Investigation of an energy nonadditivity for nonextensive systems

Abstract

We investigate the energy nonadditivity relationship E(AB) = E(A) + E(B) + E(A)E(B)which is often considered in the development of the statistical physics of nonextensive systems. It was recently found that  in this equation was not constant for a given system in a given situation and could not characterize nonextensivity for that system. In this work, we select several typical nonextensive systems and compute the behavior of  when a system changes its size or is divided into subsystems in different fashions. Three kinds of interactions are considered. It is found by a thought experiment that  depends on the system size and the interaction as expected and on the way we divide the system. However, one of the major results of this work is that, for given system,  has a minimum with respect to division position. Around this position, there is a zone in which  is more or less constant, a situation where the sizes of the subsystems are comparable. The width of this zone depends on the interaction and on the system size. We conclude that if  is considered approximately constant in this zone, the two mathematical difficulties raised in previous studies are solved, meaning that the nonadditive relationship can characterize the nonadditivity of the system as an approximation. In all the cases,  tends to zero in the thermodynamic limit (N→) as expected. Many kinds of physical systems in nature are recognized as energy nonadditive or nonextensive 1) . This includes small size systems as well as large size systems having long range interaction such as gravity. For these systems, knowing the nonadditivity or nonextensivity in energy and entropy will be very useful for understanding some physical properties and for establishing corresponding theoretical descriptions. Energy and entropy additivity is a crucial hypothesis and a cornerstone for many physical theories [1]. How to replace this basic hypothesis is obviously a puzzling problem. In view of the diversity of the nonadditive systems in nature [2,3], it is unlikely that a universally valid and simple relationship exists.In this work, we focus on a special relationship of energy nonadditivity given bywhere A and B are two subsystems of a composite system AB and E is their energy.  characterizes the energy nonadditivity of the system under consideration and should be constant for a given system in a given situation with constant E(AB). The same logic can be found in [4] for a similar nonadditivity in entropy with many examples of empirically determined  for entropy.This relationship is frequently considered in the development of statistical physics for nonextensive systems [5][6][7][8]. Ou et al. used eq. (1) as a first principle to show that the q-exponential distribution [8] can occur in small systems

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