N. D. Stein

Stochastic resonance in perspective.

SummaryWe outline the historical development of stochastic resonance (SR), a phenomenon in which the signal and/or the signal-to-noise ratio in a nonlinear system increase with increasing intensity of noise. We discuss basic theoretical ideas explaining and describing SR, and we review some revealing experimental data that place SR within the wider context of statistical physics. We emphasize the close relationship of SR to some effects that are well known in condensed-matter physics.

STOCHASTIC RESONANCE IN MONOSTABLE SYSTEMS

The first observations of noise-induced enhancements and phase shifts of a weak periodic signal-characteristics signatures of stochastic resonance (SR)-are reported for a monostable system. The results are shown to be in good agreement with a theoretical description based on linear-response theory and the fluctuation dissipation theorem. It is argued that SR is a general phenomenon that can in principle occur for any underdamped nonlinear oscillator.

Nonconventional Stochastic Resonance

It is argued, on the basis of linear response theory (LRT), that new types of stochastic resonance (SR) are to be anticipated in diverse systems, quite different from the one most commonly studied to date, which has a static double-well potential and is driven by a net force equal to the sum of periodic and stochastic terms. On this basis, three new nonconventional forms of SR are predicted, sought, found, and investigated both theoretically and by analogue electronic experiment: (a) in monostable systems; (b) in bistable systems with periodically modulated noise; and (c) in a system with coexisting periodic attractors. In each case, it is shown that LRT can provide a good quantitative description of the experimental results for sufficiently weak driving fields. It is concluded that SR is a much more general phenomenon than has hitherto been appreciated.

Stochastic resonance: Linear response and giant nonlinearity

The response of a bistable noise-driven system to a weak periodic force is investigated using linear response theory (LRT) and by analogue electronic experiment. For quasithermal systems the response, and in particular its increase with increasing noise intensityD, are described by the fluctuationdissipation relations. For smallD the low-frequency susceptibility of the systemχ(ω) has been found in explicit form allowing for both forced oscillations about the states and periodic modulation of the probabilities of fluctuational transitions between the states. It is shown, both theoretically and experimentally, that a phase lagφ between the force and the response passes through a maximum whenD is tuned through the range where stochastic resonance (SR) occurs. A giant nonlinearity of the response is shown to arise for smallD and small frequencies of the driving force. It results in the signal induced by a sinusoidal force being nearly rectangular. The range of applicability of LRT is established.

High-frequency stochastic resonance in SQUIDs

Abstract It is shown theoretically and by analogue electronic experiment that stochastic resonance (SR), in which a weak periodic signal can be optimally enhanced by the addition of noise of appropriate intensity, is to be anticipated in underdamped SQUIDs (superconducting quantum interference devices). It manifests under conditions quite unlike those needed for classical SR, which is restricted to low frequencies and confined to systems that are both overdamped and bistable. The zero-dispersion SR reported here can be expected over a vastly wider, tunable, range of high frequencies in highly underdamped SQUIDs that need not necessarily be bistable.

Zero-dispersion stochastic resonance

A new form of stochastic resonance (SR), discovered in underdamped nonlinear oscillators for which the dependence of eigenfrequency upon energy has an extremum, is investigated. Its characteristic features are identified and discussed on the basis of linear response theory and the fluctuation dissipation theorem. In common with conventional SR (in bistable systems), sharp increases in the response to a weak periodic force, and in the signal/noise ratio, occur with increasing intensity of external noise (temperature) within a certain range. Unlike conventional SR, however, the dependence of the response on frequency is strongly resonant.

Noise-induced linearisation

It is found that the response of a nonlinear dynamical system can be linearised, and its frequency dispersion diminished, by the addition of external noise of sufficient intensity. Taking as an example an overdamped bistable system driven by a low-frequency periodic field, this noise-induced linearisation is investigated through analogue electronic experiments. The wider implications are considered.

Large fluctuations in a periodically driven dynamical system.

Fluctuations in a periodically driven overdamped oscillator are studied theoretically and experimentally in the limit of low noise intensity by investigation of their prehistory. It is shown that, for small noise intensity, fluctuations to points in coordinate space that are remote from the stable states occur along paths that form narrow tubes. The tubes are centered on the optimal paths corresponding to trajectories of an auxiliary Hamiltonian system. The optimal paths themselves, and the tubes of paths around them, are visualized through measurements of the prehistory probability distribution for an electronic model. Some general features of fluctuations in nonequilibrium systems, such as singularities in the pattern of optimal paths, the corresponding nondifferentiability of the generalized nonequilibrium potential, and the feasibility of their experimental investigation, are discussed.

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.

Zero-Dispersion Stochastic Resonance in Underdamped SQUIDS

Zero-dispersion stochastic resonance (ZDSR) in underdamped SQUIDs (superconducting quantum interference devices) has been investigated theoretically and by analogue electronic experiment. It is shown that, in common with the standard form of stochastic resonance (SR) that occurs at low frequencies in overdamped bistable systems, ZDSR can result in large noise-induced enhancements of the signal/noise ratio for a weak periodic signal. Unlike standard SR, however, ZDSR is to be anticipated over a vastly wider and tunable range of high frequencies in highly underdamped SQUIDs that need not necessarily be bistable.