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From Rényi Relative Entropic Generalization to Quantum Thermodynamical Universality

It is shown that the structure of thermodynamics is "form invariant", when it is derived using maximum entropy principle for various choices of entropy and even beyond equilibrium. By the form invariance of thermodynamics, it is meant that the form of the free energy (internal energy minus the temperature times entropy) remains unaltered when all the entities entering this relation are suitably defined. The useful ingredients for this are the equilibrium entropy associated with thermal density matrix and the relative entropy between an arbitrary density matrix and the thermal density matrix. To delineate the form invariance, we consider the quantum Rényi entropic versions (indexed by a parameter $α$), i.e., Rényi entropy with appropriate internal energy and equilibrium state defined for all $α$. These results reduce to the well-known Gibbs-von Neumann results when $α\rightarrow 1$. Moreover, we show that the \textit{universality} of the Carnot statement of the second law is the consequence of the form invariance of the free energy. Further, the Clausius inequality, which is the precursor to the Carnot cycle, is also shown to hold based on the known data processing inequalities for the traditional and the sandwiched Rényi relative entropies. Thus, we find the thermodynamics of nonequilibrium state and its deviation from equilibrium together determine the thermodynamic laws.