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

Thermodynamic and Differential Entropy under a Change of Variables

1,* email and 2,#,* email
Received: 17 December 2009; in revised form: 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
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MDPI and ACS Style

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

AMA Style

Hnizdo V, Gilson MK. Thermodynamic and Differential Entropy under a Change of Variables. Entropy. 2010; 12(3):578-590.

Chicago/Turabian Style

Hnizdo, Vladimir; Gilson, Michael K. 2010. "Thermodynamic and Differential Entropy under a Change of Variables." Entropy 12, no. 3: 578-590.

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