Among various possible routes to extend entropy and thermodynamics tononequilibrium steady states (NESS), we take the one which is guided byoperational thermodynamics and the Clausius relation. In our previous study, wederived the extended Clausius relation for NESS, where the heat in the originalrelation is replaced by its "renormalized" counterpart called the excess heat,and the Gibbs-Shannon expression for the entropy by a new symmetrizedGibbs-Shannon-like expression. Here we concentrate on Markov processesdescribing heat conducting systems, and develop a new method for derivingthermodynamic relations. We first present a new simpler derivation of theextended Clausius relation, and clarify its close relation with the linearresponse theory. We then derive a new improved extended Clausius relation witha "nonlinear nonequilibrium" contribution which is written as a correlationbetween work and heat. We argue that the "nonlinear nonequilibrium"contribution is unavoidable, and is determined uniquely once we accept the(very natural) definition of the excess heat. Moreover it turns out that tooperationally determine the difference in the nonequilibrium entropy to thesecond order in the temperature difference, one may only use the previousClausius relation without a nonlinear term or must use the new relation,depending on the operation (i.e., the path in the parameter space). Thispeculiar "twist" may be a clue to a better understanding of thermodynamics andstatistical mechanics of NESS.
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