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Quasistatic dynamical systems

Published online by Cambridge University Press:  12 May 2016

NEIL DOBBS  and
MIKKO STENLUND
NEIL DOBBS
Affiliation:
Département de Physique Théorique, Université de Genève, Geneva 1211, Switzerland email [email protected]
MIKKO STENLUND
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68, Fin-00014 University of Helsinki, Finland email [email protected]
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Abstract

We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes in thermodynamics, which are idealized processes where the observed system transforms (infinitesimally) slowly due to external influence, tracing out a continuous path of thermodynamic equilibria over an (infinitely) long time span. Time evolution of states under a quasistatic dynamical system is entirely deterministic, but choosing the initial state randomly renders the process a stochastic one. In the prototypical setting where the time evolution is specified by strongly chaotic maps on the circle, we obtain a description of the statistical behavior as a stochastic diffusion process, under surprisingly mild conditions on the initial distribution, by solving a well-posed martingale problem. We also consider various admissible ways of centering the process, with the curious conclusion that the ‘obvious’ centering suggested by the initial distribution sometimes fails to yield the expected diffusion.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 8 , December 2017 , pp. 2556 - 2596
Copyright
© Cambridge University Press, 2016 

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References

Aimino, R., Hu, H., Nicol, M., Török, A. and Vaienti, S.. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete Contin. Dyn. Syst. 35(3) (2015), 793–806.CrossRefGoogle Scholar
Bakhtin, V. I.. Random processes generated by a hyperbolic sequence of mappings. I. Izv. Ross. Akad. Nauk Ser. Mat. 58(2) (1994), 40–72.Google Scholar
Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16) . World Scientific Publishing Co., Inc., River Edge, NJ, 2000.CrossRefGoogle Scholar
Baxendale, P. H.. Stability and equilibrium properties of stochastic flows of diffeomorphisms. Diffusion Processes and Related Problems in Analysis, Vol. II (Charlotte, NC, 1990) (Progress in Probability, 27) . Birkhäuser Boston, Boston, MA, 1992, pp. 3–35.Google Scholar
Billingsley, P.. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication.Google Scholar
Bressaud, X. and Liverani, C.. Anosov diffeomorphisms and coupling. Ergod. Th. & Dynam. Sys. 22(1) (2002), 129–152.Google Scholar
Chernov, N.. Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122(6) (2006), 1061–1094.Google Scholar
Chernov, N. and Dolgopyat, D.. Brownian Brownian motion. I. Mem. Amer. Math. Soc. 198(927) (2009), viii+193.Google Scholar
Conze, J.-P. and Raugi, A.. Limit theorems for sequential expanding dynamical systems on [0, 1]. Ergodic Theory and Related Fields (Contemporary Mathematics, 430) . American Mathematical Society, Providence, RI, 2007, pp. 89–121.Google Scholar
De Simoi, J. and Liverani, C.. The martingale approach after Varadhan and Dolgopyat. Hyperbolic Dynamics, Fluctuations and Large Deviations (Proceedings of Symposia in Pure Mathematics, 89) . American Mathematical Society, Providence, RI, 2015, pp. 311–339.Google Scholar
Dolgopyat, D.. Averaging and invariant measures. Mosc. Math. J. 5(3) (2005), 537–576, 742.Google Scholar
Douce, A. P.. Thermodynamics of the Earth and Planets. Cambridge University Press, Cambridge, 2011.CrossRefGoogle Scholar
Durrett, R.. Stochastic Calculus: A Practical Introduction (Probability and Stochastics Series) . CRC Press, Boca Raton, FL, 1996.Google Scholar
Gouëzel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26(1) (2006), 189–217.Google Scholar
Gupta, C., Ott, W. and Török, A.. Memory loss for time-dependent piecewise expanding systems in higher dimension. Math. Res. Lett. 20(1) (2013), 141–161.Google Scholar
Kawan, C.. Metric entropy of nonautonomous dynamical systems. Nonauton. Dyn. Syst. 1 (2014), 26–52.Google Scholar
Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28(1) (1999), 141–152.Google Scholar
Kolyada, S., Misiurewicz, M. and Snoha, Ł.. Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval. Fund. Math. 160(2) (1999), 161–181.CrossRefGoogle Scholar
Kolyada, S. and Snoha, Ł.. Topological entropy of nonautonomous dynamical systems. Random Comput. Dynam. 4(2–3) (1996), 205–233.Google Scholar
Lasota, A. and Yorke, J. A.. When the long-time behavior is independent of the initial density. SIAM J. Math. Anal. 27(1) (1996), 221–240.Google Scholar
Le Jan, Y.. On isotropic Brownian motions. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 70(4) (1985), 609–620.Google Scholar
Leonov, V. P.. On the dispersion of time means of a stationary stochastic process. Teor. Verojatnost. Primenen. 6 (1961), 93–101.Google Scholar
Lindvall, T.. Lectures on The Coupling Method. Dover Publications, Inc., Mineola, NY, 2002, Corrected reprint of the 1992 original.Google Scholar
Livšic, A. N.. Certain properties of the homology of Y-systems. Mat. Zametki 10 (1971), 555–564.Google Scholar
Livšic, A. N.. Cohomology of dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320.Google Scholar
Mandl, F.. Statistical Physics (Manchester Physics Series) , 2nd edn. John Wiley & Sons, 1988.Google Scholar
Mohapatra, A. and Ott, W.. Memory loss for nonequilibrium open dynamical systems. Discrete Contin. Dyn. Syst. 34(9) (2014), 3747–3759.Google Scholar
Nándori, P., Szász, D. and Varjú, T.. A central limit theorem for time-dependent dynamical systems. J. Stat. Phys. 146(6) (2012), 1213–1220.Google Scholar
Ott, W., Stenlund, M. and Young, L.-S.. Memory loss for time-dependent dynamical systems. Math. Res. Lett. 16(3) (2009), 463–475.Google Scholar
Rogers, L. C. G. and Williams, D.. Diffusions, Markov Processes, and Martingales. Vol. 2: Itô Calculus (Cambridge Mathematical Library) . Cambridge University Press, Cambridge, 2000. Reprint of the second (1994) edition.Google Scholar
Stenlund, M.. Non-stationary compositions of Anosov diffeomorphisms. Nonlinearity 24 (2011), 2991–3018.Google Scholar
Stenlund, M. and Sulku, H.. A coupling approach to random circle maps expanding on the average. Stoch. Dyn. 14(4) (2014),1450008; 29 pp.Google Scholar
Stenlund, M., Young, L.-S. and Zhang, H.. Dispersing billiards with moving scatterers. Comm. Math. Phys. 322(3) (2013), 909–955.Google Scholar
Stroock, D. W. and Srinivasa Varadhan, S. R.. Multidimensional Diffusion Processes (Classics in Mathematics) . Springer, Berlin, 2006. Reprint of the 1997 edition.Google Scholar
Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.CrossRefGoogle Scholar
Zhu, Y., Liu, Z., Xu, X. and Zhang, W.. Entropy of nonautonomous dynamical systems. J. Korean Math. Soc. 49(1) (2012), 165–185.Google Scholar