Non-Boltzmannian Entropies for Complex Classical Systems, Quantum Coherent States and Black Holes

Abstract

Summary. The Renyi entropy is derived as a cumulant average of the Boltzmann entropy in the same way as the Helmholtz free energy can be obtained by cumulant averaging of a Hamiltonian. Such a form of the information entropy and the principle of entropy maximum (MEP) for it are justified by the Shore–Johnson theorem. The application of MEP to the Renyi entropy gives rise to the Renyi distribution. Thermodynamic entropy in the Renyi thermostatistics increases with system complexity (gain of an order parameter η =1 − q) and reaches its maximal value at qmin. The Renyi distribution for such q becomes a pure power–law distribution. Because a power–law distribution is characteristic for self-organizing systems the Renyi entropy can be considered as a potential that drives the system to a self-organized state. The derivative of difference of entropies in the Renyi and Gibbs thermostatistics in respect to η exhibits a jump at η =0 . This permits us to consider the transfer to the Renyi thermostatistics as a peculiar kind of a phase transition into a more organized state. The last section is devoted to development of thermodynamics of coherent states of quantum systems and black holes. The entropy of the quantum field is found to be proportional to the surface area of the static source. The Bekenstein–Hawking entropy of a black hole can also be interpreted as the thermodynamic entropy of coherent states of a physical vacuum in a vicinity of a horizon surface.