Geodesic
Homogeneous Cooling with Repulsive and Attractive Long-Range Potentials
M. K. Müller1 ; S. Luding1
1 Multi Scale Mechanics, CTW, UTwente, P. O. Box 217, 7500 AE Enschede, Netherlands
Mathematical modelling of natural phenomena, Tome 6 (2011) no. 4, pp. 118-150 Cet article a éte moissonné depuis la source EDP Sciences

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The interplay between dissipation and long-range repulsive/attractive forces in homogeneous, dilute, mono-disperse particle systems is studied. The pseudo-Liouville operator formalism, originally introduced for hard-sphere interactions, is modified such that it provides very good predictions for systems with weak long-range forces at low densities, with the ratio of potential to fluctuation kinetic energy as control parameter. By numerical simulations, the theoretical results are generalized with empirical, density dependent correction-functions up to moderate densities.The main result of this study on dissipative cooling is an analytical prediction for the reduced cooling rate due to repulsive forces and for the increased rate due to attractive forces. In the latter case, surprisingly, for intermediate densities, similar cooling behavior is observed as in systems without long-range interactions. In the attractive case, in general, dissipation leads to inhomogeneities earlier and faster than in the repulsive case.
DOI : 10.1051/mmnp/20116406

M. K. Müller 1 ; S. Luding 1

1 Multi Scale Mechanics, CTW, UTwente, P. O. Box 217, 7500 AE Enschede, Netherlands 
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M. K. Müller; S. Luding. Homogeneous Cooling with Repulsive and Attractive Long-Range Potentials. Mathematical modelling of natural phenomena, Tome 6 (2011) no. 4, pp. 118-150. doi: 10.1051/mmnp/20116406

[1] M. P. Allen, D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, Oxford, 1987.

[2] J. C. Almekinders, C. Jones J. Aerosol Sci. 1999 969 971

[3] J. Barnes, P. Hut Nature 1986 446 449

[4] D. L. Blair, A. Kudrolli. Magnetized granular materials. In H. Hinrichsen and D. Wolf, editors, The Physics of Granular Media., pages 281–296, Weinheim, 2004. Wiley-VCH.

[5] J. Blum, S. Bruns, D. Rademacher, A. Voss, B. Willenberg, M. Krause Phys. Rev. Lett. 2006

[6] J. Blum, G. Wurm, S. Kempf, T. Poppe, H. Klahr, T. Kozasa, M. Rott, T. Henning, J. Dorschner, R. Schräpler, H.U. Keller, W.J. Markiewicz, I. Mann, B.A.S. Gustafson, F. Giovane, D. Neuhaus, H. Fechtig, E. Grün, B. Feuerbacher, H. Kochan, L. Ratke, A. El Goresy, G. Morfill, S.J. Weidenschilling, G. Schwehm, K. Metzler, W.-H. Ip Phys. Rev. Lett. 2000 2426 2429

[7] P. Bode, J. P. Ostriker Astrophys. J. Supplem. Series 2003 1 13

[8] A. Brahic Astronomy and Astrophysics 1977 895 907

[9] F. G. Bridges, A. Hatzes, D. N. C. Lin Nature 1984 333 335

[10] N. V. Brilliantov, T. Pöschel Phys. Rev. E 2000 2809 2814

[11] N. V. Brilliantov, T. Pöschel. Kinetic Theory of Granular Gases. Oxford University Press, Oxford, 2004.

[12] N. V. Brilliantov, C. Saluena, T. Schwager, T. Pöschel Phys. Rev. Let. 2004

[13] N. V. Brilliantov, F. Spahn, J.-M. Hertzsch, T. Pöschel Phys. Rev. E 1996 5382 5392

[14] N. F. Carnahan, K. E. Starling J. Chem. Phys. 1969 635 636

[15] J. A. Cross. Electrostatics: Principles, Problems and Applications. Adam Hilger, Bristol, 1987.

[16] S. M. Dammer, J. Werth, H. Hinrichsen. Electrostatically charged granular matter. In H. Hinrichsen D. Wolf, editors, The Physics of Granular Media., pages 255–280, Wiley-VCH, Weinheim, 2004.

[17] S. M. Dammer, D. E. Wolf Phys. Rev. E 2004

[18] J. Du Central European Journal of Physics 2005 376 381

[19] J. Duran. Sands, Powders and Grains. Springer-Verlag, New York, 2000.

