Abstract
A complete study of non-Markovian effects induced by correlated noise applied to a bistable dynamical system is presented. Starting from the exact functional integral solution of the stochastic equation, it is possible to show that the customary expansion in powers of the characteristic correlation time gives wrong asymptotic results. Other approaches based on a Fokker-Planck equation with a modified diffusion coefficient also fail in reproducing the right long-time behavior of the system. Using a generalized version of instanton calculus of functional integrals, explicit expressions of the invariant measure and transition time between stable fixed points are obtained, in the limit of small noise intensity but arbitrary correlation time. In particular, an original method for extracting the collective degrees of motion has been developed. These analytical results fit, for a large range of parameters, with numerical calculations, giving confidence in the formalism employed.
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References
H. Haken,Synergetics (Springer-Verlag, Berlin, 1983).
Z. Schuss,Theory and Applications of Stochastic Differential Equations (Wiley, New York, 1984).
R. Graham,Europhys. Lett. 2:901 (1986).
K. Ishii,Progr. Theor. Phys. (Suppl.)53:77 (1973).
J. M. Sancho, M. San Miguel, S. L. Katz, and J. D. Gunton,Phys. Rev. A 26:1589 (1982).
P. Hanggi,J. Stat. Phys. 42:105 (1986).
M. I. Freidlin and A. D. Wentzell,Random Perturbations of Dynamical Systems (Springer-Verlag, New York, 1984).
J. F. Luciani and A. D. Verga,Europhys. Lett. 4:255 (1987).
R. Rajaraman,Solitons and Instantons (North-Holland, Amsterdam, 1982), p. 390.
A. I. Larkin and V. N. Ovchinnikov,Sov. Phys. JETP 59:420 (1984).
N. G. van Kampen,Phys. Rep. 24:113 (1976).
P. Hanggi, F. Marchesoni, and P. Grigolini,Z. Phys. B 56:333 (1984).
R. Baush, H. K. Jansen, and H. Wagner,Z. Phys. B 24:113 (1976).
F. Langouche, D. Rekaerts, and E. Tirapegui,Field Theory, Quantization and Statistical Physics, E. Tirepagui, ed. (Reidel, Dordrecht, 1981), p. 295.
B. Jouvet and R. Phythian,Phys. Rev. A 19:1350 (1979).
R. V. Jensen,J. Stat. Phys. 25:183 (1981); R. V. Jensen and C. R. Oberman,Phys. Rev. Lett. 46:1547 (1981).
E. Gozzi,Phys. Rev. D 28:1922 (1983).
M. I. Freidlin,Functional Integration and Partial Differential Equations (Princeton University Press, Princeton, New Jersey, 1985).
U. Weiss,Phys. Rev. A 25:2444 (1982).
E. J. McShane,Stochastic Calculus and Stochastic Models (Academic Press, New York, 1974), p. 186.
P. Hanggi, T. J. Mroczkowski, F. Moss, and P. V. E. McClintock,Phys. Rev. A 32:695 (1985).
K. Vogel, Th. Leiber, H. Risken, P. Hänggi, and W. Schleich,Phys. Rev. A 35:4882 (1987).
R. F. Fox and R. Roy,Phys. Rev. A 35:1838 (1987).
R. F. Fox,Phys. Rev. A 33:467 (1986),34:4525 (1986).
A. G. Grin',SIAM Theory Prob. Appl. 20:442 (1975).
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Luciani, J.F., Verga, A.D. Bistability driven by correlated noise: Functional integral treatment. J Stat Phys 50, 567–597 (1988). https://doi.org/10.1007/BF01026491
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DOI: https://doi.org/10.1007/BF01026491