Abstract
We introduce a random perturbed version of the classical fidelity and we show that it converges with the same rate of decay of correlations, but not uniformly in the noise. This makes the classical fidelity unstable in the zero-noise limit.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Alves and V. Araujo, Random perturbations of nonuniformly expanding maps. Astérisque 286:25–62 (2003).
Viviane Baladi and Masato Tsujii, Anisotropic hölder and sobolev spaces for hyperbolic diffeomorphisms.Ann. Inst. Fourier 57:127–154 (2007).
V. Baladi and L.-S. Young, On the spectra of randomly perturbed expanding maps. Comm. Math. Phys. 156:355–385 (1993).
G. Benenti and G. Casati, Quantum-classical correspondence in perturbed chaotic dynamical systems. Phys. Rev. E 65:066205-1 (2002).
G. Benenti, G. Casati, and G. Velbe, Stability of classical motion under a system's perturbations. Phys. Rev. E 67:055202-1 (2003).
M. Blank and G. Keller, Random perturbations of chaotic dynamical systems: stability of the spectrum. Nonlinearity 11:1351–1364 (1998).
M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius Spectrum for Anosov Maps. Nonlinearity 15(6):1905–1973 (2002).
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Springer, Lecture Notes in Mathematics 470 (1975).
G. Casati, T. Prosen, J. Lan and B. Li Universal Decay of the Classical Loschmidt Echo of Neutrally Stable Mixing Dynamics. Physical Review Letters 94:114101 (2005).
M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, to appear in Transactions of the American Mathematical Society.
B. Eckhardt, Echoes in Classical Dynamical Systems. J. Phys. A: Math. and General 36:371–380 (2003).
S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems. Ergodic Theory and Dynamical Systems 26(1):189–217 (2006).
P.-D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems. Springer, Lecture Notes in Mathematics 1606 (1995).
Z.P. Karkuszewski, C. Jarzynski, and W.H. Zurek, Quantum chaotic environments, the butterfly effect and decoherence. Phys. Rev. Letters 89:17 (2002).
Y. Kifer, Ergodic theory of random transformations, Birkhäuser, 1986.
G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (4) Vol. XXVIII:141–152 (1999).
A. Peres. Phys. Rev. A 30:1610 (1984).
M. Viana, Stochastic Dynamics of Deterministic Systems, Brazillian Math. Colloquium, IMPA, 1997.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liverani, C., Marie, P. & Vaienti, S. Random Classical Fidelity. J Stat Phys 128, 1079–1091 (2007). https://doi.org/10.1007/s10955-007-9338-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-007-9338-5