Pesin-Type Identity for Weak Chaos

Details Pesin-Type Identity for Weak Chaos Pesin-Type Identity for Weak Chaos

2008-08-10

arxiv.org/abs/0808.1398v1

Phys. Rev. Lett. 102, 050601 (2009)

Physics

Condensed Matter

Statistical Mechanics

5 pages, 3 figures

Scientific paper

10.1103/PhysRevLett.102.050601

Pesin's identity provides a profound connection between entropy $h_{KS}$ (statistical mechanics) and the Lyapunov exponent $\lambda$ (chaos theory). It is well known that many systems exhibit sub-exponential separation of nearby trajectories and then $\lambda=0$. In many cases such systems are non-ergodic and do not obey usual statistical mechanics. Here we investigate the non-ergodic phase of the Pomeau-Manneville map where separation of nearby trajectories follows $\delta x_t= \delta x_0 e^{\lambda_{\alpha} t^{\alpha}}$ with $0<\alpha<1$. The limit distribution of $\lambda_{\alpha}$ is the inverse L{\'e}vy function. The average $< \lambda_{\alpha} >$ is related to the infinite invariant density, and most importantly to entropy. Our work gives a generalized Pesin's identity valid for systems with an infinite invariant density.

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