Abstract
We consider semigroups of operators for hierarchies of evolution equations of large particle systems, namely, of the dual BBGKY (Bogolyubov-Born-Green-Yovan) hierarchy for marginal observables and the BBGKY hierarchy for marginal distribution functions. We establish that the generating operators of the expansions for one-parameter families of operators of these hierarchies are the corresponding order cumulants (semi-invariants) of semigroups for the Liouville equations. We also apply constructed semigroups to the description of the kinetic evolution of interacting stochastic Markovian processes, modeling the microscopic evolution of soft active matter. For this purpose we consider the mean field asymptotic behavior of the semigroup generated by the dual BBGKY hierarchy for marginal observables. The constructed scaling limit is governed by the set of recurrence evolution equations, namely, by the Vlasov-type dual hierarchy. Moreover, the relationships of this hierarchy of evolution equations with the Vlasov-type kinetic equation with initial correlations are established.
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References
Banasiak, J., Arlotti, L.: Perturbations of Positive Semigroups with Applications. Springer-Verlag, London (2006)
Banasiak, J., Lachowicz, M.: Methods of Small Parameter in Mathematical Biology. Birkhäuser, Boston (2014)
Belleni-Morante, A., McBride, A.C.: Applied Nonlinear Semigroups: An Introduction. John Wiley and Sons, Inc., Chichester (1998)
Bellouquid A., Delitala M.: Mathematical Modeling of Complex Biological Systems: A Kinetic Theory Approach. Birkhäuser, Boston (2006)
Borgioli, G., Gerasimenko, V.I.: Initial-value problem of the quantum dual BBGKY hierarchy. Nuovo Cimento Soc. Ital. Fis. C. 33(1), 71–78 (2010)
Borovchenkova, M.S., Gerasimenko, V.I.: On the non-Markovian Enskog equation for granular gases. J. Phys. A: Math. Theor. 47(3), 035001 (2014)
Cercignani, C., Gerasimenko, V.I., Petrina, D.Ya.: Many-Particle Dynamics and Kinetic Equations. Kluwer Acad. Publ., Dordrecht (1997)
Gallagher I., Saint-Raymond L., Texier B. From Newton to Boltzmann: Hard Spheres and Short-range Potentials. EMS Publ. House: Zürich Lectures in Advanced Mathematics (2014)
Gerasimenko V.I.: Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinet. Relat. Models. 4(1), 385–399 (2011)
Gerasimenko, V.I.: Hierarchies of quantum evolution equations and dynamics of many-particle correlations. In: Statistical Mechanics and Random Walks: Principles, Processes and Applications.Nova Science Publ., Inc., N.Y., 233–288 (2012)
Gerasimenko, V.I.: On the approaches to the derivation of the Boltzmann equation with hardsphere collisions. Proc. Inst. Math. NASU. 10(2), 71–95 (2013)
Gerasimenko, V.I., Fedchun, Yu.Yu.: Nonperturbative solution expansions of hierarchies of evolution equations in functional derivatives. Proc. Inst. Math. NASU. 9(2), 347–375 (2012)
Gerasimenko, V.I., Fedchun, Yu.Yu.: On kinetic models for the evolution of many-entity systems in mathematical biology. J. Coupled Syst. Multiscale Dyn. 1(2), 273–279 (2013)
Gerasimenko, V.I., Polishchuk, D.O.: A nonperturbative solution of the nonlinear BBGKY hierarchy for marginal correlation operators. Math. Methods Appl. Sci. 36(17), 2311–2328 (2013)
Gerasimenko, V.I., Tsvir, Zh.A.: On quantum kinetic equations of many-particle systemsin condensed states. Physica A: Stat. Mech. Appl. 391(24), 6362–366 (2012)
Lachowicz, M.: Individually-based Markov processes modeling nonlinear systems in mathematical biology. Nonlinear Analysis: Real World Applications. 12, 2396–2408 (2011)
Marchetti M.C., Joanny J.F., Ramaswamy S., Liverpool T.B., Prost J., Rao M., Simha R.A.: Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143–1194 (2013)
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Gerasimenko, V.I., Fedchun, Y.Y. (2015). On Semigroups of Large Particle Systems and Their Scaling Asymptotic Behavior. In: Banasiak, J., Bobrowski, A., Lachowicz, M. (eds) Semigroups of Operators -Theory and Applications. Springer Proceedings in Mathematics & Statistics, vol 113. Springer, Cham. https://doi.org/10.1007/978-3-319-12145-1_10
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DOI: https://doi.org/10.1007/978-3-319-12145-1_10
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