Expoente de Lyapunov para um Gás de Lennard–Jones - CBPFIndex

Abstract
The largest Lyapunov exponent measures the sensitivity to initial conditions in dynamical
systems. Its value provides the rate of exponential growth of two initially close phase space
trajectories. In contrast with hard-sphere systems, where the theory of Lyapunov exponents
is very well known since the Krylov’s work, there is not a consensual approach for smooth
Hamiltonian systems in general. For the latter, usually the answer comes from numerical
simulations, typically using the method developed by Benettin, which relies on the formal
definition of Lyapunov exponent. One attempt to work out this issue was made with the
so called Stochastic Approach, which offers, in principle, the possibility of obtaining a theo-
retical estimate for the maximum Lyapunov exponent of smooth Hamiltonian systems with
many degrees of freedom. Based on van Kampen’s cumulant expansion, the Stochastic Ap-
proach expresses the Lyapunov exponent as a function of a few statistical properties of the
Hessian matrix of the interaction that can be calculated as suitable microcanonical averages.
We present the application of the Stochastic Approach to a three dimensional N particle
system interacting trough a Lennard–Jones 6-12 potential, with fixed temperature, for se-
veral values of densities. The parameters of the theory were calculated both analytically
and numerically, with the numerical studies performed using microcanonical Molecular Dy-
namical simulations. Our final result is the largest Lyapunov exponent as function of the
density.
Keywords: Lyapunov Exponent, Molecular Dynamics, Lennard–Jones Potential, Cumulant
Expansion.
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