Nonextensive Statistical Mechanics

2.4 Boltzmann–Gibbs Statistical Mechanics and Thermodynamics 33Replacing (2.57) in (2.55), we finally obtainp opt =e − (x−X (1) ) 22[X (2) −(X (1) ) 2 ]√2π[X(2)− (X (1) ) 2 ] . (2.58)We see that the only effect of a nonzero mean value of x is to re-center theGaussian.2.3.4 OthersA quite general situation would be to impose, in addition to∫dx p(x) = 1 , (2.59)the constraint∫dx f(x) p(x) = F , (2.60)where f (x) is some known function and F a known number. We obtainp opt =f (x)e−β∫ . (2.61)dx e−β f (x)It is clear that, by appropriately choosing f (x), we can force p opt (x) to be virtuallyany distribution we wish. For example, by choosing f (x) =|x| γ (γ ∈ R), weobtain a generic stretched exponential p opt (x) ∝ e −β|x|γ ; by choosing f (x) = ln x,we obtain for p opt (x) a power law. But the use of such procedures hardly has anyepistemological interest at all, since it provides no hint onto the underlying natureof the problem. Only choices such as f (x) = x or f (x) = x 2 are sound since suchconstraints correspond to very generic informational features, namely the locationof the center and the width of the distribution. Other choices are, unless some exceptionalfact enters into consideration (e.g., f (x) being a constant of motion of thesystem), quite ad hoc and uninteresting. Of course, this mathematical fact is by nomeans exclusive of S BG : the same holds for virtually any entropic form.2.4 Boltzmann–Gibbs StatisticalMechanics and ThermodynamicsThere are many formal manners for deriving the BG entropy and its associatedprobability distribution for thermal equilibrium. None of them uses exclusively firstprinciple arguments, i.e., arguments that entirely remain at the level of mechanics