A scaling theory of bifurcations in the symmetric weak-noise escape problem

Journal of Statistical Physics

Volumn 83, Issue 3-4, 1996, Pages 291-357

  Maier, Robert S ^a   Stein, D L ^a  

^a University of Arizona   (United States)

Author keywords

Boundary layer; Caustics; Double well; Fokker Planck equation; Lagrangian manifold; Large deviation theory; Large fluctuations; Maslov WKB method; Non Arrhenius behavior; Nongeneric caustics; Pearcey function; Singular perturbation theory; Smoluchowski equation

Indexed keywords

EID: 0030526108     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02183736     Document Type: Article

References (38)

1. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
2. V. I. Arnol'd, Critical points of smooth functions and their normal forms, Russ. Math. Surv. 30(5):1-75 (1975).
3. A. Auerbach and S. Kivelson, The path decomposition and multidimensional tunneling, Nucl. Phys. B 257[FS 14]:799-858 (1985).
4. M. V. Berry, Waves and Thorn's theorem, Adv. Phys. 25:1-26 (1976).
5. M. V. Berry and N. L. Balazs, Evolution of semiclassical quantum states in phase space, J. Phys. A 12:625-642 (1979).
6. A. J. Bray and A. J. McKane, Instanton calculation of the escape rate for activation over a potential barrier driven by colored noise, Phys. Rev. Lett. 62:493-496 (1989).
7. B. Carmeli, V. Mujica, and A. Nitzan, Dynamics of multidimensional barrier crossing in the overdamped limit, Ber. Bunsen ges. 95:319-326 (1991).
8. B. Caroli, C. Caroli, B. Roulet, and J.-F. Gouyet, A WKB treatment of diffusion in a multidimensional bistable potential, J. Stat. Phys. 22:515-536 (1980).
9. M. V. Day, Recent progress on the small parameter exit problem, Stochastics 20:121-150 (1987).
10. J. J. Duistermaat, Oscillatory integrals, Lagrange immersions, and unfolding of singularities, Commun. Pure Appl. Math. 27:207-281 (1974).
11. M. I. Dykman, P. V. E. McClintock, V. N. Smelyanskiy, N. D. Stein, and N. G. Stocks, Optimal paths and the prehistory problem for large fluctuations in noise-driven systems, Phys. Rev. Lett. 68:2718-2721 (1992).
12. M. I. Dykman, M. M. Millonas, and V. N. Smelyanskiy, Observable and hidden singular features of large fluctuations in nonequilibrium systems, Phys. Lett. A 195:53-58 (1994).
13. J.-P. Eckmann and R. Sénéor, The Maslov-WKB method for the (an-)harmonic oscillator, Arch. Rat. Mech. Anal. 61:153-173 (1976).
14. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer-Verlag, New York, 1984).
15. S. Glasstone, K. J. Laidler, and H. Eyring, The Theory of Rate Processes (McGraw-Hill, New York, 1941).
16. R. Graham, Macroscopic potentials, bifurcations and noise in dissipative systems, in Theory of Continuous Fokker-Planck Systems, F. Moss and P. V. E. McClintock, eds. (Cambridge University Press, Cambridge, 1989), Chapter 7, pp. 225-278.
17. M. C. Gutzwiller, Periodic orbits and classical quantization conditions, J. Math. Phys. 12:343-358 (1971).
18. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, Berlin, 1990).
19. H. R. Jauslin, Nondifferentiable potentials for nonequilibrium steady states, Physica A 144:179-191 (1987).
20. J. B. Keller, Corrected Bohr-Sommerfeld quantum conditions for non-separable systems, Ann. Phys. 4:180-188 (1958).
21. M. M. Kłosek-Dygas, B. M. Hoffman, B. J. Matkowsky, A. Nitzan, M. A. Ratner, and Z. Schuss, Diffusion theory of multidimensional activated rate processes: The role of anisotropy, J. Chem. Phys. 90:1141-1148 (1989).
22. H. A. Kramers, Brownian motion on a field of force and the diffusion model of chemical reactions, Physica 7:284-304 (1940).
23. R. Landauer, Motion out of noisy states, J. Stat. Phys. 53:233-248 (1988).
24. J. S. Langer, Statistical theory of the decay of metastable states. Ann. Phys. 54:258-275 (1969).
25. R. G. Littlejohn, The Van Vleck formula, Maslov theory, and phase space geometry, J. Stat. Phys. 68:7-50 (1992).
26. D. Ludwig, Persistence of dynamical systems under random perturbations, SIAM Rev. 17:605-640 (1975).
27. S.-K. Ma, Modern Theory of Critical Phenomena (Benjamin, Reading, Massachusetts, 1976).
28. R. S. Maier and D. L. Stein, The effect of focusing and caustics on exit phenomena in systems lacking detailed balance, Phys. Rev. Lett. 71:1783-1786 (1993).
29. R. S. Maier and D. L. Stein, The escape problem for irreversible systems, Phys. Rev. E 48:931-938 (1993).
30. R. S. Maier and D. L. Stein, Limiting exit location distributions in the stochastic exit problem, SIAM J. Appl. Math., provisionally accepted.
31. M. Mangel, Barrier transitions driven by fluctuations, with applications to evolution and ecology, Theoret. Popul. Biol. 45:16-40 (1994).
32. V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics (Reidel, Dordrecht, 1981).
33. T. Naeh, M. M. Kłosek, B. J. Matkowsky, and Z. Schuss, A direct approach to the exit problem, SIAM J. Appl. Math. 50:595-627 (1990).
34. L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. 91:1505-1512 (1953).
35. R. B. Paris, The asymptotic behaviour of Pearcey's integral for complex variables, Proc. R. Soc. Lond. A 432:391-426 (1991).
36. L. S. Schulman, Techniques and Applications of Path Integrations (Wiley, New York, 1981).
37. M. A. Shayman, A geometric view of the matrix Riccati equation, in The Riccati Equation, S. Bittanti, A. J. Laub, and J. C. Willems, eds. (Springer-Verlag, Berlin, 1991), pp. 89-112.
38. P. Talkner, Mean first passage times and the lifetime of a metastable state, Z. Phys. B 68:201-207 (1987).