Disambiguation of the Boltzmann-Gibbs-Shannon Entropy in Systems with Continuous States

Authors

John D. Ramshaw, Portland State UniversityFollow

Published In

Journal of Statistical Mechanics-Theory and Experiment

Document Type

Citation

Publication Date

12-1-2022

Abstract

The Boltzmann–Gibbs–Shannon entropy S_d of a system with discrete states i is straightforward and well established, but its proper generalization to systems with continuous states x is problematical. The generally accepted expression S = −∫dxρ(x)log[ρ(x)/m(x)] exhibits anomalous behavior when the probability density ρ(x) varies significantly over volumes of order v(x) = 1/m(x), and diverges when ρ(x) is singular. The traditional remedy for these deficiencies has been coarse graining (CG) over small discrete cells in x-space, but such procedures are ad hoc, arbitrary, subjective, and ultimately ambiguous. Here we propose an alternative procedure in which CG is replaced by an integral transform of ρ(x) that represents the statistical accuracy to which the value of x can be resolved or determined. The resulting unambiguous expression for S preserves the essential properties of S_d for arbitrary ρ(x), including the singular ρ(x) that occur in nonequilibrium steady states.

Rights

© 2022 IOP Publishing Ltd

Locate the Document

https://doi.org/10.1088/1742-5468/aca0e8

DOI

10.1088/1742-5468/aca0e8

Persistent Identifier

https://archives.pdx.edu/ds/psu/39229

Citation Details

Ramshaw, J. D. (2022). Disambiguation of the Boltzmann–Gibbs–Shannon entropy in systems with continuous states. Journal of Statistical Mechanics: Theory and Experiment, 2022(12), 123203.