MAXENT and the Tsallis Parameter

TL;DR

This paper explores methods to determine the Tsallis entropic parameter q within the MAXENT framework, providing solutions for systems where q varies or is fixed based on constraint equations, with applications to the Generalized Fokker-Planck Equation.

Contribution

It introduces two approaches for determining the variable or fixed q parameter in nonextensive entropy maximization depending on the form of the constraints.

Findings

Exact solutions to the static Generalized Fokker-Planck Equation derived from MAXENT.

Demonstrates how to determine q as a function of Lagrange multipliers.

Shows how additional constraints can fix the value of q.

Abstract

The nonextensive entropic measure proposed by Tsallis 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 an operator O can be written as $Tr[F^{q}O]=C$, where C may be an explicit function of the distribution function, F. 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 solutions to the static Generalized Fokker-Planck Equation (GFP) are obtained from MAXENT. 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 for all values of q trivially. 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.