Thermodynamics of an ideal generalized gas:II Means of order α\alpha

The property that power means are monotonically increasing functions of their order is shown to be the basis of the second laws not only for processes involving heat conduction but also for processes involving deformations. In an $L$-potentail equilibration the final state will be one of maximum entropy, while in an entropy equilibrium the final state will be one of minimum $L$. A metric space is connected with the power means, and the distance between means of different order is related to the Carnot efficiency. In the ideal classical gas limit, the average change in the entropy is shown to be proportional to the difference between the Shannon and R\'enyi entropies for nonextensive systems that are multifractal in nature. The $L$-potential, like the internal energy, is a Schur convex function of the empirical temperature, which satisfies Jensen's inequality, and serves as a measure of the tendency to uniformity in processes involving pure thermal conduction.Comment: 8 page

Thermodynamic analysis of the Quantum Critical behavior of Ce-lattice compounds

A systematic analysis of low temperature magnetic phase diagrams of Ce compounds is performed in order to recognize the thermodynamic conditions to be fulfilled by those systems to reach a quantum critical regime and, alternatively, to identify other kinds of low temperature behaviors. Based on specific heat ($C_m$) and entropy ($S_m$) results, three different types of phase diagrams are recognized: i) with the entropy involved into the ordered phase ($S_{MO}$) decreasing proportionally to the ordering temperature ($T_{MO}$), ii) those showing a transference of degrees of freedom from the ordered phase to a non-magnetic component, with their $C_m(T_{MO})$ jump ($\Delta C_m$) vanishing at finite temperature, and iii) those ending in a critical point at finite temperature because their $\Delta C_m$ do not decrease with $T_{MO}$ producing an entropy accumulation at low temperature. Only those systems belonging to the first case, i.e. with $S_{MO}\to 0$ as $T_{MO}\to 0$, can be regarded as candidates for quantum critical behavior. Their magnetic phase boundaries deviate from the classical negative curvature below $T\approx 2.5$\,K, denouncing frequent misleading extrapolations down to T=0. Different characteristic concentrations are recognized and analyzed for Ce-ligand alloyed systems. Particularly, a pre-critical region is identified, where the nature of the magnetic transition undergoes significant modifications, with its $\partial C_m/\partial T$ discontinuity strongly affected by magnetic field and showing an increasing remnant entropy at $T\to 0$. Physical constraints arising from the third law at $T\to 0$ are discussed and recognized from experimental results