of extended conformational<br />
transitions.<br />
Reference<br />
1. A. Battisti et al., Phys. Rev. E 79, 046206 (2009)<br />
Authors<br />
A. Tenenbaum, A. Battisti, R.G. Lalopa<br />
30<br />
V d<br />
(kJ/mole)<br />
25<br />
20<br />
15<br />
10<br />
5<br />
−4 −2 0 2 4<br />
γ (rad)<br />
Figure 2: Torsion potential of the dihedral angle.<br />
<strong>Sapienza</strong> Università di Roma 68 Dipartimento di Fisica