The generalized zeroth law of thermodynamics indicates that the physical temperature in nonextensive statistical mechanics is different from the inverse of the Lagrange multiplier, beta. This leads to modifications of some of thermodynamic relations for nonextensive systems. Here, taking the first law of thermodynamics and the Legendre transform structure as the basic premises, it is found that Clausius definition of the thermodynamic entropy has to be appropriately modified, and accordingly the thermodynamic relations proposed by Tsallis, Mendes and Plastino [Physica A 261 (1998) 534] are also to be rectified. It is shown that the definition of specific heat and the equation of state remain form invariant. As an application, the classical gas model is reexamined and, in marked contrast with the previous result obtained by Abe [Phys. Lett. A 263 (1999) 424: Erratum A 267 (2000) 456] using the unphysical temperature and the unphysical pressure, the specific heat and the equation of state are found to be similar to those in ordinary extensive thermodynamics.Comment: 17 pages. The discussion about the Legendre transform structure is modified and some additional comments are mad

The proper way of averaging is an important question with regards to Tsallis' Thermostatistics. Three different procedures have been thus far employed in the pertinent literature. The third one, i.e., the Tsallis-Mendes-Plastino (TMP) [1] normalization procedure, exhibits clear advantages with respect to earlier ones. In this work, we advance a distinct (from the TMP-one) way of handling the Lagrange multipliers involved in the extremization process that leads to Tsallis' statistical operator. It is seen that the new approach considerably simplifies the pertinent analysis without losing the beautiful properties of the Tsallis-Mendes-Plastino formalism. 95.35.+d, 05.70.Ce,

In spite of its undeniable success, there are still open questions regarding Tsallis' non-extensive statistical formalism, whose founding stone was laid in 1988 in JSTAT. Some of them are concerned with the so-called normalization problem of just how to evaluate expectation values. The Jaynes' MaxEnt approach for deriving statistical mechanics is based on the adoption of (1) a specific entropic functional form S and (2) physically appropriate constraints. The literature on non-extensive thermostatistics has considered, in its historical evolution, four possible choices for the evaluation of expectation values: (i) 1988 Tsallis-original (TO), (ii) Curado-Tsallis (CT), (iii) Tsallis-MendesPlastino (TMP), and (iv) the same as (iii), but using centered operators as constraints (OLM).The 1988 was promptly abandoned and replaced, mostly with versions ii) and iii). We will here (a) show that the 1988 is as good as any of the others, (b) demonstrate that the four cases can be easily derived from just one (any) of them, i.e., the probability distribution function in each of these four instances may be evaluated with a unique formula, and (c) numerically analyze some consequences that emerge from these four choices.PACS: 05.30.Jp