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.
Arrayás M, Casado J M, Gómez Ordóñez J, McClintock P V E, Morillo M, Stein N D 1998 Phys. Rev. Lett. 80 2273
Boltzmann L, 1904 ``On Statistical Mechanics'', address given to the Scientific Congress in St. Louis, 1904 reprinted in Theoretical Physics and Philosophical Problems ed B McGuinness (Dordrecht: Reidel) pp 159-172
Bray A J and McKane A J 1989 Phys. Rev. Lett. 62 493
Dykman M I, McClintock P V E, Stein N D and Stocks N G 1992 Phys. Rev. Lett. 68 2718
Dykman M I, Luchinsky D G, McClintock P V E and Smelyanskiy V N 1996 Phys. Rev. Lett. 77 5229
Dykman M I, Golding B, McCann L I, Smelyanskiy V N, Luchinsky D G, Mannella R and McClintock P V E 2001 Chaos 11 587
Freidlin M I and Wentzell A D 1984 Random Perturbations in Dynamical Systems (New York: Springer-Verlag).
Graham R 1989 in Noise in Nonlinear Dynamical Systems edited by F. Moss and P. V. E. McClintock (Cambridge: Cambridge University Press) vol 1, pp 225-278
Haken H 1975 Rev. Mod. Phys. 47 67
Jauslin H R 1987 J. Stat. Phys. 42 573
Khovanov I A, Luchinsky D G, Mannella R and McClintock P V E 2000 Phys. Rev. Lett 85 2100
Luchinsky D G 1997 J. Phys. A 30 L577
Luchinsky D G and McClintock P V E 1997 Nature 389 463
Luchinsky D G, Maier R S, Mannella R, McClintock P V E and Stein D L 1997 Phys. Rev. Lett. 79 3109
Luchinsky D G, Maier R S, Mannella R, McClintock P V E and Stein D L 1999 Phys. Rev. Lett. 82 1806
Luchinsky D G, Beri S, Mannella R, McClintock P V E and Khovanov I A 2002 Intern J. of Bifurcation and Chaos 12 583
Onsager L and Machlup S 1953 Phys. Rev. 91 1505-1512
Turbulence in superfluids - e.g. the superfluid states of liquid He and 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 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 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 s after the Big Bang (Dodd et al, 1998) will be reviewed.
Baurle C, Bunkov Y M, Fisher S N,Godfrin H, and Pickett G R 1996 Nature 382, 332.
Bowick M J, Chander L, Schiff E A, and Srivastava A M 1994 Science 263, 943.
Chuang I, Turok N, and Yurke B Phys. Rev. Lett. 66, 2472.
Davis S I, Hendry P C, P V E McClintock P V E and Nichol H 2001 in Quantized Vortex Dynamics and Superfluid Turbulence, ed. Barenghi C F, Vinen W F, and R J Donnelly R J, Springer, Berlin, pp 73.
Dodd M E, Hendry P C, Lawson N S, McClintock P V E, and Williams C D H 1998 Phys. Rev. Lett. 81, 3703.
Dodd M E, Hendry P C, Lawson N S, McClintock P V E, and Williams C D H 1999 J. Low Temperature Phys. 115, 89.
Donnelly RJ 1991 Quantized Vortices in Helium II, Cambridge University Press.
Hendry P C, Lawson N S, Lee R A M, McClintock P V E, and Williams C D H 1994 Nature 368, 315.
Kivotides D, Vassilicos J C, Samuels D C, and Barenghi C F 2001 Phys. Rev. Lett. 86, 3080.
Leadmeater M, Winiecki T, Samuels D C, Barenghi C F, and Adams C S 2001 Phys. Rev. Lett. 86, 1410.
McClintock P V E 1999 J. Phys.-Cond. Matter 11, 7695.
Ruutu V M H, Eltsov V B, Gill A J, Kibble T W B, Krusius M, Makhlin Y G, Placais B, Volovik G E, and Xu W 1996 Nature 382, 334.
Stalp S R, Skrbek L, and Donnelly R 1999 Phys. Rev. Lett. 82, 4831.
Svistunov B V 1995 Phys. Rev. B 52, 3647.
Tsubota M, Araki T, and Nemirowski S K 2001 Phys. Rev. B 62 11751.
W F Vinen 2000, Phys. Rev. B 61, 1410-1420.