Abstract
In this paper I will deal with the effect of external random perturbations, “noise”, on chemical systems and other open nonlinear systems. As a concrete example let us consider a CSTR. This is an open system and as such subject to external constraints, namely the concentrations of the chemical species in the feed streams, the flow rate, the stirring rate, the temperature, and the incident light intensity in the case of a photochemical reaction. These external constraints characterize the state of the environment of the open system and will, in general, fluctuate more or less strongly. Such environmental fluctuations are particularly important for natural systems; here random fluctuations are always present and their amplitude is not necessarily small as in laboratory systems. In the latter systems the experimenter will of course try to minimize the effect of random perturbations, though it is impossible to eliminate noise completely. Clearly, random external noise is ubiquitous in open systems, but this fact by itself would hardly warrant a systematic study of the effects of external fluctuations. The question is whether noise is more than a mere nuisance we have to live with. Is there any hope of finding interesting physics? The intuitive, and wrong, answer would be negative: The system averages out rapid fluctuations and the only trace of external noise would be a certain fuzziness in the state of the system. Of course, if the state of the system becomes unstable, the fluctuations initiate the departure from the unstable state. Then the dynamics of the system take over and the system evolves to a new stable state.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P.I. Kuznetsov, R.L. Stratonovic, V.I. Thikhonov: “The Effect of Electrical Fluctuations on a Valve Oscillator”, in Nonlinear Transformations of Stochastic Processes, ed. by P.I. Kuznetsov, R.L. Stratonovic, V.I. Thikhonov (Perga-mon, Oxford 1965) p. 223
R.M. May: Stability and Complexity in Model Ecosystems (University Press, Princeton, NJ 1973)
H.S. Hahn, A. Nitzan, P. Ortoleva, J. Ross: Proc. Natl. Acad. Sci. USA 71, 4067 (1974)
L. Arnold, W. Horsthemke, R. Lefever: Z. Phys. B29, 367 (1978)
W. Horsthemke, R. Lefever: Noise-Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology. Springer Series in Synergetics 15 (Springer, Berlin 1984)
J.L. Doob: Ann. Math. 43, 351 (1942)
G. Blankenship, G.C. Papanicolaou: SIAM J. Appl. Math. 34, 437 (1978)
W. Horsthemke, R. Lefever: Z. Phys. B40, 241 (1980)
H. Haken: Synergetics. An Introduction. 2nd ed. (Springer, Berlin 1977)
G. Nicolis, I. Prigogine: Self-Organization in Nonequilibrium Systems (Wiley, New York 1977)
Yu.V. Prohorov, Yu.A. Rozanov: Probability Theory (Springer, Berlin 1969)
A.R. Bulsara, W.C. Schieve, R.F. Gragg: Phys. Lett. A68, 294 (1978)
J. de la Rubia, M.G. Velarde: Phys. Lett. A69, 304 (1978)
H. Brand, A. Schenzle: J. Phys. Soc. Jpn. 48, 1382 (1980)
M. San Miguel, J.M. Sancho: Z. Phys. B43, 361 (1981)
W. Horsthemke, C.R. Doering, R. Lefever, A.S. Chi: Phys. Rev. A (to appear)
F. Moss, G.V. Weiland: Phys. Rev. A25, 3389 (1982)
R. Graham, M. Höhnerbach, A. Schenzle: Phys. Rev. Lett. 48, 1396 (1982)
S.N. Dixit, P.S. Sahni: Phys. Rev. Lett. 50, 1273 (1983)
R. Lefever, W. Horsthemke: Proc. Natl. Acad. Sci. USA 76, 2490 (1979)
