**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 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:

(15) |

(16) |

Given a population *p*_{j} of positive variants,
and an inequality like (16),
I am now prepared to define
the ``**Jensen average**'' .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 :

(17) |

(18) |

(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 ,assuming the functional form to be .

10/21/1998