We investigate the conditions under which a perfect gas subject to a three-dimensional external conservative field is in mechanical equilibrium without being in thermal equilibrium, that is, with a nonvanishing temperature gradient. We find the class of potentials for which this behavior is possible. In particular, gravitational, Coulomb, and centrifugal fields are shown to belong to this class. We obtain the functional form of the temperature field that assures the absence of convection and obtain the stability condition for mechanical equilibrium.

Topics
Equilibrium thermodynamics, Physics of gases
1.
K. Huang, Statistical Mechanics (Wiley, New York, 1987), 2nd ed.
2.
L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, London, 1959).
3.
For the one-dimensional case, see Ref. 2.
4.
See J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999), 3rd ed., Chap. 2.
5.
R. G. Barry and R. J. Chorley, Atmosphere, Weather and Climate (Routledge, New York, 2003).
6.
For an introductory treatment of stability in fluid dynamics, see, for example, D. J. Acheson, Elementary Fluid Dynamics (Clarendon, Oxford, 1990), Chap. 9.
7.
See Ref. 2, pp. 8–9. More precisely, our Eq. (32) is the three-dimensional counterpart of Eq. (4.4) in Ref. 2.
This content is only available via PDF.
© 2004 American Association of Physics Teachers.
2004
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.