K.A. Wiesenfeld, E. Knobloch: Phys. Rev. A26, 2946 (1982)
W. Ebeling, H. Engel-Herbert: Physica 104A, 378 (1980)
C. van den Broeck: J. Stat. Phys. 31, 467 (1983)
J.M. Sancho, M. San Miguel: Z. Phys. B36, 357 (1980)
K. Lindenberg, B.J. West: Physica 119A, 485 (1983)
K. Kitahara, W. Horsthemke, R. Lefever: Phys. Lett. A70, 377 (1979)
K. Kitahara, W. Horsthemke, R. Lefever, Y. Inaba: Prog. Theor. Phys. 64, 1233 (1980)
A. Schenzle, H. Brand: Phys. Rev. A20, 1628 (1979)
M. Suzuki, K. Kaneko, F. Sasagawa: Prog. Theor. Phys. 65, 828 (1981)
J.M. Sancho, M. San Miguel, S.L. Katz, J.D. Gunton: Phys. Rev. A26, 1589 (1982);
J.M. Sancho, M. San Miguel, H. Yamazaki, T. Kawakubo: Physica 116A, 560 (1982)
P. Hänggi, P. Riseborough: Phys. Rev. A27, 3379 (1983);
P. Talkner, P. Hänggi: Phys. Rev. A29, 768 (1984)
E. Ben-Jacob, D.J. Bergman, B.J. Matkowsky, Z. Schuss: Phys. Rev. A26, 2805 (1982)
E. Cortes, K. Lindenberg: Physica 123A, 99 (1984)
R. Graham: Phys. Rev. A25, 3234 (1982)
R. Lefever, J.W. Turner: in Fluctuations and Sensitivity in Nonequilibrium Systems, ed. by W. Horsthemke and D.K. Kondepudi, Springer Proceedings in Physics 1 (Springer, Berlin 1984)
E. Knobloch, K.A. Wiesenfeld: J. Stat. Phys. 33, 611 (1983)
F.J. de la Rubia, J. Garcia Sanz, M.G. Verlarde: in Nonlinear Stochastic Problems, ed. by R. Bucy, J.M.F. Moura (Reidel, Dordrecht 1983)
K.H. Hoffmann: Z. Phys. B49, 245 (1982)
M. San Miguel, J.M. Sancho: Phys. Lett. 90A, 455 (1982); M.A. Rodriguez, L. Pesquera, M. San Miguel, J.M. Sancho: preprint 1984
W. Horsthemke, R. Lefever: Phys. Lett. A (submitted)
L. Hannon: in Fluctuations and Sensitivity in Noneguilibrium Systems, ed. by W. Horsthemke and D.K. Kondepudi, Springer Proceedings in Physics 1 (Springer, Berlin 1984)
W. Horsthemke, L. Hannon: J. Chem. Phys. (in press); see also this volume
A.S. Mikhailov, L. Schimansky-Geier, W. Ebeling: Phys. Lett. 96A, 453 (1983)
A.S. Mikhailov: Phys. Lett. 73A, 143 (1979);
A.S. Mikhailov: Z. Phys. B41, 277 (1981);
A.S. Mikhailov, I.V. Uporov: Sov. Phys. JETP 52, 989 (1980)
K. Lindenberg, V. Seshadri, B.J. West: Phys. Rev. A22, 2171 (1980)
C.R. Doering, W. Horsthemke: J. Stat. Phys. (Feb. 1985)
For a review, see: J.P. Eckmann: Rev. Mod. Phys. 53, 643 (1981)
S. Kabashima, T. Kawakubo: Phys. Lett. 70A, 375 (1979);
S. Kabashima, S. Kogure, T. Kawakubo, T. Okada: J. Appl. Phys. 50, 6296 (1979)
P. DeKepper, W. Horsthemke: C. R. Acad. Sci. Paris C287, 251 (1978)
J.C. Micheau, W. Horsthemke, R. Lefever: J. Chem. Phys. (Sept. 1984)
S. Kai, T. Kai, M. Takata, K. Hirakawa: J. Phys. Soc. Jpn. 47, 1379 (1979)
T. Kawakubo, A. Yanagita, S. Kabashima: J. Phys. Soc. Jpn. 50, 1451 (1981)
H.R. Brand, S. Kai, S. Wakabayashi: preprint 1984
K. Kaminishi, R. Roy, R. Short, L. Mandel: Phys. Rev. A24, 370 (1981)
J. Smythe, F. Moss, P.V.E. McClintock: Phys. Rev. Lett. 51, 1062 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Horsthemke, W. (1984). Noise Induced Transitions. In: Vidal, C., Pacault, A. (eds) Non-Equilibrium Dynamics in Chemical Systems. Springer Series in Synergetics, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-70196-2_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-70196-2_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-70198-6
Online ISBN: 978-3-642-70196-2
eBook Packages: Springer Book Archive