Disambiguation of the Boltzmann–Gibbs–Shannon entropy in systems with continuous states

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 = −∫dx ρ(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.