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Onset of nonlinearity in thermostatted active particles models for complex systems. (English) Zbl 06118802
Summary: This paper is concerned with the derivation of a new discrete general framework of the kinetic theory, suitable for the modeling of complex systems under the action of an external force field and constrained to kept constant the mass or density, and the kinetic or activation energy. The resulting model relies on the interactions of single individuals within the population and is expressed by means of nonlinear ordinary or partial integro-differential equations. The global in time existence and uniqueness of the solution to the relative Cauchy problem are proved for which the density and the energy of the solution are preserved. A critical analysis, proposed in the last part of the paper, outlines suitable applications and research perspectives.

MSC:
92DGenetics and population dynamics
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[1] Bianca, C.; Bellomo, N.: Towards a mathematical theory of multiscale complex biological systems, Series in mathematical biology and medicine (2011) · Zbl 1286.92003
[2] Erban, R.; Othmer, H. G.: From individual to collective behavior in bacterial chemotaxis, SIAM journal of applied mathematics 65, 361-391 (2004) · Zbl 1073.35116 · doi:10.1137/S0036139903433232
[3] Levin, S. L.: Complex adaptive systems: exploring the known, the unknown and the unknowable, American mathematical society bulletin 40, 3-19 (2002) · Zbl 1015.92001
[4] Schweitzer, F.: Brownian agents and active particles, (2003) · Zbl 1140.91012
[5] Helbing, D.: Stochastic and Boltzmann-like models for behavioral changes, and their relation to game theory, Physica A 193, 241-258 (1993)
[6] Kacperski, K.; Holyst, J. A.: Opinion formation model with strong leader and external impact: a mean field approach, Physica A 269, 511-526 (1999)
[7] Weidlich, W.: Thirty years of sociodynamics. An integrated strategy of modelling in the social sciences: application to migration and urban evolution, Chaos solitons fractals 24, 45-56 (2005) · Zbl 1142.91727 · doi:10.1016/j.chaos.2004.07.022
[8] Gatignol, R.: Théorie cinétique d’un gaz à répartition discréte des vitèsses, Springer lecture notes in physics 36 (1975)
[9] Bellomo, N.; Bianca, C.; Delitala, M.: Complexity analysis and mathematical tools towards the modelling of living systems, Physics of life reviews 6, 144-175 (2009)
[10] Kerner, B.: The physics of traffic: empirical freeway pattern features, engineering applications, and theory, (2004)
[11] Bianca, C.; Coscia, V.: On the coupling of steady and adaptive velocity grids in vehicular traffic modelling, Applied mathematics letters 24, 149-155 (2011) · Zbl 1201.90050 · doi:10.1016/j.aml.2010.08.035
[12] Bellomo, N.; Bellouquid, A.: Global solution to the Cauchy problem for discrete velocity models of vehicular traffic, Journal of differential equations 252, 1350-1368 (2012) · Zbl 1230.35056 · doi:10.1016/j.jde.2011.09.005
[13] Dogbe, C.: On the modelling of crowd dynamics by generalized kinetic models, Journal of mathematical analysis and applications 387, 512-532 (2012) · Zbl 1229.90035 · doi:10.1016/j.jmaa.2011.09.007
[14] Morriss, G. P.; Dettmann, C. P.: Thermostats: analysis and application, Chaos 8, 321-336 (1998) · Zbl 0977.80002 · doi:10.1063/1.166314
[15] Ruelle, D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, Journal of statistical physics 95, 393-468 (1999) · Zbl 0934.37010 · doi:10.1023/A:1004593915069
[16] Gauss, K. F.: Uber ein neues allgemeines grundgesatz der mechanik (on a new fundamental law of mechanics), Journal für die reine und angewandte Mathematik 4, 232-235 (1829) · Zbl 004.0157cj · doi:10.1515/crll.1829.4.232
[17] Bianca, C.: On the mathematical transport theory in microporous media: the billiard approach, Nonlinear analysis: hybrid systems 4, 699-735 (2010) · Zbl 1202.37055 · doi:10.1016/j.nahs.2010.04.007
[18] Bianca, C.: Weyl-flow and the conformally symplectic structure of thermostatted billiards: the problem of the hyperbolicity, Nonlinear analysis: hybrid systems 5, 32-51 (2011) · Zbl 05850225
[19] Bianca, C.: On the existence of periodic orbits in nonequilibrium ehrenfest gas, International mathematical forum 7, 221-232 (2012) · Zbl 1256.37016
[20] Bianca, C.; Rondoni, L.: The nonequilibrium ehrenfest gas: a chaotic model with flat obstacles?, Chaos 19, 013121 (2009) · Zbl 1311.76118
[21] Bonetto, F.; Daems, D.; Lebowitz, J. L.; Ricci, V.: Properties of stationary nonequilibrium states in the thermostatted periodic Lorentz gas: the multiparticle system, Physical review E 65, 051204 (2002)
