Tsallis Statistics: Averages and a Physical Interpretation of the Lagrange Multiplier $\beta$

Tsallis has proposed a generalisation of the standard entropy, which has since been applied to a variety of physical systems. In the canonical ensemble approach that is mostly used, average energy is given by an unnromalised, or normalised, $q$-expectation value. A Lagrange multiplier $\beta$ enforces the energy constraint whose physical interpretation, however, is lacking. Here, we use a microcanonical ensemble approach and find that consistency requires that only normalised $q$-expectation values are to be used. We then present a physical interpretation of $\beta$, relating it to a physical temperature. We derive this interpretation by a different method also.