April  2013, 33(4): 1431-1450. doi: 10.3934/dcds.2013.33.1431

Glauber dynamics in continuum: A constructive approach to evolution of states

1. 

Institute of Mathematics, National Academy of Sciences of Ukraine, 01601 Kiev-4, Ukraine

2. 

Fakultät für Mathematik, Universität Bielefeld, Postfach 110 131, 33501 Bielefeld, Germany

3. 

Instytut Matematyki, Uniwersytet Marii Curie-Skłodwskiej, 20-031 Lublin, Poland

Received  September 2011 Revised  November 2011 Published  October 2012

The evolutions of states is described corresponding to the Glauber dynamics of an infinite system of interacting particles in continuum. The description is conducted on both micro- and mesoscopic levels. The microscopic description is based on solving linear equations for correlation functions by means of an Ovsjannikov-type technique, which yields the evolution in a scale of Banach spaces. The mesoscopic description is performed by means of the Vlasov scaling, which yields a linear infinite chain of equations obtained from those for the correlation function. Its main peculiarity is that, for the initial correlation function of the inhomogeneous Poisson measure, the solution is the correlation function of such a measure with density which solves a nonlinear differential equation of convolution type.
Citation: Dmitri Finkelshtein, Yuri Kondratiev, Yuri Kozitsky. Glauber dynamics in continuum: A constructive approach to evolution of states. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1431-1450. doi: 10.3934/dcds.2013.33.1431
References:
[1]

J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications,'' Springer Monographs in Mathematics. Springer-Verlag Ltd., London, 2006.  Google Scholar

[2]

R. L. Dobrushin, Y. G. Sinai and Y. M. Sukhov, Dynamical systems of statistical mechanics, in "Itogi Nauki'', VINITI (1985), 235-284; engl. transl. in "Dynamical systems. II: Ergodic theory with applications to dynamical systems and statistical mechanics'' (ed. Ya. G. Sinai), Encyclopaedia Math. Sci., Springer, Berlin Heidelberg, 1989.  Google Scholar

[3]

N. L. Garcia and T. G. Kurtz, Spatial birth and death processes as solutions of stochastic equations, ALEA Lat. Am. J. Probab. Math. Stat., 1 (2006), 281-303.  Google Scholar

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I. M. Gel'fand and G. E. Shilov, "Generalized Functions. Vol. 3: Theory of Differential Equations,'' Transl. from the Russian by E. Meinhard, Mayer Academic Press, New York-London, 1967.  Google Scholar

[5]

D. Finkelshtein, Yu. Kondratiev and O. Kutovyi, Vlasov scaling for stochastic dynamics of continuous systems, J. Stat. Phys., 141 (2010), 158-178. doi: 10.1007/s10955-010-0038-1.  Google Scholar

[6]

D. Finkelshtein, Yu. Kondratiev and O. Kutovyi, Individual based model with competition in spatial ecology, SIAM J. Math. Anal., 41 (2009), 297-317. doi: 10.1137/080719376.  Google Scholar

[7]

D. Finkelshtein, Yu. Kondratiev and O. Kutovyi, Vlasov scaling for the Glauber dynamics in continuum, Infin. Dimens. Anal. Quantum Probab. Relat. Top, 14 (2011), 537-569. doi: 10.1142/S021902571100450X.  Google Scholar

[8]

D. Finkelshtein, Yu. Kondratiev, O. Kutovyi and E. Zhizhina, An approximative approach for construction of the Glauber dynamics in continuum, Math. Nachr., 285 (2012), 223-235. doi: 10.1002/mana.200910248.  Google Scholar

[9]

D. Finkelshtein, Yu. Kondratiev and E. Lytvynov, Equilibrium Glauber dynamics of continuous particle systems as a scaling limit of Kawasaki dynamics, Random Oper. Stochastic Equations, 15 (2007), 105-126. doi: 10.1515/rose.2007.007.  Google Scholar

[10]

D. Finkelshtein, Yu. Kondratiev and M. J. Oliveira, Markov evolution and hierarchical equations in the continuum. I: One-component systems, J. Evol. Equ., 9 (2009), 197-233. doi: 10.1007/s00028-009-0007-9.  Google Scholar

[11]

Yu. Kondratiev and T. Kuna, Harmonic analysis on configuration space. I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5 (2002), 201-233. doi: 10.1142/S0219025702000833.  Google Scholar

[12]

Yu. Kondratiev and O. Kutovyi, On the metrical properties of the configuration space, Math. Nachr., 279 (2006), 774-783. doi: 10.1002/mana.200310392.  Google Scholar

[13]

Yu. Kondratiev, O. Kutovyi and R. Minlos, On non-equilibrium stochastic dynamics for interacting particle systems in continuum, J. Funct. Anal., 255 (2008), 200-227. doi: 10.1016/j.jfa.2007.12.006.  Google Scholar

[14]

Yu. Kondratiev, O. Kutovyi and R. Minlos, Ergodicity of non-equilibrium Glauber dynamics in continuum, J. Funct. Anal., 258 (2010), 3097-3116. doi: 10.1016/j.jfa.2009.09.005.  Google Scholar

[15]

