R. Mannella

Activated escape of periodically driven systems

We discuss activated escape from a metastable state of a system driven by a time-periodic force. We show that the escape probabilities can be changed very strongly even by a comparatively weak force. In a broad parameter range, the activation energy of escape depends linearly on the force amplitude. This dependence is described by the logarithmic susceptibility, which is analyzed theoretically and through analog and digital simulations. A closed-form explicit expression for the escape rate of an overdamped Brownian particle is presented and shown to be in quantitative agreement with the simulations. We also describe experiments on a Brownian particle optically trapped in a double-well potential. A suitable periodic modulation of the optical intensity breaks the spatio-temporal symmetry of an otherwise spatially symmetric system. This has allowed us to localize a particle in one of the symmetric wells. (c) 2001 American Institute of Physics.

Linear Response Theory in Stochastic Resonance

The susceptibility of an overdamped Markov system fluctuating in a bistable potential of general form is obtained by analytic solution of the Fokker-Planck equation (FPE) for low noise intensities. The results are discussed in the context of the LRT theory of stochastic resonance. They go over into recent results of Hu et al. [Phys. Lett. A 172 (1992) 21] obtained from the FPE for the case of a symmetrical potential, and they coincide with the LRT results of Dykman et al. [Phys. Rev. Lett. 65 (1990) 2606; JETP Lett. 52 (1990) 144; Phys. Rev. Lett. 68 (1992) 2985] obtained for the general case of bistable systems.

The projection approach to the Fokker-Planck equation. I: Colored gaussian noise

It is shown that the Fokker-Planck operator can be derived via a projection-perturbation approach, using the repartition of a more detailed operatorℒ into a perturbationℒ1 and an unperturbed partℒ0. The standard Fokker-Planck structure is recovered at the second order inℒ1, whereas the perturbation terms of higher order are shown to provoke the breakdown of this structure. To get rid of these higher order terms, a key approximation, local linearization (LL), is made. In general, to evaluate at the second order inℒ1 the exact expression of the diffusion coefficient which simulates the influence of a Gaussian noise with a finite correlation timeτ, a resummation up to infinite order inτ must be carried out, leading to what other authors call the best Fokker-Planck approximation (BFPA). It is shown that, due to the role of terms of higher order inℒ1, the BFPA leads to predictions on the equilibrium distributions that are reliable only up to the first order in t. The LL, on the contrary, in addition to making the influence of terms of higher order inℒ1 vanish, results in a simple analytical expression for the term of second order that is formally coincident with the complete resummation over all the orders in t provided by the Fox theory. The corresponding diffusion coefficient in turn is shown to lead in the limiting case τ→∞ to exact results for the steady-state distributions. Therefore, over the whole range 0⩽τ⩽∞ the LL turns out to be an approximation much more accurate than the global linearization proposed by other authors for the same purpose of making the terms of higher order inℒ1 vanish. In the short-τ region the LL leads to results virtually coincident with those of the BFPA. In the large-τ region the LL is a more accurate approximation than the BFPA itself. These theoretical arguments are supported by the results of both analog and digital simulation.

Thermally activated escape of driven systems: the activation energy

Thermally activated escape in the presence of a periodic external field is investigated theoretically and through analogue experiments and digital simulations. The observed variation of the activation energy for escape with driving force parameters is accurately described by the logarithmic susceptibility (LS). The frequency dispersion of the LS is shown to differ markedly from the standard linear susceptibility. Experimental data on the dispersion are in quantitative agreement with the theory. Switching between different branches of the activation energy is demonstrated for a nonsinusoidal (biharmonic) force.

Noise-Induced Spectral Narrowing in Nonlinear Oscillators

The spectral densities of the fluctuations of noise-driven underdamped nonlinear oscillators are discussed with particular reference to the large class of systems whose eigenfrequencies vary nonmonotonically with energy. It is shown by analogue electronic experiments and theoretically that, astonishingly, the widths of their spectral peaks can sometimes decrease with increasing noise intensity.

Ratchet driven by quasimonochromatic noise

The currents generated by noise-induced activation processes in a periodic potential are investigated analytically, by digital simulation and by performing analog experiments. The noise is taken to be quasimonochromatic and the potential to be a smoothed sawtooth. Two analytic approaches are studied. The first involves a perturbative expansion in inverse powers of the frequency characterizing quasimonochromatic noise and the second is a direct numerical integration of the deterministic differential equations obtained in the limit of weak noise. These results, together with the digital and analog experiments, show that the system does indeed give rise, in general, to a net transport of particles. All techniques also show that a current reversal exists for a particular value of the noise parameters.

The effect of multiplicative noise on the relaxation time of a real non-linear physical system: a comparison of experiment and theory for the random growing rate model (RGRM)

It is found that the theoretical behaviour of the relaxation time T for the random growing rate model (RGRM) under the influence of multiplicative noise of intensity Q (a monotonic decrease of T-1 towards a limiting value as Q to infinity ) differs markedly from that actually measured in electronic simulators of that system. A new electronic analogue experiment is described in which a distinct minimum in T-1(Q) has been observed for the first time, with clear evidence for an increase of T-1(Q) with Q at large Q in good qualitative agreement with an earlier analogue experiment. The discrepancy between experiment and the theoretical solutions of the (idealised) equation is attributed to the profound influence exerted by the very weak additive noise which must also, in some measure, always be present in a real physical system.

Spectral Distribution of a Double-Well Duffing Oscillator Subject to a Random Force

The power spectral density Q(ω) for an electronic circuit model of the double-well Duffing oscillator, driven by Gaussian white noise, has been measured in the limit of very low damping. Three distinct maxima and a plateau region are found in Q(ω), in excellent qualitative agreement with a recent theoretical prediction by Dykman and co-workers.

Adiabatic ac-drive as a tool for acceleration of diffusion in spatially periodic structures and of reset process in threshold devices

We have shown that both deterministic and stochastic dynamics of a spatially periodic underdamped system in the presence of, even a rather weak, adiabatic ac-drive drastically differs from that in the absence of the drive or in the presence of other kinds of driving. This suggests promising applications.

Short-time dynamics of noise-induced escape (Invited Paper)

We consider by means of the optimal fluctuation method the initial stage of the evolution of the noise-induced escape through various types of boundaries, especially concentrating on two types of the boundary - the wall and the boundary of the basin of attraction. We show in both cases that, if the damping is small enough, then the escape flux evolution possesses a remarkable property: it is it stairs-like i.e. intervals of a nearly constant flux alternate with intervals of a sharply increasing flux. This property is related to the successive increase of the number of turning points in the most probable escape path as time increases. Our results are relevant both for the absorbing and transparent boundaries. The major results of the theory are verified in computer simulations.