Abstract
Periodic orbit theory is all effective tool for the analysis of classical and quantum chaotic systems. In this paper we extend this approach to stochastic systems, in particular to mappings with additive noise. The theory is cast in the standard field-theoretic formalism and weak noise perturbation theory written in terms of Feynman diagrams. The result is a stochastic analog of the next-to-leading ħ corrections to the Gutzwiller trace formula, with long-time averages calculated from periodic orbits of the deterministic system. The perturbative corrections are computed analytically and tested numerically on a simple 1-dimensional system.
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REFERENCES
P. Cvitanović and E. Ott, Steady fast kinematic dynamos: The topological entropy bound and the effect of magnetic field diffusion, in preparation.
E. Aurell and A. D. Gilbert, Fast dynamos and determinants of singular intergral-opertors, Geophysical and Astrophysical Fluid Dynamics 73:5–32 (1993).
N. J. Balmforth, P. Cvitanović, G. R. Ierley, E. A. Spiegel, and G. Vattay, Advection of vector fields by chaotic flows, Annals of New York Academy of Sciences 706:148 (1993), Volume entitled Stochastic Process in Astrophysics.
S. Childress and A. D. Gilbert, Stretch, Twist, Fold: The Fast Dynamo, Vol. M37, Lecture Notes in Physics. New Series M. (Springer-Verlag, Berlin, 1995).
Ya. G. Sinai, Gibbs measures in ergodic theory, Russian Mathematical Surveys 166:21 (1972).
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov-Diffeomorphisms, Vol. 470, Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1975).
D. Ruelle, Thermodynamic Formalism (Addison-Wesley, Reading, 1978).
P. Cvitanović, Field theory, NORDITA lecture notes, Copenhagen, January 1983. Notes prepared by Ejnar Gyldenkerne.
P. C. Martin, E. D. Siggia, and H. A. Rose, Statistical mechanics of classical systems, Phys. Rev. A 8(1):423–437 (1973).
G. Parisi and Y. Wu, Perturbation-theory without gauge fixing, Scientia Sinica 24:483–496 (1981).
M. J. Feigenbaum and B. Hasslacher, Irrational decimations and path-integrals for external noise, Phys. Rev. Lett. 49:605–609 (1982).
M. Gutzwiller, Chaos in Classical and uantum Mechanics (Springer-Verlag, New York, 1990).
P. Gaspard and D. Alonso, h expansion for the periodic-orbit quantization of hyperbolic systems, Phys. Rev. A 47:R3468–R3471 (1993).
P. Gaspard and D. Alonso, h expansion for the periodic-orbit quantization of chaotic systems, Chaos 3:601 (1993).
P. Gaspard, h-expansion for quantum trace formulas, G. Casati and B. Chirikov, eds., Quantum Chaos between Order and Disorder, pp. 385–404 (Cambridge University Press, 1995).
G. Vattay, Bohr-Sommerfeld quantization of periodic orbits, Phys. Rev. Lett. 76:1059–1062 (1996).
P. Reimann, Noisy one-dimensional maps near a crisis 1: Weak gaussian white and colored noise, J. Stat. Phys. 82:1467–1501 (1996).
P. Reimann, Noisy one-dimensional maps near a crisis 2: General uncorrelated weak noise, J. Stat. Phys. 85:403–425 (1996).
P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, 1997).
P. Cvitanović et al. Classical and quantum chaos: A cyclist treatise. http://www.nbi.dk/ChaosBook/ (Niels Bohr Institute, Copenhagen, 1998).
N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1981).
A. Lasota and M. MacKey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics (Springer-Verlag, Berlin, 1994).
C. P. Dettmann, Traces and determinants of strongly stochastic operators, chaodyn/9806019.
P. Damgaard and H. Huffel, eds. Stochastic Quantization, (World Scientific, Singapore, 1988).
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).
P. Cvitanović, C. P. Dettmann, R. Mainieri, and G. Vattay, Trace formulas for stochastic evolution operators: Smooth conjugation method, in preparation.
L. Kadanoff and C. Tang, Escape rate from strange repellers, Proceedings of the National Academy of Science, USA 81:1276 (1984).
O. Cepas and J. Kurchan, Canonically invariant formulation of Langevin and Fokker-Planck equations, cond-mat/9706296.
M. V. Berry and J. P. Keating, A rule for quantizing chaos, J. Phys. A 23(21):4839–4849 (1990).
P. Cvitanović and G. Vattay, Entire Fredholm determinants for evaluation of semiclassical and thermodynamical spectra, Phys. Rev. Lett. 71:4138–4141 (1993).
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Cvitanović, P., Dettmann, C.P., Mainieri, R. et al. Trace Formulas for Stochastic Evolution Operators: Weak Noise Perturbation Theory. Journal of Statistical Physics 93, 981–999 (1998). https://doi.org/10.1023/B:JOSS.0000033173.38345.f9
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DOI: https://doi.org/10.1023/B:JOSS.0000033173.38345.f9