Optimal fluctuations and the control of chaos

Peter V.E. McClintock

Department of Physics, Lancaster University,
Lancaster LA1 4YB, United Kingdom

Large fluctuations are responsible for many important physical phenomena, including e.g. stochastic resonance and transport in Brownian ratchets. They usually proceed along optimal paths. Starting from Boltzmann (1904), a huge body of theory was developed during the last century; the modern understanding dates from Onsager and Machlup (1953). The introduction of the prehistory probability distribution established optimal paths as physical observables (Dykman et al, 1992), and the corresponding optimal force driving the fluctuations was measured for the first time by Luchinsky (1997). Recent developments, centered on nonequilibrium systems, will be discussed, including extensions of the work has to encompass escape from chaotic attractors (Khovanov et al, 2000; Luchinsky et al, 2002). In particular, it has been established that fluctuational escape from a chaotic attractor involves the system passing between unstable saddle cycles - thus paving the way for an analytic theory. Measurements of the optimal force can be used to determine the energy-optimal control function needed to effect escape in the deterministic system in the absence of fluctuations.

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Quantized turbulence

Peter V.E. McClintock

Department of Physics, Lancaster University,
Lancaster LA1 4YB, United Kingdom

Turbulence in superfluids - e.g. the superfluid states of liquid $^4$He and $^3$He, the electron gas in superconductors, the nucleonic fluids in neutron stars, and Bose-Einstein condensates in laser-cooled gases - is quantized. It consists of a tangle of vortex lines, each element of which is identical to every other in any given system. Apart from its intrinsic scientific interest it is of importance because (a) being in some ways a very simple form of turbulence one can hope to understand in considerable detail, and (b) it is the state believed to be created during a fast passage through a second order phase transition. Two ongoing research programmes on superfluid turbulence will be reviewed and discussed. First, the initial experiments (Davis et al, 2000) on the decay of turbulence in superfluid $^4$He at mK temperatures will be considered. The vortices are created with a electrostatically-driven vibrating grid, and detected by the use of negative ions travelling near the Landau critical velocity in isotopically pure $^4$He. Preliminary results indicate that the vortex decay rate apparently becomes temperature-independent below about 70 mK. It is believed (Vinen, 2000) that the corresponding decay mechanism may involve a Kolmogorov cascade, Kelvin waves and, ultimately, phonon creation. Secondly, the status of superfluid helium experiments modelling the GUT transition in the early universe 10$^{-35}$ s after the Big Bang (Dodd et al, 1998) will be reviewed.

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