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