Electronic Journal of Probability

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

Frank den Hollander, Francesca Nardi, and Alessio Troiani

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Abstract

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 atlow temperature in a large finite box with an open boundary. Each pair of particlesoccupying neighboring sites has a negative binding energy provided their types aredifferent, while each particle has a positive activation energy that depends onits 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 thatrepresents 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 thisregion, in the limit as the temperature tends to zero, we show that the firstentrance distribution on the set of critical droplets is uniform, compute theexpected 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.

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 2, 26 pp.

Dates
Accepted: 1 January 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062324

Digital Object Identifier
doi:10.1214/EJP.v17-1693

Mathematical Reviews number (MathSciNet)
MR2869249

Zentralblatt MATH identifier
1246.60119

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C26: Dynamic and nonequilibrium phase transitions (general)

Keywords
Multi-type particle systems Kawasaki dynamics metastable region metastable transition time critical droplet potential theory Dirichlet form capacity

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

den Hollander, Frank; Nardi, Francesca; Troiani, Alessio. Metastability for Kawasaki dynamics at low temperature with two types of particles. Electron. J. Probab. 17 (2012), paper no. 2, 26 pp. doi:10.1214/EJP.v17-1693. https://projecteuclid.org/euclid.ejp/1465062324


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References

  • Ben Arous, Gérard; Cerf, Raphaël. Metastability of the three-dimensional Ising model on a torus at very low temperatures. Electron. J. Probab. 1 (1996), no. 10, approx. 55 pp. (electronic).
  • van den Berg, M. Exit and return of a simple random walk. Potential Anal. 23 (2005), no. 1, 45–53.
  • Bovier, Anton. Metastability. Methods of contemporary mathematical statistical physics, 177–221, Lecture Notes in Math., 1970, Springer, Berlin, 2009.
  • A. Bovier, Metastability: from mean field models to spdes, to appear in a Festschrift on the occassion of the 60-th birthday of Jürgen Gärtner and the 65-th birthday of Erwin Bolthausen, Springer Proceedings in Mathematics.
  • Bovier, Anton; Eckhoff, Michael; Gayrard, Véronique; Klein, Markus. Metastability and low lying spectra in reversible Markov chains. Comm. Math. Phys. 228 (2002), no. 2, 219–255.
  • A. Bovier and F. den Hollander, Metastability – A Potential-Theoretic Approach, manuscript in preparation.
  • Bovier, A.; den Hollander, F.; Nardi, F. R. Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary. Probab. Theory Related Fields 135 (2006), no. 2, 265–310.
  • Bovier, Anton; den Hollander, Frank; Spitoni, Cristian. Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures. Ann. Probab. 38 (2010), no. 2, 661–713.
  • Bovier, Anton; Manzo, Francesco. Metastability in Glauber dynamics in the low-temperature limit: beyond exponential asymptotics. J. Statist. Phys. 107 (2002), no. 3-4, 757–779.
  • Cirillo, Emilio N. M.; Olivieri, Enzo. Metastability and nucleation for the Blume-Capel model. Different mechanisms of transition. J. Statist. Phys. 83 (1996), no. 3-4, 473–554.
  • Gaudillière, A.; den Hollander, F.; Nardi, F. R.; Olivieri, E.; Scoppola, E. Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics. Stochastic Process. Appl. 119 (2009), no. 3, 737–774.
  • A. Gaudillière, F. den Hollander, F.R. Nardi, E. Olivieri and E. Scoppola, Droplet dynamics in a two-dimensional rarified gas under Kawasaki dynamics, manuscript in preparation.
  • A. Gaudillière, F. den Hollander, F.R. Nardi, E. Olivieri and E. Scoppola, Homogeneous nucleation for two-dimensional Kawasaki dynamics, manuscript in preparation.
  • A. Gaudillière, E. Scoppola, An introduction to metastability, lecture notes for the 12th Brazilian School of Probability (pdf-file).
  • A. Gaudillière, Condensers physics applied to Markov chains, Lecture notes for the 12th Brazilian School of Probability, arXiv:0901.3053v1
  • den Hollander, Frank. Three lectures on metastability under stochastic dynamics. Methods of contemporary mathematical statistical physics, 223–246, Lecture Notes in Math., 1970, Springer, Berlin, 2009.
  • den Hollander, F.; Nardi, F. R.; Olivieri, E.; Scoppola, E. Droplet growth for three-dimensional Kawasaki dynamics. Probab. Theory Related Fields 125 (2003), no. 2, 153–194.
  • den Hollander, F.; Olivieri, E.; Scoppola, E. Metastability and nucleation for conservative dynamics. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41 (2000), no. 3, 1424–1498.
  • F. den Hollander, F.R. Nardi and A. Troiani, Kawasaki dynamics with two types of particles: stable/metastable configurations and communication heights, J. Stat. Phys. 145 (2011).
  • F. den Hollander, F.R. Nardi and A. Troiani, Kawasaki dynamics with two types of particles: critical droplets, manuscript in preparation.
  • Manzo, F.; Nardi, F. R.; Olivieri, E.; Scoppola, E. On the essential features of metastability: tunnelling time and critical configurations. J. Statist. Phys. 115 (2004), no. 1-2, 591–642.
  • Nardi, F. R.; Olivieri, E. Low temperature stochastic dynamics for an Ising model with alternating field. Disordered systems and statistical physics: rigorous results (Budapest, 1995). Markov Process. Related Fields 2 (1996), no. 1, 117–166.
  • Nardi, F. R.; Olivieri, E.; Scoppola, E. Anisotropy effects in nucleation for conservative dynamics. J. Stat. Phys. 119 (2005), no. 3-4, 539–595.
  • Neves, E. Jordão; Schonmann, Roberto H. Critical droplets and metastability for a Glauber dynamics at very low temperatures. Comm. Math. Phys. 137 (1991), no. 2, 209–230.
  • Olivieri, Enzo; Scoppola, Elisabetta. An introduction to metastability through random walks. Braz. J. Probab. Stat. 24 (2010), no. 2, 361–399.
  • Olivieri, Enzo; Vares, Maria Eulália. Large deviations and metastability. Encyclopedia of Mathematics and its Applications, 100. Cambridge University Press, Cambridge, 2005. xvi+512 pp. ISBN: 0-521-59163-5
  • Révész, Pál. Random walk in random and nonrandom environments. World Scientific Publishing Co., Inc., Teaneck, NJ, 1990. xiv+332 pp. ISBN: 981-02-0237-7