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2019 | OriginalPaper | Buchkapitel

Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics

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Abstract

The Widom-Rowlinson model is an equilibrium model for point particles in Euclidean space. It has a repulsive interaction between particles of different colors, and shows a phase transition at high intensity. Natural versions of the model can moreover be formulated in different geometries: in particular as a lattice system or a mean-field system. We will discuss recent results on dynamical Gibbs-non Gibbs transitions in this context. Main issues will be the possibility or impossibility of an immediate loss of the Gibbs property, and of full-measure discontinuities of the time-evolved models.

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Literatur
1.
Zurück zum Zitat Arguin, L.-P., Damron, M., Newman, C., Stein, D.: Uniqueness of ground states for short-range spin glasses in the half-plane. Commun. Math. Phys. 300, 641–657 (2010)MathSciNetCrossRef Arguin, L.-P., Damron, M., Newman, C., Stein, D.: Uniqueness of ground states for short-range spin glasses in the half-plane. Commun. Math. Phys. 300, 641–657 (2010)MathSciNetCrossRef
2.
Zurück zum Zitat Bovier, A.: Statistical Mechanics of Disordered Systems: A Mathematical Perspective, vol. 18. Cambridge University Press (2006) Bovier, A.: Statistical Mechanics of Disordered Systems: A Mathematical Perspective, vol. 18. Cambridge University Press (2006)
3.
Zurück zum Zitat Bricmont, J., Kuroda, K., Lebowitz, J.L.: The structure of Gibbs states and phase coexistence for nonsymmetric continuum Widom–Rowlinson models. Z. Wahrsch. Verw. Gebiete 67, 121–138 (1984) Bricmont, J., Kuroda, K., Lebowitz, J.L.: The structure of Gibbs states and phase coexistence for nonsymmetric continuum Widom–Rowlinson models. Z. Wahrsch. Verw. Gebiete 67, 121–138 (1984)
4.
Zurück zum Zitat Chayes, J.T., Chayes, L., Kotecký, R.: The analysis of the Widom-Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551–569 (1995)MathSciNetCrossRef Chayes, J.T., Chayes, L., Kotecký, R.: The analysis of the Widom-Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551–569 (1995)MathSciNetCrossRef
5.
Zurück zum Zitat Cotar, C., Jahnel, B., Külske, C.: Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures. Electron. Commun. Probab. 23, 1–12 (2018)MathSciNetCrossRef Cotar, C., Jahnel, B., Külske, C.: Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures. Electron. Commun. Probab. 23, 1–12 (2018)MathSciNetCrossRef
6.
Zurück zum Zitat Dereudre, D., Drouilet, R., Georgii, H.-O.: Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Relat. Fields 153, 643–670 (2012)MathSciNetCrossRef Dereudre, D., Drouilet, R., Georgii, H.-O.: Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Relat. Fields 153, 643–670 (2012)MathSciNetCrossRef
7.
Zurück zum Zitat van Enter, A.C.D., Ermolaev, V.N., Iacobelli, G., Külske, C.: Gibbs-non-Gibbs properties for evolving Ising models on trees. Ann. Inst. Henri Poincare Probab. Stat. 48, 774–791 (2012)MathSciNetCrossRef van Enter, A.C.D., Ermolaev, V.N., Iacobelli, G., Külske, C.: Gibbs-non-Gibbs properties for evolving Ising models on trees. Ann. Inst. Henri Poincare Probab. Stat. 48, 774–791 (2012)MathSciNetCrossRef
8.
Zurück zum Zitat van Enter, A.C.D., Fernández, R., den Hollander, F., Redig, F.: Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures. Commun. Math. Phys. 226, 101–130 (2002)MathSciNetCrossRef van Enter, A.C.D., Fernández, R., den Hollander, F., Redig, F.: Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures. Commun. Math. Phys. 226, 101–130 (2002)MathSciNetCrossRef
9.
Zurück zum Zitat van Enter, A.C.D., Fernández, R., den Hollander, F., Redig, F.: A large-deviation view on dynamical Gibbs-non-Gibbs transitions. Moscow Math. J. 10, 687–711 (2010)MathSciNetCrossRef van Enter, A.C.D., Fernández, R., den Hollander, F., Redig, F.: A large-deviation view on dynamical Gibbs-non-Gibbs transitions. Moscow Math. J. 10, 687–711 (2010)MathSciNetCrossRef
10.
Zurück zum Zitat van Enter, A.C.D., Fernández, R., Sokal, A.D.: Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72, 879–1167 (1993)MathSciNetCrossRef van Enter, A.C.D., Fernández, R., Sokal, A.D.: Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72, 879–1167 (1993)MathSciNetCrossRef
11.
