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| Titel |
Rare transitions between metastable states in the stochastic Chaffee–Infante equation. |
| VerfasserIn |
Joran Rolland, Freddy Bouchet, Eric Simonnet |
| Konferenz |
EGU General Assembly 2015
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 17 (2015) |
| Datensatznummer |
250113974
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| Publikation (Nr.) |
 EGU/EGU2015-14223.pdf |
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| Zusammenfassung |
| We present a numerical and theoretical study of the transitions in the Stochastic one
dimensional Chaffee–Infante equation. The one dimensional Chaffee–Infante equation, also
know as the Ginzburg–Landau or Allen–Cahn equation in physics, is the prototype equation
for bistability in extended systems. As such, it is the perfect model equation for the test of
numerical or theoretical methods intended at investigating metastability in more complex
stochastic partial differential equations ; typically those arising in oceanicl fluid dynamics.
Among other examples, one can think of the alternance of meander paths of the
Kuroshio current near Japan, or the switching of the thermohaline circulation in the
north Atlantic ocean. The reactive trajectories, the realisations of the dynamics that
actually evolve from one metastable state to the other, are the central events in such
studies.
The novelty and originality of our approach is the combination of theoretical approaches
with a novel numerical method, Adaptive Multilevel Splitting (AMS), for the computation of
the full distribution of reactive trajectories and all the properties of the rare transitions. AMS
is a mutation selection/selection algorithm that uses N clones dynamics of the system of
interest, and only requires N|ln(α)| iterations. Meanwhile several 1/α realisations are
required for a direct numerical simulation (with α the probability of observing a transition). It
thus becomes a very powerful method when the noise amplitude and therefore α goes to
zero.
We used the algorithm to compute the properties (escape probability, mean first passage
time, average duration of reactive trajectories, number of fronts etc.) of the transition in the
full parameter space (L,β) (with L the size of the system and β the inverse of the noise
amplitude).
There is an excelent quantitative agreement with the various theoretical approaches of the
study of metastability. All of them are asymptotic and therefore concern only specific
sections of the (L,β) plane. In the low noise limit, the reactive trajectories can be
deduced from the Freidlin–Wentzel principle of large deviations (that yields the
instanton trajectories), while the mean first passage times are derived using the
Eyring–Kramers theory. However, these approaches require that the saddles between the
metastable states are not to flat, i.e. that the size of the system is not too large. In the
large L limit, we are still able to calculate the mean first passage times using an
explicit description of the system. In the large noise limit, where the instantons are no
longer the most probable trajectories, we can predict the number of fronts of the
trajectories.
The combined numerical and theoretical approaches have proven their efficiency in this
system and are therefore very promising tools for the study of more complex systems. Indeed,
the simplest space-dependent models of the thermohaline circulation do have a relatively
simple phase space. However, this is not necessarily the case of the shallow-water or
quasi-geostrophic models which are the simplest way of describing currents like the
Kuroshio. |
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