Geometry of Chaos and Phase Transitions

Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 33)

In the previous chapters we have shown how simple concepts belonging to classical differential geometry can be successfully used as tools to build a geometric theory of chaotic Hamiltonian dynamics. Such a theory is able to describe the instability of the dynamics in classical systems consisting of a large number N of mutually interacting particles, by relating these properties to the average and the fluctuations of the curvature of the configuration space. Such a relation is made quantitative through (5.45), which provides an approximate analytical estimate of the largest Lyapunov exponent in terms of the above-mentioned geometric quantities, and which compares very well with the outcome of numerical simulations in a number of cases, three of which have been discussed in detail in Chapter 5.


Phase Transition Critical Temperature Lyapunov Exponent Chaotic Dynamics Thermodynamic Limit 
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© Springer Science+Business Media 2007

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