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

**Frank den Hollander**

*(Leiden University and EURANDOM)*

**Francesca Romana Nardi**

*(Technische Universiteit Eindhoven and EURANDOM)*

**Alessio Troiani**

*(Leiden University)*

#### 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.

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Pages: 1-26

Publication Date: January 1, 2012

DOI: 10.1214/EJP.v17-1693

#### 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). MR1423463 - van den Berg, M. Exit and return of a simple random walk.
*Potential Anal.*23 (2005), no. 1, 45--53. MR2136208 - Bovier, Anton. Metastability.
*Methods of contemporary mathematical statistical physics,*177--221, Lecture Notes in Math., 1970,*Springer, Berlin,*2009. MR2581606 - 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. MR1911735 - A. Bovier and F. den Hollander, phMetastability -- 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. MR2218873 - 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. MR2642889 - 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. MR1898856 - 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. MR1386350 - 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. MR2499857 - 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. MR2581607 - 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. MR1961341 - 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. MR1757966 - 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. MR2070109 - 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. MR1418410 - Nardi, F. R.; Olivieri, E.; Scoppola, E. Anisotropy effects in nucleation for conservative dynamics.
*J. Stat. Phys.*119 (2005), no. 3-4, 539--595. MR2149934 - 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. MR1101685 - Olivieri, Enzo; Scoppola, Elisabetta. An introduction to metastability through random walks.
*Braz. J. Probab. Stat.*24 (2010), no. 2, 361--399. MR2643571 - 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 MR2123364 - 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 MR1082348

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