Jensen average

Physicists speak of maximizing entropy, which, if we change the polarity, is like minimizing the various Jensen inequalities. As we minimize a Jensen inequality, the small values tend to get larger while the large values tend to get smaller. For each population of values there is an average value, i.e., a value that tends to get neither larger nor smaller. The average depends not only on the population, but also on the definition of entropy.

Commonly, the p_j are positive and $\sum w_j p_j$ is an energy. Typically the total energy, which will be fixed, can be included as a constraint, or we can find some other function to minimize. For example, divide both terms in (3) by the second term and get an expression which is scale invariant; i.e., scaling p leaves (15) unchanged:  

$${\sum_{j=1}^N w_j f(p_j)\over f\left( \sum_{j=1}^N w_j p_j\right) } \quad \geq \quad 1$$ (15)

Because the expression exceeds unity, we are tempted to take a logarithm and make a new function for minimization:  

$$J \eq \ln \left( \sum_j w_j f(p_j)\right)\ -\ \ln \left[ f \left( \sum_j w_j p_j\right) \right] \quad \geq \quad 0$$ (16)

Given a population p_j of positive variants, and an inequality like (16), I am now prepared to define the ``Jensen average'' $\overline{p}$.Suppose there is one element, say p_J, of the population p_j that can be given a first-order perturbation, and only a second-order perturbation in J will result. Such an element is in equilibrium and is the Jensen average $\overline{p}$: 

$$0 \eq \left. {\partial J\over\partial p_J} \right] _{p_J = \overline{p}}$$ (17)

Let f_p denote the derivative of f with respect to its argument. Inserting (16) into (17) gives

$$0 \eq {\partial J\over\partial p_J} \eq {w_J\ f_p(p_J) \ov... ...\sum_{j=1}^N w_j p_j) w_J\over f\left( \sum w_j p_j \right) }$$ (18)

Solving,

$$\overline{p} \eq p_J \eq f_p^{-1} \left( e^J f_p(\sum_{j=1}^N w_j p_j) \right)$$ (19)

But where do we get the function f, and what do we say about the equilibrium value? Maybe we can somehow derive f from the population. If we cannot work out a general theory, perhaps we can at least find the constant $\gamma$,assuming the functional form to be $f=p^\gamma$.