History: Stochastic resonance was first discovered in a study of the periodic recurrence of earth’s ice ages. The theory developed out of an effort to understand how the earth's climate oscillates periodically between two relatively stable global temperature states, one "normal" and the other an "ice age" state. The conventional explanation was that variations in the eccentricity of earth's orbital path occurred with a period of about 100,000 years and caused the average temperature to shift dramatically. The measured variation in the eccentricity had a relatively small amplitude compared to the dramatic temperature change, however, and stochastic resonance was developed to show that the temperature change due to the weak eccentricity oscillation and added stochastic variation due to the unpredictable energy output of the sun (known as the solar constant) could cause the temperature to move in a nonlinear fashion between two stable dynamic states.
Wikipedia: Stochastic resonance is observed when noise added to a system changes the system's behaviour in some fashion. More technically, SR occurs if the signal-to-noise ratio of a nonlinear system or device increases for moderate values of noise intensity [or reversely?] It often occurs in bistable systems or in systems with a sensory threshold and when the input signal to the system is "sub-threshold". For lower noise intensities, the signal does not cause the device to cross threshold, so little signal is passed through it. For large noise intensities, the output is dominated by the noise, also leading to a low signal-to-noise ratio. For moderate intensities, the noise allows the signal to reach threshold, but the noise intensity is not so large as to swamp it. Thus, a plot of signal-to-noise ratio as a function of noise intensity shows a '∩' shape.
Typical curve of output performance versus input noise magnitude, for systems capable of stochastic resonance.
For small and large noise, the performance metric (e.g., SNR, mutual information, Fisher information, correlation, discrimination index) is very small, while some intermediate nonzero noise level provides optimal performance.
The word resonance in the term stochastic resonance was originally used because the signature feature of SR is that a plot of a performance measure—such as output signal-to-noise ratio (SNR)—against input noise “intensity” has a single maximum at a nonzero value. Such a plot, as shown in Figure, has a similar appearance to frequency-dependent systems that have a maximum SNR, or output response, for some resonant frequency. However, in the case of SR, the resonance is “noise-induced” rather than at a particular frequency.
Stochastic resonance occurs in bistable systems, when a small periodic (sinusoidal) force is applied together with a large wide band stochastic force (noise). The system response is driven by the combination of the two forces that compete/cooperate to make the system switch between the two stable states. The degree of order is related to the amount of periodic function that it shows in the system response. When the periodic force is chosen small enough in order to not make the system response switch, the presence of a non-negligible noise is required for it to happen. When the noise is small very few switches occur, mainly at random with no significant periodicity in the system response. When the noise is very strong a large number of switches occur for each period of the sinusoid and the system response does not show remarkable periodicity. Between these two conditions, there exists an optimal value of the noise that cooperatively concurs with the periodic forcing in order to make almost exactly one switch per period (a maximum in the signal-to-noise ratio).
Such a favorable condition is quantitatively determined by the matching of two time scales: the period of the sinusoid (the deterministic time scale) and the Kramers rate (i.e., the inverse of the average switch rate induced by the sole noise: the stochastic time scale). Thus the term "stochastic resonance".
- Weisstein, Eric W. "Kramers Rate." From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/KramersRate.html The charachteristic escape rate from a stable state of a potential in the absence of signal.
- Benzi, R.; Sutera, A.; and Vulpiani, A. "The Mechanism of Stochastic Resonance." J. Phys. A 14, L453-L457, 1981.
- Gammaitoni, L. "Stochastic Resonance E-Print Server." http://www.umbrars.com/sr/.
- Bulsara, A. R. and Gammaitoni, L. "Tuning in to Noise." Phys. Today 49, 39-45, March 1996. A stochastic resonance is a phenomenon in which a nonlinear system is subjected to a periodic modulated signal so weak as to be normally undetectable, but it becomes detectable due to resonance between the weak deterministic signal and stochastic noise. The earliest definition of stochastic resonance was the maximum of the output signal strength as a function of noise.
Stochastic resonance was discovered and proposed for the first time in 1981 to explain the periodic recurrence of ice ages. Since then the same principle has been applied in a wide variety of systems. Nowadays stochastic resonance is commonly invoked when noise and nonlinearity concur to determine an increase of order in the system response.
The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.The classical Kramers problem (the dissociation problem): The rate of escape of a classical particle over an energy barrier is a well-posed problem as long as the potential energy features a barrier or transition-state that has to be crossed. The escape-rate problem is known to be ill-defined when the particle is trapped in a potential well which is the only point of minimum in the potential profile (which then either diverges or reaches asymptotes along the coordinated axis), see a 'brandnew', Kramers rate theory of ionization and dissociation of bound states, Alessio Zaccone, Eugene M. Terentjev,