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Making sense of the Legendre transform
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Abstract
The Legendre transform is a powerful tool in theoretical physics and plays an important role in classical mechanics,statistical mechanics, and thermodynamics. In typical undergraduate and graduate courses the motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform. We then discuss examples to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations.
© 2009 American Association of Physics Teachers
Received Thu Sep 27 00:00:00 UTC 2007
Accepted Thu Mar 26 00:00:00 UTC 2009
Acknowledgments: We thank many colleagues for fruitful discussions as well as Beate Schmittmann for critical readings of the manuscript. This work is supported in part by the U.S. National Science Foundation through Grant Nos. DMR-0705152 and DUE-0524987.
Article outline:
I. INTRODUCTION
II. THE LEGENDRE TRANSFORM AS AN ALTERNATIVE WAY TO DISPLAY INFORMATION
III. THE MATHEMATICS OF THE LEGENDRE TRANSFORM
A. A graphic-geometric approach
B. Symmetric representation of the Legendre transform
C. Properties of the extrema
D. Symmetric representation of the higher derivatives
IV. EXAMPLES OF THE LEGENDRE TRANSFORM IN SINGLE-PARTICLE MECHANICS
V. THE LEGENDRE TRANSFORM IN STATISTICAL THERMODYNAMICS
A. The route of mathematics
B. The route of physics: Interpretation of the equilibrium condition
C. How does the Legendre transform enter into thermodynamics?
VI. LEGENDRE TRANSFORM WITH MANY VARIABLES
VII. CONCLUDING REMARKS
/content/aapt/journal/ajp/77/7/10.1119/1.3119512
1.
1.C.-C. Cheng, “Maxwell’s equations in dynamics,” Am. J. Phys. 34, 622 (1966);
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1.A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua (McGraw-Hill, New York, 1980).
2.
2.K. Huang, Statistical Mechanics (Wiley, New York, 1987);
2.H. S. Robertson, Statistical Thermophysics (Prentice Hall, New York, 1997).
4.
4.M. Artigue, J. Menigaux, and L. Viennot, “Some aspects of students’ conceptions and difficulties about differentials,” Eur. J. Physiol. 11, 262–267 (1990);
http://dx.doi.org/10.1088/0143-0807/11/5/002
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.
5.In this example, , , , and are all positive. Thus, the “ axis” points downward, opposite to the “ axis.”
6.
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.
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.
8.E. Taylor and J. A. Wheeler, Spacetime Physics (Freeman, New York, 1966).
9.
9.In general may be regarded as a smooth -dimensional manifold. The eigenvalues of are the principal curvatures of this surface at .
10.
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).
http://dx.doi.org/10.1119/1.1427088
11.
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).
http://dx.doi.org/10.1119/1.1975198
12.
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.
13.We follow the notation in R. K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972).
14.
14.D. V. Schroeder, An Introduction to Thermal Physics (Addison-Wesley, Reading, MA, 2000), Fig. 5.27.
16.
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).
http://dx.doi.org/10.1016/0001-6160(80)90032-2
17.
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.
18.J. Schwinger, “The theory of quantized fields I,” Phys. Rev. 82, 914–927 (1951)
http://dx.doi.org/10.1103/PhysRev.82.914
18.For a more recent treatment, see, for example, S. Weinberg, The Quantum Theory of Fields (Cambridge U. P., Cambridge, MA, 1996).
19.
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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Abstract
The Legendre transform is a powerful tool in theoretical physics and plays an important role in classical mechanics,statistical mechanics, and thermodynamics. In typical undergraduate and graduate courses the motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform. We then discuss examples to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations.
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