[20] J. W. Eastwood, R. W. Hockney, D. Lawrence Comp. Phys. Commun. 1980 215 261

[21] M. H. Ernst Physics Reports 1981 1 171

[22] M. H. Ernst, J. R. Dorfman, W. R. Hoegy, J. M. J. Van Leeuwen Physica 1969 127 146

[23] M. H. Ernst, E. Trizac, A. Barrat J. Stat. Phys. 2006 549 586

[24] P. Eshuis, K. Van Der Weele, D. Van Der Meer, D. Lohse Phys. Rev. E 2005

[25] S. E. Esipov, T. Pöschel. J. Stat. Phys. 1997 1385 1395

[26] L. W. Esposito, J. N. Cuzzi, J. B. Holberg, E. A. Marouf, G. L. Tyler, C. C. Porco. Saturn’s rings, structure, dynamics and particle properties. In Saturn, pages 463–545, Tucson, AZ, Univ. of Arizona Press, 1984.

[27] P. P. Ewald Ann. d. Physik 1921 253 287

[28] K. B. Geerse. Application of Electrospray: from people to plants. Ph.D. thesis, Technische Universiteit Delft, 2003.

[29] I. Goldhirsch, G. Zanetti Phys. Rev. Lett. 1993 1619 1622

[30] P. Goldreich Ann. Rev. Astron. Astrophys. 1982 249 283

[31] R. Greenberg, A. Brahic. Planetary Rings. Arizona University Press, Tucson, AZ, 1984.

[32] L. Greengard, V. Rokhlin J. of Comp. Phys. 1987 325 348

[33] P. K. Haff J. Fluid Mech. 1983 401 430

[34] J.-P. Hansen, I. R. McDonald. Theory of Simple Liquids. Academic Press Ltd., London, San Diego, 1990.

[35] D. Henderson Molec. Phys. 1975 971 972

[36] O. Herbst, R. Cafiero, A. Zippelius, H. J. Herrmann, S. Luding Phys. of Fluids 2005

[37] O. Herbst, P. Müller, A. Zippelius Phys. Rev. E 2005

[38] L. Hernquist Comp. Phys. Commun. 1988 107 115

[39] R. Hoffmann. Modeling and Simulation of an Electrostatic Image Transfer. (Ph.D. thesis) Shaker Verlag, Aachen, 2004.

[40] J. S. Høye Physica A 2010 1380 1390

[41] M. Huthmann, A. Zippelius Phys. Rev. E 1997 R6275 R6278

[42] W. Kleber,, A. Lang J. of Electrostatics 1997 237 240

[43] M. Krause, J. Blum Phys. Rev. Lett. 2004

[44] C. W. J. Lemmens. An Investigation, Implementation and Comparison of 3 important Particle Simulation Techniques: PP: Particle-Particle PM: Particle-Mesh TC: Tree-Code. Report 97-46, Faculty of Technical Mathematics and Informatics, Delft, 1997.

[45] M. Linsenbühler, J. H. Werth, S. M. Dammer, H. A. Knudsen, H. Hinrichsen, K.-E. Wirth, D. E. Wolf Powder Technology 2006 124 133

[46] D. Lohse, R. Bergmann, R. Mikkelsen, C. Zeilstra, D. Van Der Meer, M. Versluis, K. Van Der Weele, M. Van Der Hoef, H. Kuipers Phys. Rev. Let. 2004

[47] J. Lowell, A. C. Rose-Innes Adv. in Phys. 1980 947 1023

[48] S. Luding T.A.S.K. Quarterly, Scientific Bulletin of Academic Computer Centre of the Technical University of Gdansk. 1998 417 443

[49] S. Luding. Collisions contacts between two particles. In H. J. Herrmann, J.-P. Hovi, S. Luding, editors, Physics of dry granular media - NATO ASI Series E350, page 285, Dordrecht, 1998. Kluwer Academic Publishers.

[50] S. Luding Pranama-J. Phys. 2005 893 902

[51] S. Luding Granular Matter 2008 235 246

[52] S. Luding Nonlinearity 2009 R101 R146

[53] S. Luding, A. Goldshtein Granular Matter 2003 159 163

[54] S. Luding, H. J. Herrmann Chaos 1999 673 681

[55] S. Luding, M. Huthmann, S. Mcnamara, A. Zippelius Phys. Rev. E 1998 3416 3425

[56] S. Luding, S. Mcnamara Granular Matter 1998 113 128

[57] S. Mcnamara Phys. of Fluids A 1993 3056 3070

[58] S. Mcnamara, W. R. Young Phys. Rev. E 1996 5089 5100

[59] S. Miller. Clusterbildung in Granularen Gasen. (in German). Ph.D. thesis, Universität Stuttgart, 2003.

