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
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show all references

References:
[1]

Springer Monographs in Mathematics. Springer-Verlag Ltd., London, 2006.  Google Scholar

[2]

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]

ALEA Lat. Am. J. Probab. Math. Stat., 1 (2006), 281-303.  Google Scholar

[4]

Transl. from the Russian by E. Meinhard, Mayer Academic Press, New York-London, 1967.  Google Scholar

[5]

J. Stat. Phys., 141 (2010), 158-178. doi: 10.1007/s10955-010-0038-1.  Google Scholar

[6]

SIAM J. Math. Anal., 41 (2009), 297-317. doi: 10.1137/080719376.  Google Scholar

[7]

Infin. Dimens. Anal. Quantum Probab. Relat. Top, 14 (2011), 537-569. doi: 10.1142/S021902571100450X.  Google Scholar

[8]

Math. Nachr., 285 (2012), 223-235. doi: 10.1002/mana.200910248.  Google Scholar

[9]

Random Oper. Stochastic Equations, 15 (2007), 105-126. doi: 10.1515/rose.2007.007.  Google Scholar

[10]

J. Evol. Equ., 9 (2009), 197-233. doi: 10.1007/s00028-009-0007-9.  Google Scholar

[11]

Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5 (2002), 201-233. doi: 10.1142/S0219025702000833.  Google Scholar

[12]

Math. Nachr., 279 (2006), 774-783. doi: 10.1002/mana.200310392.  Google Scholar

[13]

J. Funct. Anal., 255 (2008), 200-227. doi: 10.1016/j.jfa.2007.12.006.  Google Scholar

[14]

J. Funct. Anal., 258 (2010), 3097-3116. doi: 10.1016/j.jfa.2009.09.005.  Google Scholar

[15]

J. Math. Phys., 47 (2006), 17 pp. 113501. doi: 10.1063/1.2354589.  Google Scholar

[16]

World Scientific, Singapore, 1999.  Google Scholar

[17]

in "Positivity IV--Theory and Applications," 135-146, Tech. Univ. Dresden, Dresden, 2006.  Google Scholar

[18]

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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