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Second law of thermodynamics in non-extensive systems

It exists a large class of systems for which the traditional notion of extensivity breaks down. From experimental examples we induce two general hypothesis concerning such systems. In the first the existence of an internal coordinate system in which extensivity works is assumed. The second hypothesis concerns the link between this internal coordinate system and the usual thermodynamic variables. This link is represented by an extra relation between two variables pertaining to the two descriptions; to be illustrative a scaling law has been introduced relating external and internal volumes. In addition, we use an axiomatic description based on the approach of the second law of thermodynamics proposed by E. Lieb and J. Yngvason (Physics Reports 310, 1999,1). We show that it exists an entropy function satisfying the monotony of the usual thermodynamic entropy. If a state results from the association of different states, the entropy is additive under these states. However, the entropy is a non-extensive function and we give its law of transformation under a change of the external volume. The entropy is based on some reference states, a change of these states leads to an affine transformation of the entropy. To conclude we can say that the main aspect of the second law of thermodynamics survives in the case of non-extensive systems.