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Making sense of the Legendre transform
1.C.-C. Cheng, “Maxwell’s equations in dynamics,” Am. J. Phys. 34, 622 (1966);
1.A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua (McGraw-Hill, New York, 1980).
2.K. Huang, Statistical Mechanics (Wiley, New York, 1987);
2.H. S. Robertson, Statistical Thermophysics (Prentice Hall, New York, 1997).
4.M. Artigue, J. Menigaux, and L. Viennot, “Some aspects of students’ conceptions and difficulties about differentials,” Eur. J. Physiol. 11, 262–267 (1990);
4.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;
5.In this example, , , , and are all positive. Thus, the “ axis” points downward, opposite to the “ 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 .
7.See, for example, Eq. (12.7) in J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999)
7.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 may be regarded as a smooth -dimensional manifold. The eigenvalues of are the principal curvatures of this surface at .
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).
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).
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 or 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.
16.J. C. Heyraud and J. J. Métois, “Equilibrium shape of gold crystallites on a graphite cleavage surface: Surface energies and interfacial energy,” Acta Metall. 28, 1789–1797 (1980).
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;
17.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.
17.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. 82, 914–927 (1951)
18.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).
19.More complete treatments may be found in C. Itzykson and J. M. Drouffe, Statistical Field Theory (Cambridge U. P., Cambridge, MA, 1989)
19.and J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford U. P., New York, 2002).
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