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Equilibrium Phase Transition in Statistical Physics

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Abstract

A principal objective in the study of equilibrium phase transitions is to capture the transitions from one equilibrium to another and to study the nature or order of such transitions. The study of equilibrium phase transitions presented in this book involves a combination of modeling, mathematical analysis, and physical predictions.

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References

  • Aftalion, A. and Q. Du (2002). The bifurcation diagrams for the Ginzburg-Landau system of superconductivity. Phys. D 163(1-2), 94–105.

    Article  MathSciNet  MATH  Google Scholar 

  • Alikakos, N. D. and G. Fusco (1993). The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions. Indiana Univ. Math. J. 42(2), 637–674.

    Article  MathSciNet  MATH  Google Scholar 

  • Bates, P. W. and P. C. Fife (1993). The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math. 53(4), 990–1008.

    Article  MathSciNet  MATH  Google Scholar 

  • Bethuel, F., H. Brezis, and F. Hélein (1994). Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications, 13. Boston, MA: Birkhäuser Boston Inc.

    Google Scholar 

  • Cahn, J. and J. E. Hillard (1957). Free energy of a nonuniform system i. interfacial energy. J. Chemical Physics 28, 258–267.

    Google Scholar 

  • Chalkin, P. M. and T. C. Lubensky (2000). Principles of Condensed Matter Physics. Cambridge University Press; Reprint edition.

    Google Scholar 

  • Chandrasekhar, S. (1981). Hydrodynamic and Hydromagnetic Stability. Dover Publications, Inc.

    Google Scholar 

  • de Gennes, P. (1966). Superconductivity of Metals and Alloys. W. A. Benjamin.

    MATH  Google Scholar 

  • del Pino, M., P. L. Felmer, and P. Sternberg (2000). Boundary concentration for eigenvalue problems related to the onset of superconductivity. Comm. Math. Phys. 210(2), 413–446.

    Article  MathSciNet  MATH  Google Scholar 

  • Desai, R. C. and R. Kapral (2009). Self-Organized and Self-Assembled Structures. Cambridge University Press.

    Google Scholar 

  • Du, Q., M. D. Gunzburger, and J. S. Peterson (1992). Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34(1), 54–81.

    Article  MathSciNet  MATH  Google Scholar 

  • Elliott, C. M. and Z. Songmu (1986). On the Cahn-Hilliard equation. Arch. Rational Mech. Anal. 96(4), 339–357.

    Article  MathSciNet  MATH  Google Scholar 

  • Fisher, M. (1964). Specific heat of a gas near the critical point. Physical Review 136:6A, A1599–A1604.

    Google Scholar 

  • Ginzburg, V. L. (2004). On superconductivity and superfluidity (what i have and have not managed to do), as well as on the ’physical minimum’ at the beginning of the xxi century. Phys.-Usp. 47, 1155–1170.

    Article  Google Scholar 

  • Gor’kov, L. (1968). Generalization of the Ginzburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities. Sov. Phys. JETP 27, 328–334.

    Google Scholar 

  • Kleman, M. and O. D. Lavrentovich (2003). Soft Matter Physics, An Introduction. Springer.

    Google Scholar 

  • Langer, J. (1971). Theory of spinodal decomposition in allays. Ann. of Physics 65, 53–86.

    Article  Google Scholar 

  • Lin, F. H. (1996). Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure Appl. Math. 49(4), 323–359.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, T. and S. Wang (2005a). Bifurcation and stability of superconductivity. J. Math. Phys. 46(9), 095112, 31.

    Google Scholar 

  • Ma, T. and S. Wang (2005b). Bifurcation theory and applications, Volume 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.

    Google Scholar 

  • Ma, T. and S. Wang (2005d). Geometric theory of incompressible flows with applications to fluid dynamics, Volume 119 of Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society.

    Google Scholar 

  • Ma, T. and S. Wang (2008a). Dynamic model and phase transitions for liquid helium. Journal of Mathematical Physics 49:073304, 1–18.

    Google Scholar 

  • Ma, T. and S. Wang (2008b). Dynamic phase transition theory in PVT systems. Indiana University Mathematics Journal 57:6, 2861–2889.

    Google Scholar 

  • Ma, T. and S. Wang (2008c). Dynamic transitions for ferromagnetism. Journal of Mathematical Physics 49:053506, 1–18.

    Google Scholar 

  • Ma, T. and S. Wang (2008e). Superfluidity of helium-3. Physica A: Statistical Mechanics and its Applications 387:24, 6013–6031.

    Google Scholar 

  • Ma, T. and S. Wang (2009b). Cahn-hilliard equations and phase transition dynamics for binary systems. Dist. Cont. Dyn. Systs., Ser. B 11:3, 741–784.

    Google Scholar 

  • Ma, T. and S. Wang (2009c). Phase separation of binary systems. Physica A: Statistical Physics and its Applications 388:23, 4811–4817.

    Google Scholar 

  • Ma, T. and S. Wang (2009d). Phase transition and separation for mixture of liquid he-3 and he-4, in lev davidovich landau and his impact on contemporary theoretical physics, horizons in world physics, edited by a. sakaji and i. licata. 264, 107–119.

    Google Scholar 

  • Ma, T. and S. Wang (2011e). Third-order gas-liquid phase transition and the nature of andrews critical point. AIP Advances 1, 042101.

    Article  Google Scholar 

  • Nishikawa, K. and T. Morita (1998). Fluid behavior at supercritical states studied by small-angle X-ray scattering. Journal of Supercritical Fluid 13, 143–148.

    Article  Google Scholar 

  • Novick-Cohen, A. and L. A. Segel (1984). Nonlinear aspects of the Cahn-Hilliard equation. Phys. D 10(3), 277–298.

    Article  MathSciNet  Google Scholar 

  • Onuki, O. (2002). Phase transition dynamics. Combridge Univ. Press..

    Google Scholar 

  • Pismen, L. M. (2006). Patterns and Interfaces in Dissipative Dynamics. Springer, Berlin.

    MATH  Google Scholar 

  • Reichl, L. E. (1998). A modern course in statistical physics (Second ed.). A Wiley-Interscience Publication. New York: John Wiley & Sons Inc.

    MATH  Google Scholar 

  • Rubinstein, J. (2004). Topological methods in superconductivity. Not. S. Afr. Math. Soc. 35(1), 47–62. Invited papers from the 46th Annual SAMS Congress.

    Google Scholar 

  • Schmid, A. (1966). A time dependent ginzburg-landau equation and its application to the problem of resistivity in the mixed state. Phys. Kondens. Mater. 5, 302–317.

    Google Scholar 

  • Serfaty, S. (2006). Vortices in the Ginzburg-Landau model of superconductivity. In International Congress of Mathematicians. Vol. III, pp. 267–290. Eur. Math. Soc., Zürich.

    Google Scholar 

  • Shen, J. and X. Yang (2010). Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. 28(4), 1669–1691.

    Article  MathSciNet  MATH  Google Scholar 

  • Stanley, H. E. (1971). Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, New York and Oxford.

    Google Scholar 

  • Tang, Q. and S. Wang (1995). Time dependent Ginzburg-Landau equations of superconductivity. Phys. D 88(3-4), 139–166.

    Article  MathSciNet  MATH  Google Scholar 

  • Tinkham, M. (1996). Introduction to Superconductivity. McGraw-Hill, Inc.

    Google Scholar 

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Ma, T., Wang, S. (2014). Equilibrium Phase Transition in Statistical Physics. In: Phase Transition Dynamics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8963-4_3

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