[60] S. Miller, S. Luding Phys. Rev. E 2004

[61] J. M. Montanero, V. Garzò, M. Alam, S. Luding Granular Matter 2006 103 115

[62] M.-K. Müller. Untersuchung von Akkretionsscheiben mit Hilfe der Molekulardynamik. (in German). Diploma thesis, Universität Stuttgart, 2001.

[63] M.-K. Müller. Long-Range Interactions In Dilute Granular Systems. Ph.D. thesis, Universiteit Twente/Enschede, 2007.

[64] M.-K. Müller, S. Luding. Long-range interactions in ring-shaped particle aggregates. In R. García-Rojo, H.J. Herrmann, S. McNamara, editors, Powders Grains, pages 1119–1122, Balkema, Leiden, 2005.

[65] M.-K. Müller, S. Luding. Homogeneous cooling with repulsive and attractive long-range interactions. In M. Nakagawa S. Luding, editors, Powders Grains, pages 697–700, AIP Conf. Procs. #1145, 2009.

[66] M.-K. Müller, T. Winkels, K.B. Geerse, J.C.M. Marijnissen, A. Schmidt-Ott, S. Luding. Experiment and simulation of charged particle sprays. In Proceedings PARTEC 2004, Nuremberg, 2004.

[67] B. Muth, M.-K. Müller, P. Eberhard, S. Luding. Contacts between many bodies. In W. Kurnik, editor, Machine Dynamics Problems, pages 101–114, Warsaw, 2004.

[68] E. Németh. Triboelektrische Aufladung von Kunststoffen. (in German). Ph.D. thesis, Technische Universität Bergakademie Freiberg, 2003.

[69] F. Niermöller. Ladungsverteilung in Mineralgemischen und elektrostatische Sortierung nach Triboaufladung. (in German). Ph.D. thesis, Technische Universität Clausthal, 1988.

[70] J. S. Olafsen, J. S. Urbach Phys. Rev. Lett. 1998 4369

[71] J. A. G. Orza, R. Brito, T. P. C. Van Noije, M. H. Ernst Int. J. of Mod. Phys. C 1997 953 965

[72] T. Pöschel S. Luding, editors. Granular Gases, Lecture Notes in Physics 564. Springer, Berlin, 2001.

[73] R. Ramírez, T. Pöschel, N. V. Brilliantov, T. Schwager Phys. Rev. E 1999 4465 4472

[74] F. H. Ree, W. G. Hoover J. Chem. Phys. 1964 939 950

[75] T. N. Scheffler. Kollisionskühlung in elektrisch geladener granularer Materie. (in German). Ph.D. thesis, Gerhard-Mercator-Universität Duisburg, 2000.

[76] T. N. Scheffler, D. E. Wolf Granular Matter 2002 103 113

[77] T. Schwager, T. Pëoschel Phys. Rev. E 1998 650 654

[78] F. Spahn, J. Schmidt Nature 2006 614 615

[79] V. Springel, S. D. M. White, A. Jenkins, C. S. Frenk, N. Yoshida, L. Gao, J. Navarro, R. Thacker, D. Croton, J. Helly, J. A. Peacock, S. Cole, P. Thomas, H. Couchman, A. Evrard, J. Colberg, F. Pearce Nature 2005 629 636

[80] U. Trottenberg, C. W. Oosterlee, A. Schüller. Multigrid. Academic Press, San Diego, 2001.

[81] T. P. C. Van Noije, M. H. Ernst, R. Brito Physica A 1998 266 283

[82] J. H. Werth, S. M. Dammer, Z. Farkas, H. Hinrichsen, D. E. Wolf Computer Physics Communications 2002 259 262

[83] J. H. Werth, H. Knudsen, H. Hinrichsen Phys. Rev. E 2006

[84] J. H. Werth, M. Linsenbuhler, S. M. Dammer, Z. Farkas, H. Hinrichsen, K.-E. Wirth, D. E. Wolf Powder Technology 2003 106 112

[85] D. E. Wolf, T. N. Scheffler, J. Schäfer Physica A 1999 171 181

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