Thermodynamic and Differential Entropy under a Change of Variables

Vladimir Hnizdo,Michael K Gilson

doi:10.3390/e12030578

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.

Highlights

The Gibbs entropy of classical statistical thermodynamics is, apart from some non-essential constants, the differential Shannon entropy [2] of the probability density function in the phase space of the system under consideration
We have addressed the situation in which one wishes to compute ∆S for a classically treated, isothermal physical process where the phase-space probability distribution ρ(p, q) goes to ρ′ (p, q)
The momentum entropy, Sm, is not affected by the physical process, so ∆S = ∆Ss . (If the temperature T changes, there is a contribution from Sm as well, which can be computed analytically.) In practical applications, it is often preferable to compute spatial entropy in non-Cartesian coordinates, Q, but questions arise regarding the correct way to treat the entropy under a coordinate transformation because the differential

Summary

Introduction

The Gibbs entropy of classical statistical thermodynamics is, apart from some non-essential constants, the differential Shannon entropy [2] of the probability density function (pdf) in the phase space of the system under consideration. The invariance of the full entropy stems from the fact that the Jacobian of a canonical transformation equals unity, and explicit demonstration of this invariance yields a simple formula for correcting a spatial entropy computed with transformed coordinates to yield correct full entropy changes. These results have application in calculations of spatial entropy from molecular simulations when Cartesian coordinates are transformed, for example to bond-angle-torsion [4] coordinates.

Spatial Entropy in Cartesian Coordinates

Spatial Entropy under a Coordinate Transformation

Momentum Entropy under a Coordinate Transformation

Total Entropy in Terms of the Spatial Entropy in Non-Cartesian Coordinates

Conclusions