Three Lectures on Metastability Under Stochastic Dynamics

Hollander, Frank den

doi:10.1007/978-3-540-92796-9_5

Cited by 14 publications

(14 citation statements)

References 27 publications

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“…For an overview we refer the reader to two recent papers presented at the 12th Brazilian School of Probability: Gaudillière and Scoppola [14] and Gaudillière [15]. For an overview on metastability and droplet growth in a broader context, we refer the reader to the monograph by Olivieri and Vares [26], and the review papers by Bovier [3], [4], den Hollander [16], Olivieri and Scoppola [25].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Metastability for Kawasaki dynamics at low temperature with two types of particles

Hollander

Nardi

Troiani

2012

Electron. J. Probab.

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This is the first in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between neighboring particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and compute the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box.We identify the region of parameters for which the system is metastable. For this region, in the limit as the temperature tends to zero, we show that the first entrance distribution on the set of critical droplets is uniform, compute the expected transition time up to a multiplicative factor that tends to one, and prove that the transition time divided by its expectation is exponentially distributed. These results are derived under three hypotheses on the energy landscape, which are verified in the second and the third paper for a certain subregion of the metastable region. These hypotheses involve three model-dependent quantities -the energy, the shape and the number of the critical droplets -which are identified in the second and the third paper as well.MSC2010. 60K35, 82C26.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Due to the one-dimensional nature of the set ω, the variational problem in the right-hand side can be solved explicitly by elementary computations. One finds that the minimum equals 16) and is uniquely attained at h given by…”

Section: )mentioning

confidence: 99%

Metastability for Kawasaki dynamics at low temperature with two types of particles

Hollander

Nardi

Troiani

2012

Electron. J. Probab.

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“…Only sufficiently strong disturbances will put the system out of the metastable state and the system will pass into another state of equilibrium. Examples of metastability [3] can be found in biology, climatology, economics, or physics. Accordingly, an athlete working out on a stability device to improve his balance skills would exercise in a state of metastable equilibrium.…”

Section: Introductionmentioning

confidence: 99%

Metastability in plyometric training on unstable surfaces: a pilot study

Kibele

Classen

Muehlbauer

et al. 2014

BMC Sports Sci Med Rehabil

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BackgroundIn the past, plyometric training (PT) has been predominantly performed on stable surfaces. The purpose of this pilot study was to examine effects of a 7-week lower body PT on stable vs. unstable surfaces. This type of exercise condition may be denoted as metastable equilibrium.MethodsThirty-three physically active male sport science students (age: 24.1 ± 3.8 years) were randomly assigned to a PT group (n = 13) exercising on stable (STAB) and a PT group (n = 20) on unstable surfaces (INST). Both groups trained countermovement jumps, drop jumps, and practiced a hurdle jump course. In addition, high bar squats were performed. Physical fitness tests on stable surfaces (hexagonal obstacle test, countermovement jump, hurdle drop jump, left-right hop, dynamic and static balance tests, and leg extension strength) were used to examine the training effects.ResultsSignificant main effects of time (ANOVA) were found for the countermovement jump, hurdle drop jump, hexagonal test, dynamic balance, and leg extension strength. A significant interaction of time and training mode was detected for the countermovement jump in favor of the INST group. No significant improvements were evident for either group in the left-right hop and in the static balance test.ConclusionsThese results show that lower body PT on unstable surfaces is a safe and efficient way to improve physical performance on stable surfaces.

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“…Describing the evolution among competing metastable states or from a metastable state to a stable state in stochastic lattice spin systems at low temperatures is still a subject of considerable interest. We refer to [10,22,4,11] for recent monographs on the subject.…”

Section: Introductionmentioning

confidence: 99%