Zurück zum Zitat Enter, A.C.D., Külske, C., Opoku, A.A., Ruszel, W.M.: Gibbs-non-Gibbs properties for n-vector lattice and mean-field models. Braz. J. Probab. Stat. 24, 226–255 (2010)MathSciNetCrossRef Enter, A.C.D., Külske, C., Opoku, A.A., Ruszel, W.M.: Gibbs-non-Gibbs properties for n-vector lattice and mean-field models. Braz. J. Probab. Stat. 24, 226–255 (2010)MathSciNetCrossRef
12.
Zurück zum Zitat Ermolaev, V.N., Külske, C.: Low-temperature dynamics of the Curie-Weiss model: periodic orbits, multiple histories and loss of Gibbsianness. J. Stat. Phys. 141, 727–756 (2010)MathSciNetCrossRef Ermolaev, V.N., Külske, C.: Low-temperature dynamics of the Curie-Weiss model: periodic orbits, multiple histories and loss of Gibbsianness. J. Stat. Phys. 141, 727–756 (2010)MathSciNetCrossRef
13.
Zurück zum Zitat Fernández, R., den Hollander, F., Martínez, J.: Variational description of Gibbs-non-Gibbs dynamical transitions for spin-flip systems with a Kac-type interaction. J. Stat. Phys. 147, 1094–1112 (2014)MathSciNetMATH Fernández, R., den Hollander, F., Martínez, J.: Variational description of Gibbs-non-Gibbs dynamical transitions for spin-flip systems with a Kac-type interaction. J. Stat. Phys. 147, 1094–1112 (2014)MathSciNetMATH
14.
Zurück zum Zitat Gallavotti, G., Lebowitz, J.: Phase transitions in binary lattice gases. J. Math. Phys. 12, 1129–1133 (1971)CrossRef Gallavotti, G., Lebowitz, J.: Phase transitions in binary lattice gases. J. Math. Phys. 12, 1129–1133 (1971)CrossRef
15.
Zurück zum Zitat Georgii, H.-O.: Gibbs Measures and Phase Transitions. De Gruyter, New York (2011) Georgii, H.-O.: Gibbs Measures and Phase Transitions. De Gruyter, New York (2011)
16.
Zurück zum Zitat Henning, F., Kraaij, R., Külske, C.: Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction: closing the Ising gap. Bernoulli J. 25, 2051–2074 (2019)MathSciNetCrossRef Henning, F., Kraaij, R., Külske, C.: Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction: closing the Ising gap. Bernoulli J. 25, 2051–2074 (2019)MathSciNetCrossRef
17.
Zurück zum Zitat Higuchi, Y., Takei, M.: Some results on the phase structure of the two-dimensional Widom-Rowlinson model. Osaka J. Math. 41, 237–255 (2004)MathSciNetMATH Higuchi, Y., Takei, M.: Some results on the phase structure of the two-dimensional Widom-Rowlinson model. Osaka J. Math. 41, 237–255 (2004)MathSciNetMATH
18.
Zurück zum Zitat den Hollander, F., Redig, F., van Zuijlen, W.: Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions. Stoch. Process. Appl. 125, 371–400 (2015)MathSciNetCrossRef den Hollander, F., Redig, F., van Zuijlen, W.: Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions. Stoch. Process. Appl. 125, 371–400 (2015)MathSciNetCrossRef
19.
Zurück zum Zitat Jahnel, B., Külske, C.: Attractor properties for irreversible and reversible interacting particle systems. Commun. Math. Phys. 366, 139–172 (2015)MathSciNetCrossRef Jahnel, B., Külske, C.: Attractor properties for irreversible and reversible interacting particle systems. Commun. Math. Phys. 366, 139–172 (2015)MathSciNetCrossRef
20.
Zurück zum Zitat Jahnel, B., Külske, C.: The Widom-Rowlinson model under spin flip: immediate loss and sharp recovery of quasilocality. Ann. Appl. Probab. 27, 3845–3892 (2017)MathSciNetCrossRef Jahnel, B., Külske, C.: The Widom-Rowlinson model under spin flip: immediate loss and sharp recovery of quasilocality. Ann. Appl. Probab. 27, 3845–3892 (2017)MathSciNetCrossRef
21.
22.
Zurück zum Zitat Kissel, S., Külske, C.: Dynamical Gibbs-non Gibbs transitions in Curie-Weiss Widom-Rowlinson models. Markov Process. Relat. Fields 25, 379–413 (2019) Kissel, S., Külske, C.: Dynamical Gibbs-non Gibbs transitions in Curie-Weiss Widom-Rowlinson models. Markov Process. Relat. Fields 25, 379–413 (2019)
23.
Zurück zum Zitat Kissel, S., Külske, C., Rozikov, U.: Hard-core and soft-core Widom-Rowlinson models on Cayley trees. J. Stat. Mech. 4(043204), 22 (2019) Kissel, S., Külske, C., Rozikov, U.: Hard-core and soft-core Widom-Rowlinson models on Cayley trees. J. Stat. Mech. 4(043204), 22 (2019)
24.
