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Singular perturbations in noisy dynamical systems

Published online by Cambridge University Press:  31 January 2018

B. J. MATKOWSKY
B. J. MATKOWSKY*
Affiliation:
Department of Engineering Sciences & Applied Mathematics, Northwestern University, Evanston, IL 60208, USA email: b-matkowsky@northwestern.edu
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Abstract

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Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of ‘spurious solutions’, which led some to conclude that this was ‘the failure of MAE’. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.

Keywords

Singular perturbationsasymptotic expansionsJWKB methods
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 4 , August 2018 , pp. 570 - 593
Copyright
Copyright © Cambridge University Press 2018 

References

[1] Ackerberg, A. & O'Malley, R. (1970) Boundary layer problems exhibiting resonance. Stud. Appl. Math. 49, 277–295.Google Scholar
[2] Arrhenius, S. (1889) Uber die Reactionsgeschwindigkeit bei der Inversion von Rohrzucker Durch Sauern. Z. Phys. Chem. 4, 226–248.CrossRefGoogle Scholar
[3] Ben-Jacob, E., Bergman, D., Matkowsky, B. & Schuss, Z. (1988) Transitions from the equilibrium state of a hysteretic Josephson junction induced by self-generated shot noise. Phys. Lett. 99A, 343–347.Google Scholar
[4] Ben-Jacob, E., Bergman, D., Imry, J., Matkowsky, B. & Schuss, Z. (1983) Shot noise effect on the nonzero voltage state of the hysteretic Josephson junction. Appl. Phys. Lett. 42, 1045–1047.Google Scholar
[5] Ben-Jacob, E., Bergman, D., Matkowsky, B. & Schuss, Z. (1986) Master equation approach to shot noise in Josephson junctions. Phys. Rev. B 34, 1572–1581.Google Scholar
[6] Ben-Jacob, E., Bergman, D., Matkowsky, B. & Schuss, Z. (1982) The lifetime of Oscillatory States. Phys. Rev. A26, 2805–2816.Google Scholar
[7] Brillouin, L. (1926) La Mecanique Ondulatoire de ShrodingerUne Methode Generale de Resolution par Approximations Successives. Compt. Rend. Acad. Sci. 183, 24–26.Google Scholar
[8] Brown, R. (1828) A brief account of microscopic observations made in the months of June, July and August contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Phil. Mag. 4, 161–173.Google Scholar
[9] Chapman, S. J. & Matkowsky, B. (2013) unpublished.Google Scholar
[10] Dygas, M., Matkowsky, B. & Schuss, Z. (1986) A singular perturbation approach to non-Markovian escape rate problems. SIAM J. Appl. Math. 46, 265–298.Google Scholar
[11] Dygas, M., Matkowsky, B. & Schuss, Z. (1985) On the non-Markovian theory of activated rate processes in the small friction limit. J. Chem. Phys. 83, 597–600.CrossRefGoogle Scholar
[12] Dygas, M., Matkowsky, B. & Schuss, Z. (1986) A singular perturbation approach to non-Markovian escape rate problems with state dependent friction. J. Chem. Phys. 84, 3731–3738.Google Scholar
[13] Dynkin, E. (1965) Markov Processes, Springer, Berlin.Google Scholar
[14] Einstein, A. (1905) Uber die von der molekularkinetischen theorie der warme geforderte bewegung von in ruhenden flussigkeiten suspendierten teilichen. Ann. Phys. 17, 549–560; see also Einstein, A. (1956), Investigations on the Theory of the Brownian Movement, Ed. Furth, R., Dover Publications.Google Scholar
[15] Fokker, A. (1914) Die Mittlere Energie Rotlierender Elektrische Dipole in Stralungsfeld. Ann. Phys. 43, 810–820.Google Scholar
[16] Friedrichs, K. & Wasow, W. (1946) Singular perturbations of nonlinear oscillations. Duke Math. J. 13, 367–381.Google Scholar
[17] Grasman, J. & Matkowsky, B. (1977) A variational approach to singular perturbation problems with turning points for ordinary and partial differential equations. SIAM J. Appl. Math. 32, 588–597.Google Scholar
[18] Habetler, G. & Matkowsky, B. (1975) Uniform asymptotic expansions in transport theory with small mean free paths. J. Mathl. Phys. 16, 846–854.Google Scholar
[19] Ito, K. (1950) On a formula concerning stochastic differentials. Nagoya Math. J. 3, 55–65.Google Scholar
[20] Jeffreys, H. (1924) On certain approximate solutions of linear differential equations of the second order. Proc. London Math. Soc. 23, 428–436.Google Scholar
