Geodesic
The asymptotic limit of an integro-differential equation modelling complex systems
Izvestiya. Mathematics , Tome 78 (2014) no. 6, pp. 1105-1119.

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This paper is devoted to the asymptotic analysis of a mathematical framework that has recently been proposed for modelling complex systems in the applied sciences under the action of an external force field. This framework consists in an integro-differential kinetic equation coupled with a Gaussian isokinetic thermostat. The asymptotic limit obtained here using low-field scaling shows the emergence of diffusive behaviour on a macroscopic scale.
Keywords: integro-differential equation, low-field limit, velocity-jump process, kinetic theory.
Mots-clés : active particles 
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C. Bianca; M. Ferrara; L. Guerrini. The asymptotic limit of an integro-differential equation modelling complex systems. Izvestiya. Mathematics , Tome 78 (2014) no. 6, pp. 1105-1119. http://geodesic.mathdoc.fr/item/IM2_2014_78_6_a3/

[1] C. Bianca, “Kinetic theory for active particles modelling coupled to Gaussian thermostats”, Appl. Math. Sci. (Ruse), 6:13-16 (2012), 651–660 | MR | Zbl

[2] C. Bianca, “An existence and uniqueness theorem to the Cauchy problem for thermostatted-KTAP models”, Int. J. Math. Anal. (Ruse), 6:17-20 (2012), 813–824 | MR | Zbl

[3] C. Bianca, “Onset of nonlinearity in thermostatted active particles models for complex systems”, Nonlinear Anal. Real World Appl., 13:6 (2012), 2593–2608 | DOI | MR | Zbl

[4] C. Bianca, M. Ferrara, L. Guerrini, “High-order moments conservation in thermostatted kinetic models”, J. Global Optim., 58:2 (2014), 389–404 | DOI | MR

[5] C. Bianca, “Thermostatted kinetic equations as models for complex systems in physics and life sciences”, Phys. Life Rev., 9:4 (2012), 359–399 | DOI

[6] C. Bianca, “Thermostatted models – Multiscale analysis and tuning with real-world systems data”, Phys. Life Rev., 9:4 (2012), 418–425 | DOI

[7] N. Bellomo, C. Bianca, M. Delitala, “Complexity analysis and mathematical tools towards the modelling of living systems”, Phys. Life Rev., 6:3 (2009), 144–175 | DOI

[8] C. Bianca, “Existence of stationary solutions in kinetic models with Gaussian thermostats”, Math. Methods Appl. Sci., 36:13 (2013), 1768–1775 | DOI | MR | Zbl

[9] B. Derrida, “Non-equilibrium steady states: fluctuations and large deviations of the density and of the current”, J. Stat. Mech. Theory Exp., 2007, no. 7, P07023, 45 pp. (electronic) | MR

[10] A. Dickson, A. R. Dinner, “Enhanced sampling of nonequilibrium steady states”, Annu. Rev. Phys. Chem., 61 (2010), 441–459 | DOI

[11] J.-P. Eckmann, C.-A. Pillet, L. Rey-Bellet, “Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures”, Comm. Math. Phys., 201:3 (1999), 657–697 | DOI | MR | Zbl

[12] A. Fruleux, R. Kawai, K. Sekimoto, “Momentum transfer in nonequilibrium steady states”, Phys. Rev. Lett., 108:16 (2012), 160601 | DOI

[13] M. F. Gelin, D. S. Kosov, “Asymptotic nonequilibrium steady-state operators”, Phys. Rev. E (3), 80:2 (2009), 022101, 4 pp. | DOI | MR

[14] H. Ge, M. Qian, H. Qian, “Stochastic theory of nonequilibrium steady states. Part II. Applications in chemical biophysics”, Phys. Rep., 510:3 (2012), 87–118 | DOI

[15] D. Hurowitz, S. Rahav, D. Cohen, “The non-equilibrium steady state of sparse systems with non-trivial topology”, EPL, 98:2 (2012), 20002, 6 pp. | DOI

[16] Ch. Kwon, P. Ao, “Nonequilibrium steady state of a stochastic system driven by a nonlinear drift force”, Phys. Rev. E, 84:6 (2011), 061106 | DOI

[17] E. Tjhung, M. E. Cates, D. Marenduzzo, “Nonequilibrium steady states in polar active fluids”, Soft Matter, 7:16 (2011), 7453–7464 | DOI

[18] F. A. C. C. Chalub, P. A. Markowich, B. Perthame, C. Schmeiser, “Kinetic models for chemotaxis and their drift-diffusion limits”, Monatsh. Math., 142:1–2 (2004), 123–141 | DOI | MR | Zbl

[19] F. Chalub, Y. Dolak-Struss, P. Markowich, D. Oeltz, C. Schmeiser, A. Soreff, “Model hierarchies for cell aggregation by chemotaxis”, Math. Models Methods Appl. Sci., 16:7 (2006), 1173–1197 | DOI | MR | Zbl

[20] Y. Dolak, C. Schmeiser, “Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms”, J. Math. Biol., 51:6 (2005), 595–615 | DOI | MR | Zbl

[21] R. Erban, H. G. Othmer, “From individual to collective behaviour in bacterial chemotaxis”, SIAM J. Appl. Math., 65:2 (2004/05), 361–391 (electronic) | DOI | MR | Zbl

[22] F. Filbet, P. Laurençot, B. Perthame, “Derivation of hyperbolic models for chemosensitive movement”, J. Math. Biol., 50:2 (2005), 189–207 | DOI | MR | Zbl

[23] N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, “Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems”, Math. Models Methods Appl. Sci., 20:7 (2010), 1179–1207 | DOI | MR | Zbl

