Ivan Latella, Agustín Pérez-Madrid
The local thermodynamics of a system with long-range interactions in d dimensions is studied using the mean field approximation. Long-range interactions are introduced through pair interaction potentials that decay like a power law in the interparticle distance. We compute the local entropy, Helmholtz free energy and grand potential per particle in the microcanonical, canonical and grand canonical ensembles, respectively. The one-particle distribution function approach is also considered. From the local entropy per particle we obtain the local equation of state of the system by using the condition of local thermodynamic equilibrium. This equation of state turns out to be the ideal gas equation of state regardless of the pair interaction potential characterizing the long-range interactions. By volume integration of the relation between the different thermodynamic potentials at the local level, we find the corresponding equation satisfied by the potentials at the global level. It is shown that the potential energy enters as a thermodynamic variable which modifies the global thermodynamic potentials. As a result, we find a generalized Gibbs-Duhem equation which relates the potential energy to the intensive variables of the system. For the marginal case where the power of the decaying interaction potential is equal to the dimension of the space, the usual Gibbs-Duhem equation is recovered. As examples of application of this equation, we consider spatially uniform interaction potentials and the self-gravitating gas. We also point out a close relationship with the thermodynamics of small systems.
View original:
http://arxiv.org/abs/1302.4903
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