Generalised information and entropy measures in physics

Abstract

The formalism of statistical mechanics can be generalised by starting from more general measures of information than the Shannon entropy and maximising those subject to suitable constraints. We discuss some of the most important examples of information measures that are useful for the description of complex systems. Examples treated are the Rényi entropy, Tsallis entropy, Abe entropy, Kaniadakis entropy, Sharma–Mittal entropies, and a few more. Important concepts such as the axiomatic foundations, composability and Lesche stability of information measures are briefly discussed. Potential applications in physics include complex systems with long-range interactions and metastable states, scattering processes in particle physics, hydrodynamic turbulence, defect turbulence, optical lattices, and quite generally driven nonequilibrium systems with fluctuations of temperature.

Keywords:

• measures of information
• entropy
• generalised statistical mechanics
• complex systems

Notes

1. An exception to this rule is the Fisher information, which depends on gradients of the probability density but will not be discussed here.