[21] Kath, W., Knessl, C. & Matkowsky, B. (1987) Variational approach to nonlinear singularly perturbed boundary value problems. Stud. Appl. Math. 77, 61–88.Google Scholar
[22] Klosek-Dygas, M., Matkowsky, B. & Schuss, Z. (1988) Colored noise in dynamical systems. SIAM J. Appl. Math. 48, 425–441.Google Scholar
[23] Klosek-Dygas, M, Matkowsky, B. & Schuss, Z. (1989) Colored noise in activated rate processes. J. Stat. Phys. 54, 1309–1320.CrossRefGoogle Scholar
[24] Klosek-Dygas, M., Matkowsky, B. & Schuss, Z. (1989) First order dynamics driven by rapid Markovian jumps. SIAM J. Appl. Math. 49, 1811–1833.CrossRefGoogle Scholar
[25] Knessl, C., Matkowsky, B., Schuss, Z. & Tier, C. (1986) A singular perturbation approach to first passage times for Markov jump processes. J. Stat. Phys. 42, 169–184.Google Scholar
[26] Kolmogorov, A. (1931) Uber der Analytischen Methoden in Wahrscheinlichkeits Rechnung. Math. Ann. 104, 415–458.Google Scholar
[27] Kramers, H. (1940) Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304.Google Scholar
[28] Kramers, H. (1926) Wellenmechanik und Halbzaklige Quantiszeiung. Z. Phys. 39, 828–840.Google Scholar
[29] Langevin, P. (1908) Sur la Theorie du Mouvement Brownien. i Comptes Rendus Acad. Sci. 146, 530–533.Google Scholar
[30] Lee, J. & Ward, M. (1994) On the asymptotic and numerical analysis of exponentially Ill-conditioned singularly perturbed boundary value problems. Stud. Appl. Math. 94, 271–326.Google Scholar
[31] Levinson, N. (1950) The first boundary value problem for εΔu + Aux + Buy + Cu = D for small ε. Ann. Math. 51, 428–445.Google Scholar
[32] MacGillivray, A. (1997) A method for incorporating transcendentally small terms into the method of matched asymptotic expansions. Stud. Appl. Math. 99, 285–310.Google Scholar
[33] Matkowsky, B. (1970) unpublished.Google Scholar
[34] Matkowsky, B. (1975) On boundary layer problems exhibiting resonance. Siam Rev. 17, 82–100.Google Scholar
[35] Matkowsky, B. & Schuss, Z. (1977) The exit problem for randomly perturbed dynamical systems. SIAM J. Appl. Math. 33, 365–382.Google Scholar
[36] Matkowsky, B. & Schuss, Z. (1981) Eigenvalues of the Fokker–Planck operator and the approach to equilibrium for diffusions in potential fields. SIAM J. Appl. Math. 40, 242–254.Google Scholar
[37] Matkowsky, B. & Schuss, Z. (1982) Diffusion across characteristic boundaries. SIAM J. Appl. Math. 42, 822–834.Google Scholar
[38] Matkowsky, B., Schuss, Z. & Ben-Jacob, E. (1982) A singular perturbation approach to Kramers diffusion problem. SIAM J. Appl. Math. 42, 835–849.Google Scholar
[39] Matkowsky, B., Schuss, Z. & Tier, C. (1983) Diffusion across characteristic boundaries with critical points. SIAM J. Appl. Math. 43, 673–695.Google Scholar
[40] Matkowsky, B., Schuss, Z., Knessl, C., Tier, C. & Mangel, M. (1984) Asymptotic solution of the Kramers Moyal equation and first passage times for Markov jump processes. Phys. Rev. A29, 3359–3369.Google Scholar
[41] Perrin, J. (1908) Mouvement brownien et Realite Moleculaire. Ann. Chim. Phys. 18, 5–114.Google Scholar
[42] Planck, M. (1917) Uber Einen Satz der Statischen Dynamik und Seinen Erweiterung Quantentheorie. Sitz. ber. Pruss. Akad. Wiess. 24, 324–341.Google Scholar
[43] Poincare, H. (1886) Sur les Integrales Irregulieres: Des Equations Lineares. Acta Math. 8, 295–344.CrossRefGoogle Scholar
[44] Prandtl, L. (1905) Uber Flussigkrecitsbewgung bei sehr kleiner Reibung. In Verhandlungen des 3rd International Mathematiker Kongress, Heidelberg, pp. 484–491.Google Scholar
[45] Schuss, Z. & Matkowsky, B. (1979) The exit problem: A new approach to diffusion across potential barriers. SIAM J. Appl. Math. 35, 604–623.Google Scholar
[46] Smoluchowski, M. (1906) Zur Kinetischen Theorie und der Brownschen Molekularbewegung Suspensionen. Ann. Phys. 21, 756–780.Google Scholar
[47] von Kampen, N. (1981) Stochastic Processes in Chemical Physics, North Holland, Amsterdam.Google Scholar
[48] Wasow, W. (1942) Boundary Layer Problems in the Theory of ODEs, Ph.D Thesis, NYU.Google Scholar
[49] Wentzel, G. (1926) Eine Verallgeneinerung der Quanten Bedingungen fur die Zwecke der Wellen Mechanek. Z. Pys. 38, 518–529.Google Scholar
[50] Wiener, N. (1923) Differential space. Stud. Appl. Math. 2, 131–174.Google Scholar