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A toy MCT model for multiple Glass transitions: double swallow tail singularity. (English) Zbl 1301.82061
Summary: We propose a toy model to describe in the frame of Mode Coupling Theory multiple glass transitions. The model is based on the postulated simple form for static structure factor as a sum of two delta-functions. This form makes it possible to solve the MCT equations in almost analytical way. The phase diagram is governed by two swallow tails resulting from two \(A_4\) singularities and includes liquid-glass transition and multiple glasses. The diagram has much in common with those of binary and quasibinary systems.

82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
Full Text: DOI
[1] Parisi, G.; Zamponi, F., Rev. Mod. Phys., 82, 789, (2010)
[2] Wolynes, P. G.; Lubchenko, V., Structural glasses and supercooled liquids: theory, experiment and applications, (2012), Wiley
[3] Berthier, L.; Biroli, G., Rev. Mod. Phys., 83, 587, (2011)
[4] Biroli, G.; Garrahan, J. P., J. Chem. Phys., 138, 12A301, (2013)
[5] Bengtzelius, U.; Götze, W.; Sjolander, A., J. Phys. C, 17, 5917, (1984)
[6] Götze, W.; Sjorgen, L., Rep. Prog. Phys., 55, 241, (1992)
[7] Das, S. P., Rev. Mod. Phys., 76, 785, (2004)
[8] Götze, W., Complex dynamics of Glass-forming liquids: A mode-coupling theory, (2009), Oxford University Press Oxford · Zbl 1162.82003
[9] Fuchs, M.; Götze, W.; Mayer, M. A., Phys. Rev. E, 58, 3384, (1998)
[10] Appignanesi, G. A.; Rodriguez, J. A., J. Phys. Condens. Matter, 21, 203103, (2009)
[11] Kob, W., (Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter, (2003), Springer-Verlag Berlin)
[12] Fabbian, L.; Götze, W.; Sciortino, F.; Tartaglia, P.; Thiery, F., Phys. Rev. E, 59, R1347, (1999)
[13] Dawson, K.; Foffi, G.; Fuchs, M.; Götze, W.; Sciortino, F.; Sperl, M.; Tartaglia, P.; Voigtmann, Th.; Zaccarelli, E., Phys. Rev. E, 63, 011401, (2000)
[14] Foffi, G.; Dawson, K. A.; Sciortino, F.; Tartaglia, P., Phys. Rev. E, 63, 031501, (2001)
[15] Sciortino, F., Nat. Mater., 1, 145, (2002)
[16] Arnold, V. I., Catastrophe theory, (1992), Springer Berlin
[17] Arnold, V. I., Singularities of caustics and wave fronts, Math. Appl., vol. 62, (1990), Kluwer Dordrecht · Zbl 0734.53001
[18] Gilmore, R., Catastrophe theory for scientists and engineers, (1981), Wiley NY · Zbl 0497.58001
[19] Götze, W.; Sperl, M., Phys. Rev. E, 66, 011405, (2002)
[20] Flach, S.; Götze, W.; Sjögren, L., Z. Phys. B, Condens. Matter, 87, 29, (1992)
[21] Götze, W.; Voigtmann, Th., Phys. Rev. E, 67, 021502, (2003)
[22] Foffi, G.; Götze, W.; Sciortino, F.; Tartaglia, P.; Voigtmann, Th., Phys. Rev. E, 69, 011505, (2004)
[23] Voigtmann, Th., Europhys. Lett., 96, 36006, (2011)
[24] Ryzhov, V. N.; Stishov, S. M., Phys. Rev. E, 67, R010201, (2003)
[25] Ryzhov, V. N.; Stishov, S. M.; Ryzhov, V. N.; Stishov, S. M., Zh. Eksp. Teor. Fiz., J. Exp. Theor. Phys., 95, 710, (2002)
[26] Fomin, Yu. D.; Ryzhov, V. N.; Tareyeva, E. E., Phys. Rev. E, 74, 041201, (2006)
[27] Ryltsev, R. E.; Chtchelkatchev, N. M.; Ryzhov, V. N., Phys. Rev. Lett., 110, 025701, (2013)
[28] Fomin, Yu. D.; Tsiok, E. N.; Ryzhov, V. N., J. Chem. Phys., 135, 124512, (2011)
[29] Fomin, Yu. D.; Tsiok, E. N.; Ryzhov, V. N., J. Chem. Phys., 135, 234502, (2011)
[30] Sperl, M.; Zaccarelli, E.; Sciortino, F.; Kumar, P.; Stanley, H. E., Phys. Rev. Lett., 104, 145701, (2010)
[31] Sperl, M., Prog. Theor. Phys. Suppl., 184, 209, (2010)
[32] Ryzhov, V. N.; Tareyeva, E. E.; Fomin, Yu. D., Theor. Math. Phys., 167, 645, (2011)
[33] Das, G.; Gnan, N.; Sciortino, F.; Zaccarelli, E., J. Chem. Phys., 138, 134501, (2013)
[34] Imhof, A.; Dhont, J. K.G., Phys. Rev. Lett., 75, 1662, (1995)
[35] Imhof, A.; Dhont, J. K.G., Phys. Rev. E, 52, 6344, (1995)
[36] Hansen, J. P.; McDonald, I. R., Theory of simple liquids, (1986), Academic Press NY
[37] Vainberg, M. M.; Trenogin, V. A., Theory of branching of solutions of non-linear equations, Monographs and Textbooks on Pure and Applied Mathematics, (1974), Noordhoff International Publishing Leyden · Zbl 0274.47033
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