Rényi Entropy and Free Energy.

The Rényi entropy is a generalization of the usual concept of entropy which depends on a parameter q. In fact, Rényi entropy is closely related to free energy. Suppose we start with a system in thermal equilibrium and then suddenly divide the temperature by q. Then the maximum amount of work the system can perform as it moves to equilibrium at the new temperature divided by the change in temperature equals the system's Rényi entropy in its original state. This result applies to both classical and quantum systems. Mathematically, we can express this result as follows: the Rényi entropy of a system in thermal equilibrium is without the 'q-1-derivative' of its free energy with respect to the temperature. This shows that Rényi entropy is a q-deformation of the usual concept of entropy.

The Fundamental Theorem of Natural Selection.

Suppose we have n different types of self-replicating entity, with the population Pi of the ith type changing at a rate equal to Pi times the fitness fi of that type. Suppose the fitness fi is any continuous function of all the populations P1,…,Pn. Let pi be the fraction of replicators that are of the ith type. Then p=(p1,…,pn) is a time-dependent probability distribution, and we prove that its speed as measured by the Fisher information metric equals the variance in fitness. In rough terms, this says that the speed at which information is updated through natural selection equals the variance in fitness. This result can be seen as a modified version of Fisher's fundamental theorem of natural selection. We compare it to Fisher's original result as interpreted by Price, Ewens and Edwards.