Determining the Tsallis parameter via maximum entropy

The nonextensive entropic measure proposed by Tsallis [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] introduces a parameter, q, which is not defined but rather must be determined. The value of q is typically determined from a piece of data and then fixed over the range of interest. On the other hand, from a phenomenological viewpoint, there are instances in which q cannot be treated as a constant. We present two distinct approaches for determining q depending on the form of the equations of constraint for the particular system. In the first case the equations of constraint for the operator Ô can be written as Tr(F(q)Ô)=C, where C may be an explicit function of the distribution function F. We show that in this case one can solve an equivalent maxent problem which yields q as a function of the corresponding Lagrange multiplier. As an illustration the exact solution of the static generalized Fokker-Planck equation (GFPE) is obtained from maxent with the Tsallis enropy. As in the case where C is a constant, if q is treated as a variable within the maxent framework the entropic measure is maximized trivially for all values of q. Therefore q must be determined from existing data. In the second case an additional equation of constraint exists which cannot be brought into the above form. In this case the additional equation of constraint may be used to determine the fixed value of q.