Nonextensive Statistical Mechanics

8.2 Frequently Asked Questions 313(e) Is the zeroth principle of thermodynamics valid at the quasi-stationary statesof long-range-interacting Hamiltonian systems, and in nonextensive statistical mechanics?This important question was raised up to me for the first time by Oscar Nassifde Mesquita [808]. The question concerns whether the zeroth principle of thermodynamicsand thermometry are consistent with nonextensive statistical mechanics.Such questioning has already been addressed in a couple of dozens of papers thatare available in the literature. It has been recently raised once again, this time byNauenberg [809]. He concludes, among many other critiques, that it is not possibleto have thermalization between systems with different values of q. It appears tobe exactly the opposite which is factually shown in [810], where his critique isrebutted. One of the crucial points that is unfortunately missed in [809], concernsdiscussion of “weak coupling” in Hamiltonian systems. Indeed, if we call c thecoupling constant associated with long range interactions (i.e., 0 ≤ α/d ≤ 1), wehave that lim N→∞ lim c→0 cÑ = 0, whereas lim c→0 lim N→∞ cÑ diverges. No suchanomaly exists for short-range interactions (i.e., α/d > 1). Indeed, in this simplercase, we have that lim N→∞ lim c→0 cÑ = lim c→0 lim N→∞ cÑ = 0. The nonuniformconvergence that, for long-range interactions, exists at this level possibly isrelated to the concomitant nonuniform convergence associated with the t →∞andN →∞limits discussed previously in this paper. These subtleties probably playan important role in the present question.The strict verification of the zeroth principle of thermodynamics demands checkingthe transitivity of the concept of temperature through successive thermal contactsbetween three, initially disconnected systems, A, B, and C. Such a study is inprogress [811] for the paradigmatic HMF model (which corresponds to infinitelylong-rangeinteractions). As a first step, two (equal) systems, A and B, are put intothermal contact. The Hamiltonian is given by (see Fig. 8.1)H =N∑i=1++ l(L A i )22N∑i=1N∑k=1(L B i )22+ 1 N+ 1 NN∑i=1N∑[1 − cos(θi A − θ j A )]j=1N∑i=1N∑[1 − cos(θi B − θ j B )] (8.3)j=1[1 − cos(θ Ak − θ B k )] .As we see, there are long-range interactions within system A and within systemB, but only short-range interactions connecting systems A and B through the couplingconstant l. See in Fig. 8.2 the time evolution of the temperatures of A andB. We verify that, after the thermal contact being established, the two temperaturesmerge into an intermediate one, as they would do if they were at thermal equilibrium...but they are not!. Indeed, those are quasi-stationary states. Only later, the twosystems go together towards thermal equilibrium. A discussion such as the present