Yu. Kondratiev, O. Kutovyi and E. Zhizhina, Nonequilibrium Glauber-type dynamics in continuum, J. Math. Phys., 47 (2006), 17 pp. 113501. doi: 10.1063/1.2354589.  Google Scholar

[16]

D. Ruelle, "Statistical Mechanics: Rigorous Results,'' World Scientific, Singapore, 1999.  Google Scholar

[17]

H. R. Thieme and J. Voigt, Stochastic semigroups: Their construction by perturbation and approximation, in "Positivity IV--Theory and Applications," 135-146, Tech. Univ. Dresden, Dresden, 2006.  Google Scholar

[18]

F. Trèves, "Ovcyannikov Theorem and Hyperdifferential Operators,'' Notas de Matemática, No. 46 Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro, 1968.  Google Scholar

show all references

References:
[1]

J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications,'' Springer Monographs in Mathematics. Springer-Verlag Ltd., London, 2006.  Google Scholar

[2]

R. L. Dobrushin, Y. G. Sinai and Y. M. Sukhov, Dynamical systems of statistical mechanics, in "Itogi Nauki'', VINITI (1985), 235-284; engl. transl. in "Dynamical systems. II: Ergodic theory with applications to dynamical systems and statistical mechanics'' (ed. Ya. G. Sinai), Encyclopaedia Math. Sci., Springer, Berlin Heidelberg, 1989.  Google Scholar

[3]

N. L. Garcia and T. G. Kurtz, Spatial birth and death processes as solutions of stochastic equations, ALEA Lat. Am. J. Probab. Math. Stat., 1 (2006), 281-303.  Google Scholar

[4]

I. M. Gel'fand and G. E. Shilov, "Generalized Functions. Vol. 3: Theory of Differential Equations,'' Transl. from the Russian by E. Meinhard, Mayer Academic Press, New York-London, 1967.  Google Scholar

[5]

D. Finkelshtein, Yu. Kondratiev and O. Kutovyi, Vlasov scaling for stochastic dynamics of continuous systems, J. Stat. Phys., 141 (2010), 158-178. doi: 10.1007/s10955-010-0038-1.  Google Scholar

[6]

D. Finkelshtein, Yu. Kondratiev and O. Kutovyi, Individual based model with competition in spatial ecology, SIAM J. Math. Anal., 41 (2009), 297-317. doi: 10.1137/080719376.  Google Scholar

[7]

D. Finkelshtein, Yu. Kondratiev and O. Kutovyi, Vlasov scaling for the Glauber dynamics in continuum, Infin. Dimens. Anal. Quantum Probab. Relat. Top, 14 (2011), 537-569. doi: 10.1142/S021902571100450X.  Google Scholar

[8]

D. Finkelshtein, Yu. Kondratiev, O. Kutovyi and E. Zhizhina, An approximative approach for construction of the Glauber dynamics in continuum, Math. Nachr., 285 (2012), 223-235. doi: 10.1002/mana.200910248.  Google Scholar

[9]

D. Finkelshtein, Yu. Kondratiev and E. Lytvynov, Equilibrium Glauber dynamics of continuous particle systems as a scaling limit of Kawasaki dynamics, Random Oper. Stochastic Equations, 15 (2007), 105-126. doi: 10.1515/rose.2007.007.  Google Scholar

[10]

D. Finkelshtein, Yu. Kondratiev and M. J. Oliveira, Markov evolution and hierarchical equations in the continuum. I: One-component systems, J. Evol. Equ., 9 (2009), 197-233. doi: 10.1007/s00028-009-0007-9.  Google Scholar

[11]

Yu. Kondratiev and T. Kuna, Harmonic analysis on configuration space. I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5 (2002), 201-233. doi: 10.1142/S0219025702000833.  Google Scholar

[12]

Yu. Kondratiev and O. Kutovyi, On the metrical properties of the configuration space, Math. Nachr., 279 (2006), 774-783. doi: 10.1002/mana.200310392.  Google Scholar

[13]

Yu. Kondratiev, O. Kutovyi and R. Minlos, On non-equilibrium stochastic dynamics for interacting particle systems in continuum, J. Funct. Anal., 255 (2008), 200-227. doi: 10.1016/j.jfa.2007.12.006.  Google Scholar

[14]

Yu. Kondratiev, O. Kutovyi and R. Minlos, Ergodicity of non-equilibrium Glauber dynamics in continuum, J. Funct. Anal., 258 (2010), 3097-3116. doi: 10.1016/j.jfa.2009.09.005.  Google Scholar

[15]

Yu. Kondratiev, O. Kutovyi and E. Zhizhina, Nonequilibrium Glauber-type dynamics in continuum, J. Math. Phys., 47 (2006), 17 pp. 113501. doi: 10.1063/1.2354589.  Google Scholar

[16]

D. Ruelle, "Statistical Mechanics: Rigorous Results,'' World Scientific, Singapore, 1999.  Google Scholar

[17]

H. R. Thieme and J. Voigt, Stochastic semigroups: Their construction by perturbation and approximation, in "Positivity IV--Theory and Applications," 135-146, Tech. Univ. Dresden, Dresden, 2006.  Google Scholar

[18]

F. Trèves, "Ovcyannikov Theorem and Hyperdifferential Operators,'' Notas de Matemática, No. 46 Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro, 1968.  Google Scholar

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