Dykman M.I., Smelyanskiy V.N., Luchinsky D.G., McClintock P.V.E., Stein N.D., Mannella R.

Rare fluctuations and optimal paths
in systems driven by weak noise
M.I. Dykman , V.N. Smelyanskiy ,
D.G. Luchinsky , P.V.E. McClintock , N.D. Stein ,
R. Mannella
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48823, USA;
School of Physics and Chemistry, Lancaster University, Lancaster, LA1 4YB, UK;
Dipartimento di Fisica, Università di Pisa, Piazza Torricelli 2, 56100 Pisa, Italy

In physical systems driven by weak noise, the interesting and important events, e.g. transitions between coexisting periodic attractors, often occur as the result of large rare fluctuations. Such fluctuations, out in the far wings of the probability distribution, can be studied theoretically [1] in terms of optimal paths which define the most probable trajectory bringing the system to a given state far from an attractor. The role of optimal paths in the theory of fluctuations is similar to the role of dynamical trajectories in mechanics. There is also a deep analogy between optimal fluctuational paths and rays in optics, or extreme paths in quantum mechanics. Optimal paths are physically real and have been observed experimentally through measurements of the prehistory probability distribution [2]. Similar to the pattern of extreme paths the pattern of optimal paths is complicated and may have singularities. However, singularities of the most general type, caustics, may not be encountered by optimal paths [3]. Instead, there occur singularities of a different sort, switching lines [3,4] that start from cusp points or at unstable fixed points. Efforts are in progress to observe experimentally the onset of singularities in the pattern of optimal paths, and related phenomena. The results, and their wider implications, will be described and discussed.

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3. M.I. Dykman, V.N. Smelyanskiy, and M.M. Millonas, Phys. Lett. A, 195 (1994) 53.
4. R. Graham, in: Noise in Nonlinear Dynamical Systems, ed. F. Moss and P.V.E. McClintock, Cambridge University Press, 1 (1989) 225.