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.