Temporal evolution, directionality of time and irreversibility

The aim of the present interdisciplinary review is to carry out a comparative analysis of the notions of thermodynamic entropy, information entropy and entropy of non-equilibrium states and flow of entropy from a critical perspective. The problems of temporal evolution and time directionality are discussed in this context as well. The interrelation of these notions is studied with focusing on the non-equilibrium entropy. The paper is aimed to clarify the notion of entropy, entropy production and its generalizations. The Boltzmann, Gibbs, von Neumann, Shannon, Renyi, Tsallis and others types of entropy are considered concisely. The notions of the steady state, local state and local equilibrium are analyzed thoroughly to expose similarities and dissimilarities of various approaches to the definition of the entropy of non-equilibrium states. The extremal principles for entropy and entropy production are analyzed and discussed in this connection. The purpose of the present study is to elucidate certain aspects of the non-equilibrium statistical mechanics, namely the principal role of the correct description of the temporal evolution of a system and the corresponding procedure of averaging. We also touch tersely the intriguing problems of directionality of time and causality as well as relevance of constructal law that accounts for the phenomenon of evolution.