Correlation functions as used in dynamical systems give the average of
products of observables at different times. For (multiple) mixing systems the
correlation functions approach in the long time limit the products of the
averages of (more than) two observables, and for observables defined by
relatively smooth functions
this can be characterised by the rate of decay of (multiple) correlations.
The papers below express properties such as
diffusion and escape rates of
the Lorentz gas and open systems
in terms of correlations, and in some cases use correlation decay results to
show that these properties are well defined. See the animation below.
- Microscopic chaos from Brownian motion? C. P. Dettmann, E. G. D. Cohen and H. van Beijeren, Nature 401, 875-875 (1999) ps.gz (1.0M when uncompressed) arxiv
- The existence of Burnett coefficients in the periodic Lorentz gas, N. I. Chernov and C. P. Dettmann, Physica A 279, 37-44 (2000) ps arxiv
- Microscopic chaos and diffusion C. P. Dettmann and E. G. D. Cohen, J. Stat. Phys. 101, 775-817 (2000) ps.gz (28 pages; 2.1M when uncompressed) arxiv
- Note on chaos and diffusion C. P. Dettmann and E. G. D. Cohen, J. Stat. Phys. 103, 589-599 (2001) ps arxiv
- The Burnett expansion of the periodic Lorentz gas, C. P. Dettmann, Ergod. Th. Dyn. Sys. 23, 481-491 (2003) ps arxiv
- Peeping at chaos: Nondestructive monitoring of chaotic systems by measuring long-time escape rates L. A. Bunimovich and C. P. Dettmann, EPL, 80 40001 (2007). pdf arxivanimation (6.1M)
- Product of n independent uniform random variables, C. P. Dettmann and O. Georgiou, Stat. Prob. Lett., 79, 2501-2503 (2009). pdf
- New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis, C. P. Dettmann, J. Stat. Phys. 146 181-204 (2012). pdf arxiv animation (4.8M)
- Escape through a time-dependent hole in the doubling map, A. L. P. Livorati, O. Georgiou, C. P. Dettmann and E. D. Leonel, Phys. Rev. E, 89 052913 (2014). arxiv pdf
- Book chapter: The Lorentz gas as a paradigm for nonequilibrium stationary states, C. P. Dettmann, pp 315-365 in Hard ball systems and the Lorentz gas (edited by D. Szasz), Encyclopaedia of Mathematical Sciences Vol 101 (Springer, 2000). Full size version, 50 pages pdf. Environmental microscopic version, 25 pages pdf.
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