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

Abstract

The Boltzmann–Gibbs–Shannon (BGS) entropy Sd of a system with discrete states is inherently non-negative, and attains its minimum value of zero only when the system is known with certainty to occupy a particular state. In contrast, the BGS entropy S of a system with continuous states can be negative with no lower bound. This disparity is traced to the fact that the generalisation of Sd to obtain S is based on an implicit assumption which becomes conceptually inconsistent when the continuous probability density varies too rapidly with . This analysis suggests an alternative generalisation of Sd which results in a fully consistent and inherently non-negative entropy . The resulting expression for is observed to be algebraically equivalent to a conventional coarse-grained BGS entropy, but with the essential difference that the previously arbitrary cell sizes are now well defined and are no longer ambiguous. The non-equilibrium time dependence of is well known to be thermodynamically anomalous, whereas that of is shown to be consistent with the expected behaviour of the thermodynamic entropy and its irreversible production rate in both conservative and dissipative systems with mixing behaviour.