Abstract
A trace formula is derived for the Fokker–Planck equation associated with Itô stochastic differential equations describing noisy time-continuous nonlinear dynamical systems. In the weak-noise limit, the trace formula provides estimations of the eigenvalues of the Fokker–Planck operator on the basis of the Pollicott–Ruelle resonances of the noiseless deterministic system, which is assumed to be non-bifurcating. At first order in the noise amplitude, the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo-energy of the Onsager–Machlup–Freidlin–Wentzell scheme.
Similar content being viewed by others
REFERENCES
G. Nicolis, Introduction to Nonlinear Science (Cambridge University Press, Cambridge, 1995).
G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977).
N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
W. Horsthemcke and R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology (Springer-Verlag, Berlin, 1984).
C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, New York, 1990).
P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62:251 (1990).
L. Arnold, Random Dynamical Systems (Springer-Verlag, Berlin, 1998).
G. Nicolis and I. Prigogine, Proc. Natl. Acad. Sci. (USA) 68:2102 (1971).
G. Nicolis, J. Statist. Phys. 6:195 (1972).
G. Nicolis and M. Malek Mansour, Prog. Theor. Phys. Suppl. 64:249 (1978).
F. Baras, M. Malek Mansour, and C. Van Den Broeck, J. Statist. Phys. 28:577 (1982).
F. Baras, J. E. Pearson, and M. Malek Mansour, J. Chem. Phys. 93:5747 (1990).
F. Baras, Topics in Nonequilibrium Statistical Mechanics of Reactive Systems (Thèse d'Agrégation de l'Enseignement Supérieur, Université Libre de Bruxelles, février 2001).
M. Mareschal and A. De Wit, J. Chem. Phys. 96:2000 (1992).
W. Vance and J. Ross, J. Chem. Phys. 105:479 (1996).
L. Onsager and S. Machlup, Phys. Rev. 91:1505 (1953).
S. Machlup and L. Onsager, Phys. Rev. 91:1512 (1953).
G. L. Eyink, J. Statist. Phys. 61:533 (1990).
M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer-Verlag, Berlin, 1984).
R. Graham and T. Tél, Phys. Rev. A 31:1109 (1985).
R. S. Maier and D. L. Stein, Phys. Rev. Lett. 71:1783 (1993).
M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990).
M. Gutzwiller, in Chaos and Quantum Physics, M.-J. Giannoni, A. Voros, and J. Zinn-Justin, eds., Les Houches 1989, Session LII (North-Holland, Amsterdam, 1991), pp. 201-250.
H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, UK, 1999).
G. Nicolis and P. Gaspard, Chaos, Solitons & Fractals 4:41 (1994).
P. Cvitanovi? and B. Eckhardt, J. Phys. A: Math. Gen. 24:L237 (1991).
M. Pollicott, Invent. Math. 81:413 (1985); 85:147 (1986).
D. Ruelle, Phys. Rev. Lett. 56:405 (1986); J. Statist. Phys. 44:281 (1986); J. Diff. Geom. 25:117 (1987); Commun. Math. Phys. 125:239 (1989).
P. Gaspard, Chaos, Scattering, and Statistical Mechanics (Cambridge University Press, Cambridge, 1998).
C. P. Dettmann, Phys. Rev. E 59:5231 (1999).
P. Cvitanovi?, C. P. Dettmann, R. Mainieri, and G. Vattay, J. Statist. Phys. 93:981 (1998).
P. Cvitanovi?, C. P. Dettmann, R. Mainieri, and G. Vattay, Nonlinearity 12:939 (1999).
P. Cvitanovi?, C. P. Dettmann, N. Søndergaard, G. Vattay, and G. Palla, Phys. Rev. E 60:3936 (1999).
G. Palla, G. Vattay, A. Voros, N. Søndergaard, and C. P. Dettmann, Foundations of Physics 31:641 (2001).
N. Søndergaard, G. Palla, G. Vattay, and A. Voros, J. Statist. Phys. 101:385 (2000).
R. Kubo, K. Matsuo, and K. Kitahara, J. Statist. Phys. 9:51 (1973).
K. Kitahara, Adv. Chem. Phys. 29:85 (1975).
A. Suárez, J. Ross, B. Peng, K. L. C. Hunt, and P. M. Hunt, J. Chem. Phys. 102:4563 (1995).
P. Gaspard, D. Alonso, and I. Burghardt, Adv. Chem. Phys. 90:105 (1995).
R. G. Littlejohn, J. Math. Phys. 31:2952 (1990).
R. Graham, Phys. Rev. A 25:3234 (1982).
R. Graham and T. Tél, Phys. Rev. A 33:1322 (1986).
R. Graham and T. Tél, Phys. Rev. A 35:1328 (1987).
P. Gaspard, G. Nicolis, A. Provata, and S. Tasaki, Phys. Rev. E 51:74 (1995).
R. Artuso, E. Aurell, and P. Cvitanovi?, Nonlinearity 3:325, 361 (1990).
M. Dykman, X. Chu, and J. Ross, Phys. Rev. E 48:1646 (1993).
D. Gonze, J. Halloy, and A. Goldbeter, Robustness of Circadian Rhythms with Respect to Molecular Noise (preprint, Université Libre de Bruxelles, 2001).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gaspard, P. Trace Formula for Noisy Flows. Journal of Statistical Physics 106, 57–96 (2002). https://doi.org/10.1023/A:1013167928166
Issue Date:
DOI: https://doi.org/10.1023/A:1013167928166