New Kinds of Entropy

Abstract

Two new entropies are offered in this paper. Neither belongs to the traditional conservative statistical mechanics. The first recognizes that the discovery that there are states of granular or glassy systems which can be restricted by an appropriate experimental pathway implies that an entropy can be defined for them by deriving an appropriate Boltzmann equation in which the pathway to the steady state has an increasing entropy until the state characterized by a compactivity X=(∂V)/(∂S) is reached. The second problem is to consider a state where energy flows in and flows out, leaving a steady state, e.g., a steady turbulent flow. Time dependent correlation functions are well defined and hence a probability of the history of the system is defined. There are many ad hocstudies of such systems. We ask whether such systems will possess an entropy defined by the usual ∫PlogPbut now Pis the probability of the entire history of the system and the integral is over all histories.