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Such changes in isolated systems are irreversible in the sense that while such a change will occur spontaneously whenever the system is prepared in the same way, the reverse change will practically never occur spontaneously within the isolated system; this is a large part of the content of the [[second law of thermodynamics]]. Truly perfectly isolated systems do not occur in nature, and always are artificially prepared.
===In a gravitational field===
One may consider a system contained in a very tall adiabatically isolating vessel with rigid walls initially containing a thermally heterogeneous distribution of material, left for a long time under the influence of a steady gravitational field, along its tall dimension, due to an outside body such as the earth. It will settle to a state of uniform temperature throughout, though not of uniform pressure or density, and perhaps containing several phases. It is then in internal thermal equilibrium and even in thermodynamic equilibrium. This means that all local parts of the system are in mutual radiative exchange equilibrium. This means that the temperature of the system is spatially uniform.<ref name="Planck 1914 40" /> This is so in all cases, including those of non-uniform external force fields. For an externally imposed gravitational field, this may be proved in macroscopic thermodynamic terms, by the calculus of variations, using the method of Langrangian multipliers.<ref>Gibbs, J.W. (1876/1878), pp. 144-150.</ref><ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), pp. 127–130.</ref><ref>Münster, A. (1970), pp. 309–310.</ref><ref>Bailyn, M. (1994), pp. 254-256.</ref><ref>{{Cite journal | doi=10.1175/1520-0469(2004)061<0931:OMEP>2.0.CO;2| bibcode=2004JAtS...61..931V| issn=1520-0469| year=2004| volume=61| pages=931–936| title=On Maximum Entropy Profiles| last1=Verkley| first1=W. T. M.| last2=Gerkema| first2=T.| journal=Journal of the Atmospheric Sciences| issue=8| doi-access=free}}</ref><ref>Akmaev, R.A. (2008). On the energetics of maximum-entropy temperature profiles, ''Q. J. R. Meteorol. Soc.'', '''134''':187–197.</ref> Considerations of kinetic theory or statistical mechanics also support this statement.<ref>Maxwell, J.C. (1867).</ref><ref>Boltzmann, L. (1896/1964), p. 143.</ref><ref>Chapman, S., Cowling, T.G. (1939/1970), Section 4.14, pp. 75–78.</ref><ref>[[J. R. Partington|Partington, J.R.]] (1949), pp. 275–278.</ref><ref>Coombes, C.A., Laue, H. (1985). A paradox concerning the temperature distribution of a gas in a gravitational field, ''Am. J. Phys.'', '''53''': 272–273.</ref><ref>Román, F.L., White, J.A., Velasco, S. (1995). Microcanonical single-particle distributions for an ideal gas in a gravitational field, ''Eur. J. Phys.'', '''16''': 83–90.</ref><ref>Velasco, S., Román, F.L., White, J.A. (1996). On a paradox concerning the temperature distribution of an ideal gas in a gravitational field, ''Eur. J. Phys.'', '''17''': 43–44.</ref>
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