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.
REFERENCES
1.
C.-C.
Cheng
, “Maxwell’s equations in dynamics
,” Am. J. Phys.
34
, 622
(1966
);A. L.
Fetter
and J. D.
Walecka
, Theoretical Mechanics of Particles and Continua
(McGraw-Hill
, New York
, 1980
).2.
3.
J. W.
Cannon
, “Connecting thermodynamics to students’ calculus
,” Am. J. Phys.
72
, 753
–757
(2004
).4.
M.
Artigue
, J.
Menigaux
, and L.
Viennot
, “Some aspects of students’ conceptions and difficulties about differentials
,” Eur. J. Physiol.
11
, 262
–267
(1990
);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
)or Eq. (7.136) in
H.
Goldstein
, Classical Mechanics
, 2nd ed. (Addison-Wesley
, Reading, MA
, 1980
).8.
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. 5.27.15.
J.
Landau
, S. G.
Lipson
, L. M.
Määttänen
, L. S.
Balfour
, and D. O.
Edwards
, “Interface between superfluid and solid
,” Phys. Rev. Lett.
45
, 31
–35
(1980
).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
;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 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.
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
).© 2009 American Association of Physics Teachers.
2009
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.