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2019 | OriginalPaper | Buchkapitel

3. Equilibrium Phase Transitions in Statistical Physics

verfasst von : Tian Ma, Shouhong Wang

Erschienen in: Phase Transition Dynamics

Verlag: Springer International Publishing

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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. First, the standard model for a thermodynamic system is derived using the recently discovered potential-descending principle (PDP), addressed in Chap. 7. Second, all equilibrium phase transitions are fully characterized by three basic theorems. Third, the dynamical law of fluctuations is derived, and we show that the standard model, together with the dynamic law of fluctuation, offers correct information for critical exponents. Fourth, the theory is applied to gas-liquid transitions, ferromagnetism, binary systems, superconductivity, and liquid heliums, leading to various physical predictions and insights to the underlying physical problems.

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Vorheriges Kapitel Dynamic Transition Theory

Nächstes Kapitel Fluid Dynamics

The linear operator \(L_\lambda =-\left (\frac {\partial ^2F(\overline u; \lambda )}{\partial u_i, \partial u_j}\right )\) is always self-adjoint, and all eigenvalues of L _λ are real.

We used the classical Cahn–Hilliard modeling here for the conserved state function u; see (7.1.10) and the remark given there.

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

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.MathSciNetCrossRef

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

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

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

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.MathSciNetCrossRef

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.MathSciNetCrossRef

Ehrenfest, P. (1933). Phasenumwandlungen im ueblichen und erweiterten Sinn, classifiziert nach den entsprechenden Singularitaeten des thermodynamischen Potentiales. NV Noord-Hollandsche Uitgevers Maatschappij.

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

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.CrossRef

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.

Kleman, M. and O. D. Laverntovich (2007). Soft matter physics: an introduction. Springer Science & Business Media.

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

Liu, R., T. Ma, S. Wang, and J. Yang (2017a). Dynamic theory of fluctuations and critical exponents of thermodynamic phase transitions. hal-01674269.

Liu, R., T. Ma, S. Wang, and J. Yang (2019). Thermodynamical potentials of classical and quantum systems. Discrete & Continuous Dynamical Systems - B; see also hal-01632278(2017) 24, 1411.

Ma, T., R. Liu, and J. Yang (2018). Physical World from the Mathematical Point of View: Statistical Physics and Critical Phase Transition Theory (in Chinese). Science Press, Beijing.

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

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.

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.CrossRef

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

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

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

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

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.

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

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.

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

Ma, T. and S. Wang (2017a). Dynamic law of physical motion and potential-descending principle. J. Math. Study 50:3, 215–241; see also HAL preprint: hal--01558752.

Ma, T. and S. Wang (2017b). Dynamical theory of thermodynamic phase transitions. Hal preprint: hal-01568201.

Ma, T. and S. Wang (2017e). Topological Phase Transitions I: Quantum Phase Transitions. Hal preprint: hal-01651908.

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.CrossRef

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

Onuki, O. (2002). Phase transition dynamics. Cambridge Univ. Press..CrossRef

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

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

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.

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

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

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

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

Titel: Equilibrium Phase Transitions in Statistical Physics
verfasst von: Tian Ma
Shouhong Wang
Verlag: Springer International Publishing
Buch: Phase Transition Dynamics
Print ISBN: 978-3-030-29259-1

Electronic ISBN: 978-3-030-29260-7

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-29260-7_3

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