Renyi Entropy and Free Energy

TL;DR

This paper explores the relationship between Renyi entropy and free energy, revealing that Renyi entropy can be derived from the temperature derivative of free energy, applicable to classical and quantum systems.

Contribution

It establishes a mathematical link between Renyi entropy and free energy, showing Renyi entropy as a q-deformation of standard entropy based on thermodynamic principles.

Findings

Renyi entropy equals the negative q-derivative of free energy w.r.t. temperature.

The relationship holds for both classical and quantum systems.

Renyi entropy generalizes the concept of entropy through a q-parameter.

Abstract

The Renyi entropy is a generalization of the usual concept of entropy which depends on a parameter q. In fact, Renyi 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 do as it moves to equilibrium at the new temperature, divided by the change in temperature, equals the system's Renyi entropy in its original state. This result applies to both classical and quantum systems. Mathematically, we can express this result as follows: the Renyi entropy of a system in thermal equilibrium is minus the "1/q-derivative" of its free energy with respect to temperature. This shows that Renyi entropy is a q-deformation of the usual concept of entropy.