[24] A. Bellouquid, “On the asymptotic analysis of kinetic models towards the compressible Euler and acoustic equations”, Math. Models Methods Appl. Sci., 14:6 (2004), 853–882 | DOI | MR | Zbl

[25] C. Bianca, C. Dogbe, “A mathematical model for crowd dynamics: multiscale analysis, fluctuations and random noise”, Nonlinear Stud., 20:3 (2013), 349–373 | MR | Zbl

[26] L. L. Bonilla, J. S. Soler, “High-field limit for the Vlasov–Poisson–Fokker–Planck system: a comparison of different perturbation methods”, Math. Models Methods Appl. Sci., 11:8 (2001), 1457–1468 | DOI | MR | Zbl

[27] N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, “On the asymptotic theory from microscopic to macroscopic growing tissue models: an overview with perspectives”, Math. Models Methods Appl. Sci., 22:1 (2012), 1130001, 37 pp. | DOI | MR | Zbl

[28] A. Bellouquid, C. Bianca, “Modelling aggregation-fragmentation phenomena from kinetic to macroscopic scales”, Math. Comput. Modelling, 52:5–6 (2010), 802–813 | DOI | MR | Zbl

[29] M. Lachowicz, “Micro and meso scales of description corresponding to a model of tissue invasion by solid tumours”, Math. Models Methods Appl. Sci., 15:11 (2005), 1667–1683 | DOI | MR | Zbl

[30] H. G. Othmer, T. Hillen, “The diffusion limit of transport equations derived from velocity-jump processes”, SIAM J. Appl. Math., 61:3 (2000), 751–775 (electronic) | DOI | MR | Zbl

[31] H. G. Othmer, S. R. Dunbar, W. Alt, “Models of dispersal in biological systems”, J. Math. Biol., 26:3 (1988), 263–298 | DOI | MR | Zbl

[32] H. G. Othmer, T. Hillen, “The diffusion limit of transport equations. II. Chemotaxis equations”, SIAM J. Appl. Math., 62:4 (2002), 1222–1250 (electronic) | DOI | MR | Zbl

[33] G. P. Morriss, C. P. Dettmann, “Thermostats: analysis and application”, Chaos, 8:2 (1998), 321–336 | DOI | Zbl

[34] D. Ruelle, “Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics”, J. Statist. Phys., 95:1–2 (1999), 393–468 | DOI | MR | Zbl

[35] K. F. Gauß, “Über ein neues allgemeines Grundgesetz der Mechanik”, J. Reine Angew. Math., 1829:4 (1829), 232–235 | DOI | Zbl

[36] C. Bianca, L. Fermo, “Bifurcation diagrams for the moments of a kinetic type model of keloid-immune system competition”, Comput. Math. Appl., 61:2 (2011), 277–288 | DOI | MR | Zbl

[37] C. Bianca, M. Pennisi, “The triplex vaccine effects in mammary carcinoma: a nonlinear model in tune with SimTriplex”, Nonlinear Anal. Real World Appl., 13:4 (2012), 1913–1940 | DOI | MR | Zbl

[38] C. Bianca, F. Pappalardo, M. Pennisi, M. A. Ragusa, “Persistence analysis in a Kolmogorov-type model for cancer-immune system competition”, AIP Conf. Proc., 1558 (2013), 1797–1800 | DOI

[39] R. M. Solow, “A contribution to the theory of economic growth”, Quart. J. Econom., 70:1 (1956), 65–94 | DOI

[40] T. W. Swan, “Economic growth and capital accumulation”, Econ. Rec., 32:2 (1956), 334–361 | DOI

[41] R. B. Banks, Growth and diffusion phenomena. Mathematical frameworks and applications, Texts Appl. Math., 14, Springer-Verlag, Berlin, 1994, xx+455 pp. | DOI | MR | Zbl

[42] M. Ferrara, L. Guerrini, “On the dynamics of a three-sector growth model”, Int. Rev. Econ., 55:3 (2008), 275–283 | DOI

[43] C.-J. Dalgaard, H. Strulik, “Energy distribution and economic growth”, Resour. Energy Econ., 33:4 (2011), 782–797 | DOI

[44] C. Bianca, M. Ferrara, L. Guerrini, “Hopf bifurcations in a delayed-energy-based model of capital accumulation”, Appl. Math. Inf. Sci., 7:1 (2013), 139–143 | DOI | MR

[45] C. Bianca, M. Ferrara, L. Guerrini, “The Cai model with time delay: existence of periodic solutions and asymptotic analysis”, Appl. Math. Inf. Sci., 7:1 (2013), 21–27 | DOI | MR

[46] C. Bianca, L. Guerrini, “On the Dalgaard–Strulik model with logistic population growth rate and delayed-carrying capacity”, Acta Appl. Math., 128:1 (2013), 39–48 | DOI | MR | Zbl

[47] M. A. Ragusa, “Embeddings for Morrey–Lorentz Spaces”, J. Optim Theory Appl., 154:2 (2012), 491–499 | DOI | MR | Zbl

[48] M. A. Ragusa, “Necessary and sufficient condition for a $VMO$ function”, Appl. Math. Comput., 218:24 (2012), 11952–11958 | DOI | MR | Zbl

[49] M. A. Ragusa, “Commutators of fractional integral operators on Vanishing–Morrey spaces”, J. Global Optim., 40:1–3 (2008), 361–368 | DOI | MR | Zbl

[50] C. Bianca, “Modeling complex systems by functional subsystems representation and thermostatted-KTAP methods”, Appl. Math. Inf. Sci., 6:3 (2012), 495–499