of a Brownian particle in a periodically modulated asymmetric double-well potential. In the deterministic case
, the following behaviour is observed: |
For small amplitudes  , the particle remains in the
same potential well at all times, and the area enclosed by its path (over one period) is of order  . |
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For larger amplitudes, the particle switches wells twice per period. Thus, the enclosed area is of order one. |
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The small-amplitude regime: If the
amplitude of the modulation is too small to allow for transitions in
the absence of noise and the noise intensity is below threshold, then
the hysteresis area is typically of order . |
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The large-amplitude regime: If the amplitude of the modulation is large enough to allow for transitions (even in the absence of noise) and the noise intensity is below threshold, then the hysteresis area is typically close to the deterministic value which is of order one. |
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The large-noise regime: If the noise
intensity is above threshold, then the hysteresis area is typically of
order one. The remarkable fact is that this typical value is smaller
than the static hysteresis area and does not depend on the
amplitude or the speed of modulation . |
 | ||
| Deterministic and random hysteresis cycles in (a) the small-amplitude regime, (b) the large-amplitude regime, (c) the large-noise regime. | ||
The threshold value for the noise intensity shows a power-law
dependence on the speed and the amplitude of the modulation. We study the scaling of the typical
hysteresis area as a function of and (for noise intensities below threshold) and with 
(for noise intensities above threshold). The
probability of atypical behaviour is found to be exponentially small.
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Nils Berglund and Barbara Gentz The effect of additive noise on dynamical hysteresis Nonlinearity 15, 605–632 (2002) |
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Nils Berglund and Barbara Gentz Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems J. Phys. A: Math. Gen. 35, 2057–2091 (2002) |
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Nils Berglund and Barbara Gentz Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations Stoch. Dyn. 2, 327–356 (2002) |
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Nils Berglund and Barbara Gentz Noise-Induced Phenomena in Slow-Fast Dynamical Systems. A Sample-Paths Approach Probability and its Applications, Springer-Verlag (2005) |
Last modified: 29 October 2006, Barbara Gentz