9.6.4: Evaluating the Dual Variable

So far we are considering the dual variable \(\beta\) to be an independent variable. If we start with a known value \(\widetilde{G}\), we want to use \(G\) as an independent variable and calculate \(\beta\) in terms of it. In other words, we need to invert the function \(G(\beta)\), or find \(\beta\) such that Equation 9.9 is satisfied.

This task is not trivial; in fact most of the computational difficulty associated with the Principle of Maximum Entropy lies in this step. If there are a modest number of states and only one constraint in addition to the equation involving the sum of the probabilities, this step is not hard, as we will see. If there are more constraints this step becomes increasingly complicated, and if there are a large number of states the calculations cannot be done. In the case of more realistic models for physical systems, this summation is impossible to calculate, although the general relations among the quantities other than \(p(A_i)\) remain valid.

To find \(\beta\), start with Equation 9.12 for \(p(A_i)\), multiply it by \(g(A_i)\) and by \(2^{\alpha}\), and sum over the probabilities. The left hand side becomes \(G(\beta)2^{\alpha}\), because neither \(\alpha\) nor \(G(\beta)\) depend on \(i\). We already have an expression for \(\alpha\) in terms of \(\beta\) (Equation 9.14), so the left hand side becomes \(\sum_{i} G(\beta)2^{−\beta g(A_i)}\). The right \(i\) hand side becomes \(\sum_i g(A_i)2^{−\beta g(A_i)}\). Thus,

If this equation is multiplied by \(2^{\beta G(\beta)}\), the result is

\(0 = f(\beta) \tag{9.22}\)

where the function \(f(\beta)\) is