# Stochastic Resonance

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Sometimes adding the right amount of noise can increase the signal-to-noise ratio of a signal; this is called *stochastic resonance*. In this Demonstration you can follow the time evolution of a particle inside a bistable potential with a low-amplitude, periodic forcing. Usually, that forcing is not enough to switch the particle from one stability basin to the other. But a small amount of random noise makes the particle oscillate between the two basins, following the periodic forcing.

Contributed by: Alejandro Luque Estepa (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Consider the following stochastic equation:

,

where stands for a Wiener process and represents the noise level.

Now consider potentials of the form , composed of a stationary part with two minima at and and a periodic forcing with amplitude and period . If is small enough, will oscillate around either or , without ever switching to the other.

But what happens if one increases the noise amplitude ? Then there is some probability that will jump from one basin to the other. If the noise level is just right, will follow the periodic forcing and oscillate between and with period . This is what we mean by stochastic resonance.

In more general terms, there is stochastic resonance whenever adding noise to a system improves its performance or, in the language of signal processing, increases its signal-to-noise ratio. Note that the noise amplitude cannot be too large or the system can become completely random.

Further reading:

A. Bulsara and L. Gammaitoni, "Tuning in to Noise," *Physics Today,* 49(3), 1996 pp. 39-45.
L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni,* *"Stochastic Resonance*,*"* Reviews of Modern Physics,* 70(1), 1998 pp. 223-287.
S. Herrmann and P. Imkeller, "Stochastic Resonance*,*"* *in the *Encyclopedia of Mathematical Physics,* Amsterdam: Elsevier, 2006.
F. Marchesoni,* *"Order out of Noise*,*"* Physics,* 2(23), 2009.

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