Renyi entropy as a statistical entropy for complex systems

To describe a complex system, we propose using the Renyi entropy depending on the parameter $q$ $(0<q\le1)$ and passing into the Gibbs–Shannon entropy at $q=1$. The maximum principle for the Renyi entropy yields a Renyi distribution that passes into the Gibbs canonical distribution at $q=1$. The thermodynamic entropy of the complex system is defined as the Renyi entropy for the Renyi distribution. In contrast to the usual entropy based on the Gibbs–Shannon entropy, the Renyi entropy increases as the distribution deviates from the Gibbs distribution {(}the deviation is estimated by the parameter $\eta=1$ – $q)$ and reaches its maximum at the maximum possible value $\eta_{\max}$. As this occurs, the Renyi distribution becomes a power-law distribution. The parameter $\eta$ can be regarded as an order parameter. At $\eta=0$, the derivative of the thermodynamic entropy with respect to $\eta$ exhibits a jump, which indicates a kind of phase transition into a more ordered state. The evolution of the system toward further order in this phase state is accompanied by an entropy gain. This means that in accordance with the second law of thermodynamics, a natural evolution in the direction of self-organization is preferable.