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Entropy 2010, 12(3), 578-590; doi:10.3390/e12030578
Article

Thermodynamic and Differential Entropy under a Change of Variables

1,*  and 2,#,*
1 Health Effects Laboratory Division, National Institute for Occupational Safety and Health, Morgantown, West Virginia 26505, USA 2 Center for Advanced Research in Biotechnology, University of Maryland Biotechnology Institute, Rockville, Maryland 20850, USA # Current Address: Skaggs School of Pharmacy and Pharmaceutical Sciences, University of California San Diego, 9500 Gilman Drive, MC 0736, La Jolla, California 92093, USA.
* Authors to whom correspondence should be addressed.
Received: 17 December 2009 / Revised: 27 February 2010 / Accepted: 2 March 2010 / Published: 16 March 2010
(This article belongs to the Special Issue Configurational Entropy)
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Abstract

The differential Shannon entropy of information theory can change under a change of variables (coordinates), but the thermodynamic entropy of a physical system must be invariant under such a change. This difference is puzzling, because the Shannon and Gibbs entropies have the same functional form. We show that a canonical change of variables can, indeed, alter the spatial component of the thermodynamic entropy just as it alters the differential Shannon entropy. However, there is also a momentum part of the entropy, which turns out to undergo an equal and opposite change when the coordinates are transformed, so that the total thermodynamic entropy remains invariant. We furthermore show how one may correctly write the change in total entropy for an isothermal physical process in any set of spatial coordinates.
Keywords: thermodynamic entropy; Shannon entropy; spatial entropy; non-Cartesian coordinates; canonical transformation; Jacobian thermodynamic entropy; Shannon entropy; spatial entropy; non-Cartesian coordinates; canonical transformation; Jacobian
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Hnizdo, V.; Gilson, M.K. Thermodynamic and Differential Entropy under a Change of Variables. Entropy 2010, 12, 578-590.

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