Steady-state thermodynamics for heat conduction: microscopic derivation.

@article{Komatsu2008SteadystateTF,
  title={Steady-state thermodynamics for heat conduction: microscopic derivation.},
  author={Teruhisa S. Komatsu and Naoko Nakagawa and Shin-ichi Sasa and Hal Tasaki},
  journal={Physical review letters},
  year={2008},
  volume={100 23},
  pages={
          230602
        }
}
Starting from microscopic mechanics, we derive thermodynamic relations for heat conducting nonequilibrium steady states. The extended Clausius relation enables one to experimentally determine nonequilibrium entropy to the second order in the heat current. The associated Shannon-like microscopic expression of the entropy is suggestive. When the heat current is fixed, the extended Gibbs relation provides a unified treatment of thermodynamic forces in the linear nonequilibrium regime. 

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