Speaker: Piergiulio Tempesta

Title: Group theory and entropies for complex systems

Abstract: We will show that an intrinsic group-theoretical
structure is at the heart of the notion of entropy. This structure
emerges when imposing the requirement of composability of an entropy
with respect to the union of two statistically independent systems. A
generalization of the celebrated Shannon-Khinchin set of
axioms is proposed, obtained by replacing the additivity axiom by
composability.

The theory of formal groups offers a natural language for our
group-theoretical approach to generalized entropies. In this
settings, the known entropies can be encoded into a general trace-form
class, the universal-group entropy (so called due to its relation with
the Lazard universal formal group of algebraic topology).

We shall also prove that Renyi's entropy is the first example of a new
class of non trace-form entropies, of potential interest in
the theory of complex systems, that we call the Z-entropies.
Each of them is composable and, in particular, generalizes
simultaneously the entropies of Boltzmann and Renyi (recovered in
suitable limits).

The information theoretical content of the proposed generalized
entropies is shown to be a byproduct of their underlying group
structure.