Abstract
The problem of escape from a domain of attraction is applied to the case of discrete dynamical systems possessing stable and unstable fixed points. In the presence of noise, the otherwise stable fixed point of a nonlinear map becomes metastable, due to noise-induced hopping events, which eventually pass the unstable fixed point. Exact integral equations for the moments of the first passage time variable are derived, as well as an upper bound for the first moment. In the limit of weak noise, the integral equation for the first moment, i.e., the mean first passage time (MFPT), is treated, both numerically and analytically. The exponential leading part of the MFPT is given by the ratio of the noise-induced invariant probability at the stable fixed point and unstable fixed point, respectively. The evaluation of the prefactor is more subtle: It is characterized by a jump at the exit boundaries, which is the result of a discontinuous boundary layer function obeying an inhomogeneous integral equation. The jump at the boundary is shown to be always less than one-half of the maximum value of the MFPT. On the basis of a clear-cut separation of time scales, the MFPT is related to the escape rate to leave the domain of attraction and other transport coefficients, such as the diffusion coefficient. Alternatively, the rate can also be obtained if one evaluates the current-carrying flux that results if particles are continuously injected into the domain of attraction and captured beyond the exit boundaries. The two methods are shown to yield identical results for the escape rate of the weak noise result for the MFPT, respectively. As a byproduct of this study, we obtain general analytic expressions for the invariant probability of noisy maps with a small amount of nonlinearity.
Similar content being viewed by others
References
G. H. Weiss, ed.,J. Stat. Phys. 42:1–246 (1986).
P. Talkner and P. HÄnggi,Phys. Rev. A 29:768 (1984).
P. Talkner and D. Ryter, inNoise in Physical Systems and l/f Noise, M. Savelli, G. Lecoy, and J. P. Nougier, eds. (Elsevier, New York, 1983).
R. M. May,Nature 261:459 (1976).
M. J. Feigenbaum,Los Alamos Science (Summer 1980).
S. Grossmann, inNon-Equilibrium Cooperative Phenomena in Physics and Related Fields, Vol. B116,Discrete Nonlinear Dynamics, M. G. Velarde, ed. (Plenum Press, New York, 1984), pp. 413–435.
P. Bergé, Y. Pomeau, and Ch. Vidal,L'Ordre dans le Chaos (Hermann, Paris, 1984); H. G. Schuster,Deterministic Chaos (VCH, Weinheim, 1984).
J. Guckenheimer and P. Holmes,Nonlinear Oscillation, Dynamical Systems, and Bifurcations of Vector Fields (Springer, Berlin, 1983).
L. D. Landau and E. M. Lifschitz,Mechanics (Pergamon Press, New York, 1957).
T. Geisel and J. Nierwetberg,Phys. Rev. Lett. 48:7 (1982);Phys. Rev. A 29:2305 (1984).
M. Schell, S. Fraser, and R. Kapral,Phys. Rev. A 26:504 (1982).
H. Fujisaka and S. Grossmann,Z. Physik B 48:261 (1982);Phys. Rev. A 26:1779 (1982).
R. F. Voss,Phys. Rev. Lett. 50:1329 (1983).
I. Goldhirsch, Y. Imry, G. Wasserman, and E. Ben-Jacob,Phys. Rev. B 29:1218 (1984).
R. L. Kautz,J. Appl. Phys. 52:3528 (1981);52:6241 (1981); D. D'Humieres, M. R. Beasly, B. A. Huberman, and A. Libchaber,Phys. Rev. A 26:3483 (1982);Phys. Rev. Lett. 50:1328 (1982).
R. F. Miracky, M. H. Devoret, and J. Clarke,Phys. Rev. A 31:2509 (1985).
F. T. Arecchi and F. Lisi,Phys. Rev. Lett. 49:94 (1982); F. T. Arecchi and A. Califano,Phys. Lett. 101A:443 (1984); F. T. Arecchi, R. Badii, and A. Politi,Phys. Rev. A 32:402 (1985).
C. Greborgi, E. Ott, and J. A. Yorke,Phys. Rev. Lett. 48:1507 (1982).
F. T. Arecchi, R. Badii, and A. Politi,Phys. Lett. 103A:3 (1984).
R. V. Jensen and E. R. Jessup,J. Stat. Phys. 43:369 (1986).
E. G. Gwinn and R. M. Westervelt,Phys. Rev. Lett. 54:1613 (1985).
L. Glass and R. Perez,Phys. Rev. Lett. 48:1772 (1982); M. J. Feigenbaum, L. P. Kadanoff, and S. J. Shenker,Physica 5D:370 (1982); D. Rand, S. Ostlund, J. Sethna, and E. Siggia,Phys. Rev. Lett. 49:132 (1982);Physica 6D:303 (1984).
M. H. Jensen, P. Bak, and T. Bohr,Phys. Rev. Lett. 50:1637 (1983);Phys. Rev. A 30:1960; 1970 (1984).
P. Bak, T. Bohr, and M. H. Jensen,Physica Scripta T9:50 (1985).
S. Fraser, E. Celarier, and R. Kapral,J. Stat. Phys. 33:341 (1983).
S. Grossmann and S. Thomae,Z. Naturforsch. 32A:1353 (1977); H. Fujisaka and T. Yamada,Z. Naturforsch. 33A:1455 (1978); S.-J. Chang and J. Wright,Phys. Rev. A 23:1419 (1981); Y. Oono and Y. Takahashi,Progr. Theor. Phys. 63:1804 (1980).
H. Haken and G. Mayer-Kress,Phys. Lett. 84A:159 (1981);Z. Physik B 43:185 (1981).
H. Haken and A. Wunderlin,Z. Physik B 46:181 (1982).
P. HÄnggi and P. Talkner,Phys. Rev. A 32:1934 (1985).
J. Troe,J. Chem. Phys. 66:4745 (1977).
G. H. Weiss and A. Szabo,Physica 119A:569 (1983).
C. Knessl, B. J. Matkowsky, Z. Schuss, and C. Tier,J. Stat. Phys. 42:169 (1986).
I. N. Bronstein and K. A. Semendyayev,Handbook of Mathematics (Van Nostrand, New York, 1986).
B. J. Matkowsky and Z. Schuss,SIAM J. Appl. Math. 33:365 (1977);40:242 (1981);42:835 (1982); inLecture Notes in Mathematics, No. 985,Asymptotic Analysis II, F. ver Hulst, ed. (Springer-Verlag, Berlin, 1983), pp. 2–34.
Z. Schuss and B. J. Matkowsky,SIAM J. Apl. Math. 35:604 (1979).
H. A. Kramers,Physica 7:284 (1940).
P. HÄnggi,J. Stat. Phys. 42:105 (1986); Addendum,44:1003 (1986).
P. HÄnggi, H. Grabert, P. Talkner, and H. Thomas,Phys. Rev. A 29:371 (1984).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Talkner, P., HÄnggi, P., Freidkin, E. et al. Discrete dynamics and metastability: Mean first passage times and escape rates. J Stat Phys 48, 231–254 (1987). https://doi.org/10.1007/BF01010408
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01010408