The world according to Rényi: thermodynamics of fractal systems

We discuss a basic thermodynamic properties of systems with multifractal structure. This is possible by extending the notion of Gibbs–Shannon’s entropy into more general framework Rényi’s information entropy. We show a connection of Rényi’s parameter q with the multifractal singularity spectrum f (α) and clarify a relationship with the Tsallis–Havrda–Charvat entropy. Finally, we generalize Hagedorn’s statistical theory and apply it to high–energy particle collisions. INTRODUCTION One of the fundamental observations of information theory is that the most general functional form for the mean transmitted information (i.e., information measure) is that of Rényi. Although Rényi’s information measure offers perhaps the most general and conceptually cleanest setting for the entropy, it has not found so far as much applicability as its Shannon’s counterpart. To clarify the position of Rényi’s entropy in physics, we resort to systems with a multifractal structure. Such systems are very important and highly diverse, including phase transitions, turbulent flow of fluids, irregularities in heartbeat, population dynamics, chemical reactions, plasma physics, and most recently the motion of groups and clusters of stars. We shall argue that for the aforementioned the Rényi parameter is connected via a Legendre transformation with the multifractal singularity spectrum. To put some flesh on bones we generalize Hagedorn’s statistical theory and subsequently apply to a differential cross section in high–energy scattering experiments. More thorough investigation will be published elsewhere. RÉNYI’S ENTROPY Motivation From information theory follows that the most general information entropy is that of Rényi [1]. In discrete cases where the probability distribution P = {pn} the Rényi entropy is defined as Iq(P ) = 1 (1−q) log2 (