Zurück zum Zitat Kissel, S., Külske, C.: Dynamical Gibbs-non Gibbs transitions for hard-core and soft-core Widom-Rowlinson models on the lattice (in preparation) Kissel, S., Külske, C.: Dynamical Gibbs-non Gibbs transitions for hard-core and soft-core Widom-Rowlinson models on the lattice (in preparation)
25.
Zurück zum Zitat Kozlov, O.K.: A Gibbs description of a system of random variables. Problemy Peredaci Informacii 10, 94–103 (1974)MathSciNetMATH Kozlov, O.K.: A Gibbs description of a system of random variables. Problemy Peredaci Informacii 10, 94–103 (1974)MathSciNetMATH
26.
Zurück zum Zitat Kraaij, R., Redig, F., van Zuijlen, W.: A Hamilton-Jacobi point of view on mean-field Gibbs-non Gibbs transitions (2017). arXiv:1711.03489 Kraaij, R., Redig, F., van Zuijlen, W.: A Hamilton-Jacobi point of view on mean-field Gibbs-non Gibbs transitions (2017). arXiv:​1711.​03489
27.
Zurück zum Zitat Külske, C.: Metastates in disordered mean-field models: random field and Hopfield models. J. Stat. Phys. 88, 1257–1293 (1997)MathSciNetCrossRef Külske, C.: Metastates in disordered mean-field models: random field and Hopfield models. J. Stat. Phys. 88, 1257–1293 (1997)MathSciNetCrossRef
28.
Zurück zum Zitat Külske, C., Le Ny, A.: Spin-flip dynamics of the Curie-Weiss model: loss of Gibbsianness with possibly broken symmetry. Commun. Math. Phys. 271, 431–454 (2007)MathSciNetCrossRef Külske, C., Le Ny, A.: Spin-flip dynamics of the Curie-Weiss model: loss of Gibbsianness with possibly broken symmetry. Commun. Math. Phys. 271, 431–454 (2007)MathSciNetCrossRef
29.
Zurück zum Zitat Külske, C., Opoku, A.A.: The posterior metric and the goodness of Gibbsianness for transforms of Gibbs measures. Electron. J. Probab. 13, 1307–1344 (2008)MathSciNetCrossRef Külske, C., Opoku, A.A.: The posterior metric and the goodness of Gibbsianness for transforms of Gibbs measures. Electron. J. Probab. 13, 1307–1344 (2008)MathSciNetCrossRef
30.
Zurück zum Zitat Külske, C., Redig, F.: Loss without recovery of Gibbsianness during diffusion of continuous spins. Probab. Theory Relat. Fields 135, 428–456 (2006)MathSciNetCrossRef Külske, C., Redig, F.: Loss without recovery of Gibbsianness during diffusion of continuous spins. Probab. Theory Relat. Fields 135, 428–456 (2006)MathSciNetCrossRef
31.
Zurück zum Zitat Liggett, T.: Interacting Particle Systems. Springer-Verlag, New York (1985)CrossRef Liggett, T.: Interacting Particle Systems. Springer-Verlag, New York (1985)CrossRef
32.
33.
Zurück zum Zitat Mazel, A., Stuhl, I., Suhov, Y.: A classical WR model with q particle types. J. Stat. Phys. 159, 1040–1086 (2015)MathSciNetCrossRef Mazel, A., Stuhl, I., Suhov, Y.: A classical WR model with q particle types. J. Stat. Phys. 159, 1040–1086 (2015)MathSciNetCrossRef
34.
Zurück zum Zitat Newman, C.M., Stein, D.: Spin Glasses and Complexity. Princeton University Press (2013) Newman, C.M., Stein, D.: Spin Glasses and Complexity. Princeton University Press (2013)
35.
Zurück zum Zitat Rozikov, U.A.: Gibbs Measures on Cayley Trees. World Sci. Publ, Singapore (2013) Rozikov, U.A.: Gibbs Measures on Cayley Trees. World Sci. Publ, Singapore (2013)
36.
Zurück zum Zitat Ruelle, D.: Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27, 1040–1041 (1971)CrossRef Ruelle, D.: Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27, 1040–1041 (1971)CrossRef
37.
Zurück zum Zitat Ruelle, D.: Statistical Mechanics: Rigorous Results. World Scientific, River Edge, NJ (1999) Ruelle, D.: Statistical Mechanics: Rigorous Results. World Scientific, River Edge, NJ (1999)
38.
39.
Zurück zum Zitat Widom, B., Rowlinson, J.S.: New model for the study of liquid-vapor phase transition. J. Chem. Phys. 52, 1670–1684 (1970)CrossRef Widom, B., Rowlinson, J.S.: New model for the study of liquid-vapor phase transition. J. Chem. Phys. 52, 1670–1684 (1970)CrossRef
Metadaten
Titel
Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics
verfasst von
Christof Külske
Copyright-Jahr
2019
DOI
https://doi.org/10.1007/978-3-030-29077-1_1