FOUNDATIONS OF ENTROPY IN COMPLEX SYSTEMS

Jan Korbel (Personal webpage)

In this lecture, we focus on the foundations of entropy for complex systems. By starting with the basic notion of multiplicity, i.e., the number of microstates corresponding to a mesostate, we use the famous Boltzmann formula to derive entropic functionals for several types of systems. In addition to the standard Boltzmann-Gibbs-Shannon entropy based on multinomial multiplicity, we derive several other entropies, including those for systems with hidden multiplicity, structure-forming systems, and driven dissipative systems. Within this framework, we derive a consistent thermodynamic framework for these systems. This includes equilibrium (MaxEnt) distribution and thermodynamic potentials. Further, we discuss the connection to other axiomatic approaches, including Shannon-Khinchin, Shore-Johnson, Lieb-Yngvason, or Hannel-Thurner axiomatics. Finally, we discuss the connection to related concepts, such as the principle of Maximum caliber or non-equilibrium thermodynamics.