Making sense of the Legendre transform

American Journal of Physics

Volumn 77, Issue 7, 2009, Pages 614-622

  Zia, R K P ^a   Redish, Edward F ^b   McKay, Susan R ^c  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

^b Bldg 142   (United States)

^c University of Maine   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67649160915     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.3119512     Document Type: Article

References (19)

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2. K. Huang, Statistical Mechanics (Wiley, New York, 1987); H. S. Robertson, Statistical Thermophysics (Prentice Hall, New York, 1997).
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4. M. Artigue, J. Menigaux, and L. Viennot, " Some aspects of students' conceptions and difficulties about differentials.," Eur. J. Physiol. 11, 262-267 (1990); 10.1088/0143-0807/11/5/002 E. F. Redish, " Problem solving and the use of math in physics courses.," to be published in Proceedings of the Conference, World View on Physics Education in 2005: Focusing on Change, Delhi, India, August 21-26, 2005; 〈 www.physics.umd.edu/perg/papers/redish/ IniaMath.pdf 〉.
5. In this example, s, x, F, and G are all positive. Thus, the " G axis" points downward, opposite to the " F axis."
6. This restriction can be lifted, especially if physical quantities with dimensions (for example, the Hamiltonian) are studied. In that case, we must keep more careful track of the units, such as [s] = [F] [x-1].
7. See, for example, Eq. (12.7) in J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999) or Eq. (7.136) in H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, MA, 1980).
8. E. Taylor and J. A. Wheeler, Spacetime Physics (Freeman, New York, 1966).
9. In general F may be regarded as a smooth M -dimensional manifold. The eigenvalues of m l F (x) are the principal curvatures of this surface at x.
10. For systems in nonequilibrium stationary states, negative responses can be easily achieved. See, for example, R. K. P. Zia, E. L. Praestgaard, and O. G. Mouritsen, " Getting more from pushing less: Negative specific heat and conductivity in nonequilibrium steady states.," Am. J. Phys. 70, 384-392 (2002). 10.1119/1.1427088
11. This sort of construction is attributed to Born. See, for example, the discussion in W. W. Bowley, " Legendre transforms, Maxwell's relations, and the Born diagram in fluid dynamics.," Am. J. Phys. 37, 1066-1067 (1969). 10.1119/1.1975198
12. These partial derivatives are taken with the understanding that all other variables are held fixed. It is common (and reasonable) to consider derivatives with F or G held fixed. In this article we avoid discussing such complications.
13. We follow the notation in R. K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972).
14. D. V. Schroeder, An Introduction to Thermal Physics (Addison-Wesley, Reading, MA, 2000), Fig. 5.27.
15. J. Landau, S. G. Lipson, L. M. Määttänen, L. S. Balfour, and D. O. Edwards, " Interface between superfluid and solid He4.," Phys. Rev. Lett. 45, 31-35 (1980). 10.1103/PhysRevLett.45.31
16. J. C. Heyraud and J. J. Ḿtois, " Equilibrium shape of gold crystallites on a graphite cleavage surface: Surface energies and interfacial energy.," Acta Metall. 28, 1789-1797 (1980). 10.1016/0001-6160(80)90032-2
17. See, for example, M. Wortis, " Equilibrium Crystal Shapes and Interfacial Phase Transitions.," in Chemistry and Physics of Solid Surfaces, edited by, R. Vanselow, (Springer, New York, 1988), Vol. VII, pp. 367-406; and R. K. P. Zia, " Anisotropic Surface Tension and Equilibrium Crystal Shapes.," in Progress in Statistical Mechanics, edited by, C. K. Hu, (World Scientific, River Edge, NJ, 1988), pp. 303-357. See also R. K. P. Zia and J. E. Avron, " Total surface energy and equilibrium shapes: Exact results for the d=2 Ising crystal.," Phys. Rev. B 25, 2042-2045 (1982). The connection between anisotropic surface energy and the minimizing shape was first established over a century ago by G. Wulff, " Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Krystallflächen., " Z. Krystal. Mineral. 34, 449-530 (1901).
18. J. Schwinger, " The theory of quantized fields I.," Phys. Rev. 0031-899X 10.1103/PhysRev.82.914 82, 914-927 (1951) and J. Schwinger, " The theory of quantized fields II.," Phys. Rev. 91, 713-728 (1953). For a more recent treatment, see, for example, S. Weinberg, The Quantum Theory of Fields (Cambridge U. P., Cambridge, MA, 1996).
19. A recent text containing chapters on statistical fields is M. Kardar, Statistical Physics of Fields (Cambridge U. P., Cambridge, MA, 2007). More complete treatments may be found in C. Itzykson and J. M. Drouffe, Statistical Field Theory (Cambridge U. P., Cambridge, MA, 1989) and J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford U. P., New York, 2002).