Can an ideal gas feel the shape of its container?

American Journal of Physics

Volumn 65, Issue 8, 1997, Pages 739-743

  Gutiérrez, Gonzalo ^a   Yáñez, Julio M ^a  

^a PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE   (Chile)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0039268975     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18644     Document Type: Article

References (32)

1. M. Kac, "Can one hear the shape of a drum?," Am. Math. Mon. 73, 1-23 (1966); see also M. Kac, Enigmas of Chance. An Autobiography (Harper and Row, New York, 1985).
2. M. Kac, "Can one hear the shape of a drum?," Am. Math. Mon. 73, 1-23 (1966); see also M. Kac, Enigmas of Chance. An Autobiography (Harper and Row, New York, 1985).
3. In this work only Dirichlet boundary conditions will be used, for example, fixed edges for a membrane or zero electromagnetic field on the walls for a resonant cavity. Also, Neumann boundary conditions may be used. This means zero normal derivative on the boundary, for example, the pressure in the acoustical problem.
4. See, for example, D. A. McQuarrie, Statistical Thermodynamics (Harper and Row, New York, 1976), p. 82.
5. M. I. Molina, "Ideal gas in a finite container," Am. J. Phys. 64, 503-505 (1996).
6. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Analysis of Operators (Academic, New York, 1978), Vol. IV, p. 260.
7. H. Weyl, "Über die asymptotische Verteilung der Eigenwerte," Göttingen Nachr. 110-117 (1911); H. Weyl, "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)," Math. Ann. 71, 441-479 (1912). See also H. Weyl, "Ramifications, old and new, of the eigenvalue problem," Bull. Am. Math. Soc. 58, 115-139 (1950). Today these results are a classic, for example, see Ref. 20, Chap. VI, Sec. 4 for a different approach from the Weyl one, and see Ref. 5, Chap. XIII. 15, for a more modern version. For a more accessible and explicit proof in two particular cases, see R. H. Lambert, "Density of states in a Sphere and Cylinder," Am. J. Phys. 36, 417-420 (1968).
8. H. Weyl, "Über die asymptotische Verteilung der Eigenwerte," Göttingen Nachr. 110-117 (1911); H. Weyl, "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)," Math. Ann. 71, 441-479 (1912). See also H. Weyl, "Ramifications, old and new, of the eigenvalue problem," Bull. Am. Math. Soc. 58, 115-139 (1950). Today these results are a classic, for example, see Ref. 20, Chap. VI, Sec. 4 for a different approach from the Weyl one, and see Ref. 5, Chap. XIII. 15, for a more modern version. For a more accessible and explicit proof in two particular cases, see R. H. Lambert, "Density of states in a Sphere and Cylinder," Am. J. Phys. 36, 417-420 (1968).
9. H. Weyl, "Über die asymptotische Verteilung der Eigenwerte," Göttingen Nachr. 110-117 (1911); H. Weyl, "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)," Math. Ann. 71, 441-479 (1912). See also H. Weyl, "Ramifications, old and new, of the eigenvalue problem," Bull. Am. Math. Soc. 58, 115-139 (1950). Today these results are a classic, for example, see Ref. 20, Chap. VI, Sec. 4 for a different approach from the Weyl one, and see Ref. 5, Chap. XIII. 15, for a more modern version. For a more accessible and explicit proof in two particular cases, see R. H. Lambert, "Density of states in a Sphere and Cylinder," Am. J. Phys. 36, 417-420 (1968).
10. H. Weyl, "Über die asymptotische Verteilung der Eigenwerte," Göttingen Nachr. 110-117 (1911); H. Weyl, "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)," Math. Ann. 71, 441-479 (1912). See also H. Weyl, "Ramifications, old and new, of the eigenvalue problem," Bull. Am. Math. Soc. 58, 115-139 (1950). Today these results are a classic, for example, see Ref. 20, Chap. VI, Sec. 4 for a different approach from the Weyl one, and see Ref. 5, Chap. XIII. 15, for a more modern version. For a more accessible and explicit proof in two particular cases, see R. H. Lambert, "Density of states in a Sphere and Cylinder," Am. J. Phys. 36, 417-420 (1968).
11. Å. Pleijel, "A study of certain Green's functions with applications in the theory of vibrating membranes," Ark. Mat. 2, 553-569 (1954).
12. H. P. McKean and I. M. Singer, "Curvature and the eigenvalues of the Laplacian," J. Diff. Geom. 1, 43-69 (1967).
13. R. T. Waechter, "On hearing the shape of a drum: An extension to higher dimensions," Proc. Cambridge Philos. Soc. 72, 439-447 (1972).
14. K. Stewartson and R. T. Waechter, "On hearing the shape of a drum: Further results," Proc. Cambridge Philos. Soc. 69, 353-363 (1971); M. Berger, "Geometry of the spectrum. I," Proc. Symp. Pure Math. 27, 129-152 (1975); M. H. Protter, "Can one hear the shape of a drum? revisited," SIAM Rev. 29, 185-197 (1987).