[22] Chernov, N. I.; Eyink, G. L.; Lebowitz, J. L.; Sinai, Ya.G.: Derivation of ohm’s law in a deterministic mechanical model, Physical review letters 70, 2209-2212 (1993)
[23] Jepps, O. G.; Rondoni, L.: Deterministic thermostats, theories of nonequilibrium systems and parallels with the ergodic condition, Journal of physics A: mathematical theory 43, 133001 (2010) · Zbl 1195.82048 · doi:10.1088/1751-8113/43/13/133001
[24] Klages, R.: Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics, Advanced series in nonlinear dynamics 24 (2007) · Zbl 1127.82002
[25] Webb, J. R. L.: Existence of positive solutions for a thermostat model, Nonlinear analysis. Real world applications 13, 923-938 (2012) · Zbl 1238.34033
[26] Bagland, V.: Well-posedness and large time behaviour for the non-cutoff Kac equation with a Gaussian thermostat, Journal of statistical physics 138, 838-875 (2010) · Zbl 1187.82101 · doi:10.1007/s10955-009-9872-4
[27] Wennberg, B.; Wondmagegne, Y.: Stationary states for the Kac equation with a Gaussian thermostat, Nonlinearity 17, 633-648 (2004) · Zbl 1049.76058 · doi:10.1088/0951-7715/17/2/016
[28] Wennberg, B.; Wondmagegne, Y.: The Kac equation with a thermostatted force field, Journal of statistical physics 124, 859-880 (2006) · Zbl 1134.82041 · doi:10.1007/s10955-005-9020-8
[29] Bianca, C.: An existence and uniqueness theorem to the Cauchy problem for thermostatted-KTAP models, International journal of mathematical analysis 6, 813-824 (2012) · Zbl 1250.35174
[30] Bianca, C.: Kinetic theory for active particles modelling coupled to Gaussian thermostats, Applied mathematical sciences 6, 651-660 (2012) · Zbl 1250.82025
[31] Bianca, C.: On the modelling of space dynamics in the kinetic theory for active particles, Mathematical and computer modelling 51, 72-83 (2010) · Zbl 1190.82032
[32] Woodcock, L. V.: Isothermal molecular dynamics calculations for liquid salts, Chemical physics letters 10, 257-261 (1971)
[33] Lanczos, C.: The variational principles of mechanics, (1979) · Zbl 0037.39901
[34] Evans, D. J.; Morriss, G. P.: Statistical mechanics of nonequilibrium fluids, (1990) · Zbl 1145.82301
[35] Bellomo, N.; Bianca, C.; Mongioví, M. S.: On the modeling of nonlinear interactions in large complex systems, Applied mathematics letters 23, 1372-1377 (2010) · Zbl 1197.92002 · doi:10.1016/j.aml.2010.07.001
[36] Gramani, L.: On the modeling of granular traffic flow by the kinetic theory for active particles. Trend to equilibrium and macroscopic behaviour, International journal of nonlinear mechanics 44, 263-268 (2009) · Zbl 1203.90044 · doi:10.1016/j.ijnonlinmec.2008.11.008
[37] Arkeryd, L.: On the Boltzmann equation. I. existence, Archive for rational mechanics and analysis 45, 1-16 (1972) · Zbl 0245.76059
[38] Bianca, C.: Mathematical modelling for keloid formation triggered by virus: malignant effects and immune system competition, Mathematical models and methods in applied sciences 21, 389-419 (2011) · Zbl 1218.35236 · doi:10.1142/S021820251100509X
[39] Bianca, C.; Pennisi, M.: The triplex vaccine effects in mammary carcinoma: a nonlinear model in tune with simtriplex, Nonlinear analysis. Real world applications 13, 1913-1940 (2012) · Zbl 06118750
[40] Jorcyk, C. L.; Kolev, M.; Tawara, K.; Zubik-Kowal, B.: Experimental versus numerical data for breast cancer progression, Nonlinear analysis. Real world applications 13, 78-84 (2012) · Zbl 1238.92021
[41] Qi, J.; Ding, Y.; Zhu, Y.; Wu, Y.: Kinetic theory approach to modeling of cellular repair mechanisms under genome stress, Plos ONE 6 (2011)
[42] Qi, J. P.; Zhu, Y.; Ding, Y. S.: A mathematical framework for cellular repair mechanisms under genomic stress based on kinetic theory approach, Applied mechanics and materials 52--54, 7-12 (2011)
[43] Ye, J.; Feng, E.; Yin, H.; Xiu, Z.: Modelling and well-posedness of a nonlinear hybrid system in fed-batch production of 1, 3-propanediol with open loop glycerol input and ph logic control, Nonlinear analysis. Real world applications 12, 364-376 (2011) · Zbl 1201.92032 · doi:10.1016/j.nonrwa.2010.06.022
[44] Bellouquid, A.; Bianca, C.: Modelling aggregation--fragmentation phenomena from kinetic to macroscopic scales, Mathematical and computer modelling 52, 802-813 (2010) · Zbl 1202.82065 · doi:10.1016/j.mcm.2010.05.010
[45] Bellouquid, A.; De Angelis, E.: From kinetic models of multicellular growing systems to macroscopic biological tissue models, Nonlinear analysis. Real world applications 12, 1111-1122 (2011) · Zbl 1203.92020 · doi:10.1016/j.nonrwa.2010.09.005
[46] Lachowicz, M.: Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear analysis. Real world applications 12, 2396-2407 (2011) · Zbl 1225.93101 · doi:10.1016/j.nonrwa.2011.02.014