15. K. Stewartson and R. T. Waechter, "On hearing the shape of a drum: Further results," Proc. Cambridge Philos. Soc. 69, 353-363 (1971); M. Berger, "Geometry of the spectrum. I," Proc. Symp. Pure Math. 27, 129-152 (1975); M. H. Protter, "Can one hear the shape of a drum? revisited," SIAM Rev. 29, 185-197 (1987).
16. K. Stewartson and R. T. Waechter, "On hearing the shape of a drum: Further results," Proc. Cambridge Philos. Soc. 69, 353-363 (1971); M. Berger, "Geometry of the spectrum. I," Proc. Symp. Pure Math. 27, 129-152 (1975); M. H. Protter, "Can one hear the shape of a drum? revisited," SIAM Rev. 29, 185-197 (1987).
17. In some cases, especially when we have a domain with smooth borders, we can calculate explicitly some of the coefficients of the sum S(t) in Eq. (17). Thus in the two-dimensional case we can easily interpret them as the area, perimeter and curvature. However, when we go to higher dimensions a direct interpretation is a nontrivial task (see Refs. 8, 9, and 10). Note that in two dimensions there are some domains whose eigenvalues are known explicitly, so that it is possible to estimate in a simple manner the sum (9), getting the general result (12) [or (13) when it corresponds]. In particular, Ref. 17 recovers the asymptotic expansion of S(t) in the case of a square, equilateral triangle (see also Ref. 21), 30-60-90 triangle, 45-45-90 triangle (see also Ref. 18) and a narrow annular region [this last result is due to Gottlieb (Ref. 22)]. In the case of a circle, Stewartson and Waechter (Ref. 10), using the associated Green's function, could calculate the first six terms of the asymptotic expansion of S(t), whereas Sleeman and Zayed in Ref. 23 got the first three terms for an annular region. In the three-dimensional case, Waechter (Ref. 9) found the six first terms of (17) for a sphere, a rectangular parallelepiped and a cylinder. Sleeman and Zayed got the first four terms for a cylindrical shell.
18. Note that the error in Kac's formula (12) when there are domains with polygonal boundary is exponentially small, whereas when the domains have smooth boundary this error is polynomially small. See P. B. Bailey and F. H. Brownell, "Removal of the Log Factor in the Asymptotic Estimates of Polygonal Membrane Eigenvalues," J. Math. Anal. Appl. 4, 212-239 (1962). See also Refs. 8 and 10.
19. C. Gordon, D. L. Webb, and S. Wolpert, "One cannot hear the shape of a drum," Bull. Am. Math. Soc. 27, 134-138 (1992); J. Milnor had already showed, in 1964, an example of two different domains with the same spectrum, but they were two 16-dimensional tori!
20. S. Sridhar and A. Kudrolli, "Experiments on Not 'Hearing the Shape' of Drums," Phys. Rev. Lett. 72, 2175-2178 (1994).
21. In fact, the expansion for p(E) for the wave equation is true provided the wavelength is much smaller than the length scale of the container. The equivalent statement in the case of the ideal gas is that the mean free path has to be much smaller than the smallest length scale of the container.
22. In the case of Ref. 4, the author did not obtain the same coefficient because of the approximation used.
23. K. P. McHale, "Eigenvalues of the Laplacian, 'Can You Hear the Shape of a Drum?'" Master's Project (Mathematics Department), University of Missouri - Columbia, 1994 (unpublished). See also Refs. 9 and 23.
24. H. P. W. Gottlieb, "Eigenvalues of the Laplacian for rectilinear regions," J. Aust. Math. Soc., Ser. B 29, 270-281 (1988).
25. H. P. W. Gottlieb, "Eigenvalues of the Laplacian with Neumann boundary conditions," J. Aust. Math. Soc., Ser. B 26, 293-309 (1985).
26. R. Courant and D. Hubert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1, Chap. II, Sec. 5.5.
27. E. M. E. Zayed and A. I. Younis, "On hearing the shape of rectilinear regions," J. Math. Phys. 35, 3490-3496 (1994).
28. H. P. W. Gottlieb, "Hearing the Shape of an Annular Drum," J. Aust. Math. Soc., Ser. B 24, 435-438 (1983).
29. E. M. E. Zayed, "An inverse eigenvalue problem for the Laplace operator," Lecture Notes in Mathematics 964 (Springer-Verlag, Berlin, 1982), pp. 718-726; B. D. Sleeman and E. M. E. Zayed, "Trace formulae for the eigenvalues of the Laplacian," J. Appl. Math. Phys. 35, 106-115 (1984).
30. E. M. E. Zayed, "An inverse eigenvalue problem for the Laplace operator," Lecture Notes in Mathematics 964 (Springer-Verlag, Berlin, 1982), pp. 718-726; B. D. Sleeman and E. M. E. Zayed, "Trace formulae for the eigenvalues of the Laplacian," J. Appl. Math. Phys. 35, 106-115 (1984).
31. P. Bérard, "Domaines Plans Isospectraux à la Gordon-Webb-Wolpert (une preuve terre à terre)," preprint, Universite de Grenoble, 1991. See also S. J. Chapman, "Drums That Sound the Same," Am. Math. Monthly 102, 124-138 (1995).
32. P. Bérard, "Domaines Plans Isospectraux à la Gordon-Webb-Wolpert (une preuve terre à terre)," preprint, Universite de Grenoble, 1991. See also S. J. Chapman, "Drums That Sound the Same," Am. Math. Monthly 102, 124-138 (1995).